Research Article On the Spectral Properties of the...
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Research Article On the Spectral Properties of the Weighted Mean Difference Operator (, V; Δ) over the Sequence Space ℓ 1 P. Baliarsingh 1 and S. Dutta 2 1 Department of Mathematics, School of Applied Sciences, KIIT University, Bhubaneswar 751024, India 2 Department of Mathematics, Utkal University, Vanivihar, Bhubaneswar 751004, India Correspondence should be addressed to P. Baliarsingh; [email protected] Received 21 December 2013; Accepted 30 April 2014; Published 16 June 2014 Academic Editor: Baruch Cahlon Copyright © 2014 P. Baliarsingh and S. Dutta. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In the present work the generalized weighted mean difference operator (, V; Δ) has been introduced by combining the generalized weighted mean and difference operator under certain special cases of sequences = ( ) and V =(V ). For any two sequences and V of either constant or strictly decreasing real numbers satisfying certain conditions the difference operator (, V; Δ) is defined by ((, V; Δ)) =∑ =0 V ( − −1 ) with =0 for all <0. Furthermore, we compute the spectrum and the fine spectrum of the operator (, V; Δ) over the sequence space ℓ 1 . In fact, we determine the spectrum, the point spectrum, the residual spectrum, and the continuous spectrum of this operator on the sequence space ℓ 1 . 1. Introduction, Preliminaries, and Definitions Let = ( ) and V =(V ) be two bounded sequences of either constant or strictly decreasing positive real numbers such that ̸ =0 and V ̸ =0 for all , and lim →∞ = , (1) lim →∞ V − V +1 V −1 − V =1 (for strictly decreasing sequence). (2) By ℓ 1 and V , we denote the spaces of all absolutely summable and p-bounded variation series, respectively. Also, by ℓ ∞ , , and 0 , we denote the spaces of all bounded, conver- gent, and null sequences, respectively. e main perception of this paper is to introduce the weighted mean difference operator (, V; Δ) as follows. Let = ( ) be any sequence in ℓ 1 , and we define the weighted mean difference transform (, V; Δ) of by ( (, V; Δ) ) = ∑ =0 V ( − −1 ) ( ∈ N 0 ), (3) where N 0 denotes the set of nonnegative integers and we assume throughout that any term with negative subscript is zero. Instead of writing (3), the operator (, V; Δ) can be expressed as a lower triangular matrix ( ), where = { { { { { { { V , ( = ) , (V − V +1 ), (0 ≤ ≤ − 1) , 0, ( > 0) . (4) Equivalently, in componentwise the triangle ( ) can be represented by Hindawi Publishing Corporation International Journal of Analysis Volume 2014, Article ID 786437, 7 pages http://dx.doi.org/10.1155/2014/786437
Transcript of Research Article On the Spectral Properties of the...
Research ArticleOn the Spectral Properties of the Weighted Mean DifferenceOperator 119866(119906 V Δ) over the Sequence Space ℓ
1
P Baliarsingh1 and S Dutta2
1 Department of Mathematics School of Applied Sciences KIIT UniversityBhubaneswar 751024 India
2Department of Mathematics Utkal University VaniviharBhubaneswar 751004 India
Correspondence should be addressed to P Baliarsingh pbmath10gmailcom
Received 21 December 2013 Accepted 30 April 2014 Published 16 June 2014
Academic Editor Baruch Cahlon
Copyright copy 2014 P Baliarsingh and S Dutta This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
In the presentwork the generalizedweightedmean difference operator119866 (119906 V Δ)has been introduced by combining the generalizedweightedmean and difference operator under certain special cases of sequences 119906 = (119906119896) and V = (V119896) For any two sequences 119906 andV of either constant or strictly decreasing real numbers satisfying certain conditions the difference operator119866 (119906 V Δ) is defined by(119866(119906 V Δ)119909)119896 = sum
119896
119894=0119906119896V119894(119909119894 minus 119909119894minus1) with 119909119896 = 0 for all 119896 lt 0 Furthermore we compute the spectrum and the fine spectrum of the
operator119866 (119906 V Δ) over the sequence space ℓ1 In fact we determine the spectrum the point spectrum the residual spectrum andthe continuous spectrum of this operator on the sequence space ℓ
1
1 Introduction Preliminaries and Definitions
Let 119906 = (119906119896) and V = (V119896) be two bounded sequences of eitherconstant or strictly decreasing positive real numbers such that119906119896 = 0 and V119896 = 0 for all 119896 and
lim119896997888rarrinfin
119906119896 = 119906 (1)
lim119896997888rarrinfin
10038161003816100381610038161003816100381610038161003816
V119896 minus V119896+1V119896minus1 minus V119896
10038161003816100381610038161003816100381610038161003816
= 1 (for strictly decreasing sequence)
(2)
By ℓ1 and 119887V119901 we denote the spaces of all absolutelysummable and p-bounded variation series respectively Alsoby ℓinfin 119888 and 1198880 we denote the spaces of all bounded conver-gent and null sequences respectively The main perceptionof this paper is to introduce the weighted mean differenceoperator 119866(119906 V Δ) as follows
Let 119909 = (119909119896) be any sequence in ℓ1 and we define theweighted mean difference transform 119866(119906 V Δ)119909 of 119909 by
(119866 (119906 V Δ) 119909)119896 =
119896
sum
119894=0
119906119896V119894 (119909119894 minus 119909119894minus1) (119896 isin N0) (3)
where N0 denotes the set of nonnegative integers and weassume throughout that any term with negative subscript iszero Instead of writing (3) the operator 119866(119906 V Δ) can beexpressed as a lower triangular matrix (119892119899119896) where
119892119899119896 =
119906119896V119896 (119896 = 119899)
119906119896 (V119896 minus V119896+1) (0 le 119896 le 119899 minus 1)
0 (119896 gt 0)
(4)
Equivalently in componentwise the triangle (119892119899119896) can berepresented by
Hindawi Publishing CorporationInternational Journal of AnalysisVolume 2014 Article ID 786437 7 pageshttpdxdoiorg1011552014786437
2 International Journal of Analysis
(119892119899119896) =((
(
1199060V0 0 0 0 0 sdot sdot sdot
1199061 (V0 minus V1) 1199061V1 0 0 0 sdot sdot sdot
1199062 (V0 minus V1) 1199062 (V1 minus V2) 1199062V2 0 0 sdot sdot sdot
1199063 (V0 minus V1) 1199063 (V1 minus V2) 1199063 (V2 minus V3) 1199063V3 0 sdot sdot sdot
d
))
)
(5)
Themain objective of this paper is to determine the spec-trum of the operator 119866(119906 V Δ) over the basic sequence spaceℓ1 The operator 119866(119906 V Δ) has been studied by Polat et al[1] in detail by introducing the difference sequence spacesℓinfin(119906 V Δ) 1198880(119906 V Δ) and 119888(119906 V Δ) In the existing literatureseveral researchers have been actively engaged in findingthe spectrum and fine spectrum of different bounded linearoperators over various sequence spaces The spectrum ofweighted mean operator has been studied by Rhoades [2]whereas that of the difference operator Δ over the sequencespaces ℓ119901 for 0 lt 119901 lt 1 and 119888 1198880 has been studied byAltay andBasar [3 4] Kayaduman and Furkan [5] have determined thefine spectrum of the difference operator Δ over the sequencespaces ℓ1 and 119887V119901 and on generalizing these results Srivastavaand Kumar [6 7] have determined the fine spectrum of theoperator Δ ] over the sequence spaces ℓ1 and 1198880 where (]119896) isa sequence of either constant or strictly deceasing sequence ofreals satisfying certain conditions Dutta and Baliarsingh [8ndash10] have computed the spectrum of the operator Δ119903] (119903 isin N0)and Δ
2 over the sequence spaces ℓ1 1198880 and 1198880 respectivelyThe fine spectrum of the generalized difference operators119861(119903 119904) and 119861(119903 119904 119905) over the sequence spaces ℓ119901 119887V119901 and ℓ1119887V has been studied by Bilgic and Furkan [11 12] respectivelyRecently the spectrum of some particular limitationmatricesin certain sequence spaces has been studied by [13ndash15] andmany others
Let 119883 and 119884 be Banach spaces and 119879 119883 997888rarr 119884 be abounded linear operator ByR(119879) we denote the range of 119879that is
R (119879) = 119910 isin 119884 119910 = 119879119909 119909 isin 119883 (6)
By119861(119883) we denote the set of all bounded linear operatorson 119883 into itself If 119883 is any Banach space and 119879 isin 119861(119883) thenthe adjoint 119879lowast of 119879 is a bounded linear operator on the dual119883lowast of 119883 defined by (119879
lowast120601)(119909) = 120601(119879119909) for all 120601 isin 119883
lowast and119909 isin 119883 with 119879 = 119879
lowast
Let 119883 = 0 be a normed linear space over the complexfield and 119879 119863(119879) 997888rarr 119883 be a linear operator where 119863(119879)
denotes the domain of119879With119879 for a complex number 120582 weassociate an operator 119879120582 = (119879minus120582119868) where 119868 is called identityoperator on 119863(119879) and if 119879120582 has an inverse we denote it by119879minus1
120582 that is
119879minus1
120582= (119879 minus 120582119868)
minus1 (7)
and is called the resolvent operator of 119879 Many propertiesof 119879120582 and 119879
minus1
120582depend on 120582 and the spectral theory is
concerned with those properties We are interested in theset of all 120582rsquos in the complex plane such that 119879minus1
120582exists119879minus1
120582
is boundeddomain of 119879minus1120582
is dense in 119883 Now we state thefollowing results which are essential for our investigation
Definition 1 (see [16 page 371]) Let119883 = 0 be normed linearspace over the complex field and 119879 119863(119879) 997888rarr 119883 be a linearoperator with domain 119863(119879) sube 119883 A regular value of 119879 is acomplex number 120582 such that
(R1) 119879minus1120582
exists
(R2) 119879minus1120582
is bounded
(R3) 119879minus1120582
is defined on a set which is dense in 119883
The resolvent set 120588(119879119883) of 119879 is the set of all regularvalues of 119879 Its complement 120590(119879119883) = C 120588(119879119883) in thecomplex plane C is called the spectrum of 119879 Furthermorethe spectrum 120588(119879119883) is partitioned into three disjoint sets asfollows
(I) Point Spectrum 120590119901(119879119883) It is the set of all 120582 isin C suchthat (R1) does not hold The elements of 120590119901(119879119883) arecalled eigen values of 119879
(II) Continuous Spectrum 120590119888(119879119883) It is the set of all 120582 isin C
such that (R1) holds and satisfies (R3) but does notsatisfy (R2)
(III) Residual Spectrum 120590119903(119879119883) It is the set of all 120582 isin
C such that (R1) holds but does not satisfy (R3)Condition (R2) may or may not hold
Goldbergrsquos Classification of the Operator 119879120582 (see [17] page 58ndash71) Let119883 be a Banach space and119879120582 = (119879minus120582119868) isin 119861(119883) where120582 is a complex number Again let 119877(119879120582) and 119879
minus1
120582denote the
range and inverse of the operator 119879120582 respectively Then thefollowing possibilities may occur
(A) 119877(119879120582) = 119883
(B) 119877(119879120582) = 119877(119879120582) = 119883
(C) 119877(119879120582) =119883
and
(1) 119879120582 is injective and 119879minus1
120582is continuous
(2) 119879120582 is injective and 119879minus1
120582is discontinuous
(3) 119879120582 is not injective
International Journal of Analysis 3
Taking permutations (A) (B) (C) and (1) (2) (3) we getnine different states These are labelled by 11986011198602 1198603 1198611 11986121198613 1198621 1198622 and 1198623 If 120582 is a complex number such that 119879120582 isin
1198601 or 119879120582 isin 1198611 then 120582 is in the resolvent set 120588(119879119883) of 119879on 119883 The other classifications give rise to the fine spectrumof 119879 We use 120582 isin 1198612120590(119879119883) which means the operator 119879120582 isin
1198612 that is 119877(119879120582) = 119877(119879120582) = 119883 and 119879120582 is injective but 119879minus1
120582is
discontinuous similarly are the others
Lemma 2 (see [17 page 59]) A linear operator 119879 has a denserange if and only if the adjoint 119879lowast is one to one
Lemma 3 (see [17 page 60]) The adjoint operator 119879lowast is ontoif and and only if 119879 has a bounded inverse
Let 119875 119876 be two nonempty subsets of the space 119908 of allreal or complex sequences and let 119860 = (119886119899119896) be an infinitematrix of complex numbers 119886119899119896 where 119899 119896 isin N0 For every119909 = (119909119896) isin 119875 we write
119860119899 (119909) = sum
119896
119886119899119896119909119896 (119899 isin N) (8)
The sequence 119860119909 = (119860119899(119909)) if it exists is called thetransformation of 119909 by the matrix 119860 Infinite matrix 119860 isin
(119875 119876) if and only if 119860119909 isin 119876 whenever 119909 isin 119875
Lemma 4 (see [18 page 126]) Thematrix119860 = (119886119899119896) gives riseto a bounded linear operator 119879 isin 119861(ℓ1) from ℓ1 to itself if andonly if the supremum ℓ1 norms of the columns of119860 is bounded
Lemma 5 (see [19] Theorem 2) If 119886119896(0) = 0 for all 119896 isin N0then the inverse of the difference operator 119861(119886[119898]) is given bya lower triangular Toeplitz matrix 119862 = (119888119899119896) as follows
119888119899119896 =
1
119886119899 (0) (119896 = 119899)
(minus1)119899minus119896
prod119899
119895=119896119886119895 (0)
times119863(119896)
119899minus119896(119886 [119898]) (0 le 119896 le 119899 minus 1)
0 (119896 gt 119899)
(119899 119896 isin N0)
(9)
where (119861(119886[119898])119909)119896 = 119886119896(0)119909119896 + 119886119896minus1(1)119909119896minus1 + sdot sdot sdot +
119886119896minus119898(119898)119909119896minus119898 (119896 isin N0) and
119863(119896)
119899(119886 [119898]) =
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119886119896 (1) 119886119896+1 (0) 0 sdot sdot sdot 0 0 sdot sdot sdot 0
119886119896 (2) 119886119896+1 (1) 119886119896+2 (0) sdot sdot sdot 0 0 sdot sdot sdot 0
119886119896 (119898) 119886119896+1 (119898 minus 1) 119886119896+2 (119898 minus 2) sdot sdot sdot 119886119898minus1 (1) 119886119898 (0) sdot sdot sdot 0
0 119886119896+1 (119898) 119886119896+2 (119898 minus 1) sdot sdot sdot 119886119898minus1 (2) 119886119898 (1) sdot sdot sdot 0
d
0 0 sdot sdot sdot 119886119899minus119898 (119898) sdot sdot sdot sdot sdot sdot sdot sdot sdot 119886119899minus1 (1)
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(10)
2 The Spectrum of the Operator 119866(119906 V Δ)
over the Sequence ℓ1
In this sectionwe compute the spectrum the point spectrumthe continuous spectrum and the residual spectrum of thedifference matrix 119866(119906 V Δ) on the sequence space ℓ1
Theorem 6 The operator 119866(119906 V Δ) ℓ1 997888rarr ℓ1 is a linearoperator and
119866(119906 V Δ)(ℓ1ℓ1) = sup119899
10038161003816100381610038161199061198991003816100381610038161003816 (
119899minus1
sum
119896=0
1003816100381610038161003816V119896 minus V119896+11003816100381610038161003816 +
1003816100381610038161003816V1198991003816100381610038161003816)
(11)
Proof Proof of this theorem follows from Lemma 4 and inparticular cases
(i) If (119906119896) is a strictly decreasing sequence and (V119896) is aconstant sequence of positive reals (say V119896 = V for all119896) then
119866 (119906 V Δ)(ℓ1ℓ1) = 1199060V (12)
(ii) If (119906119896) and (V119896) are strictly decreasing sequences ofpositive reals then
119866 (119906 V Δ)(ℓ1ℓ1) = 1199060V0 (13)
(iii) If (119906119896) and (V119896) are constant sequences of positivereals then
119866 (119906 V Δ)(ℓ1ℓ1) = 119906V (14)
(iv) If (119906119896) is a constant sequence and (V119896) is a strictlydecreasing sequence of positive reals then
119866 (119906 V Δ)(ℓ1ℓ1) = 119906V0 (15)
4 International Journal of Analysis
Theorem7 The spectrumof119866(119906 V Δ) over the sequence spaceℓ1 is given by
120590 (119866 (119906 V Δ) ℓ1) = 120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
ge 1 (119896 isin N0)
(16)
Proof The proof is divided into four parts as followsLet (119906119896) and (V119896) be two bounded sequences of positive
reals satisfying (1) and (2) and 120582 isin C such that |(120582 minus
119906119896V119896+1)(120582 minus 119906119896V119896)| lt 1 for all 119896 isin N0 then (119866(119906 V Δ) minus 120582119868)
is a triangle and hence has an inverse that is
(119866 (119906 V Δ) minus 120582119868)minus1
=
(((((((((((((((((
(
1
1199060V0 minus 1205820 0 0 sdot sdot sdot
minus1199061 (V0 minus V1)
prod1
119895=0(119906119895V119895 minus 120582)
1
1199061V1 minus 1205820 0 sdot sdot sdot
1199062 (V0 minus V1) (120582 minus 1199061V2)
prod2
119895=0(119906119895V119895 minus 120582)
minus1199062 (V1 minus V2)
prod2
119895=1(119906119895V119895 minus 120582)
1
1199062V2 minus 1205820 sdot sdot sdot
minus1199063 (V0 minus V1) (120582 minus 1199061V2) (120582 minus 1199062V3)
prod3
119895=0(119906119895V119895 minus 120582)
1199063 (V1 minus V2) (120582 minus 1199062V3)
prod3
119895=1(119906119895V119895 minus 120582)
minus1199063 (V2 minus V3)
prod3
119895=2(119906119895V119895 minus 120582)
1
1199063V3 minus 120582sdot sdot sdot
d
)))))))))))))))))
)
(17)
In general by using Lemma 5 we observe that
119863(119896)
119899(119906 V 120582) =
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119906119896+1 (V119896 minus V119896+1) (119906119896+1V119896+1 minus 120582) 0 sdot sdot sdot 0
119906119896+2 (V119896 minus V119896+1) 119906119896+2 (V119896+1 minus V119896+2) (119906119896+2V119896+2 minus 120582) sdot sdot sdot 0
119906119896+3 (V119896 minus V119896+1) 119906119896+3 (V119896+1 minus V119896+2) 119906119896+3 (V119896+2 minus V119896+3) sdot sdot sdot 0
d
119906119899 (V119896 minus V119896+1) 119906119899 (V119896+1 minus V119896+2) 119906119899 (V119896+2 minus V119896+3) sdot sdot sdot 119906119899 (V119899minus1 minus V119899)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= (V119896 minus V119896+1) (V119896+1 minus V119896+2) sdot sdot sdot 119906119899 (V119899minus1 minus V119899) times (119906119896+1 minus(119906119896+1V119896+1 minus 120582)
(V119896+1 minus V119896+2))
times (119906119896+2 minus(119906119896+2V119896+2 minus 120582)
(V119896+2 minus V119896+3)) times sdot sdot sdot times (119906119899minus1 minus
(119906119899minus1V119899minus1 minus 120582)
(V119899minus1 minus V119899))
(18)
Therefore for all 119899 = 119896 ((119866(119906 V Δ)minus120582119868)minus1)119899119896 = 1(119906119899V119899minus120582)
for all 119896 gt 119899 ((119866(119906 V Δ) minus 120582119868)minus1)119899119896 = 0 and for all 0 le 119896 lt 119899
one can deduce that
((119866 (119906 V Δ) minus 120582119868)minus1)119899119896
=(V119896 minus V119896+1) 119906119899
(119906119896V119896 minus 120582) (119906119899V119899 minus 120582)
119899minus1
prod
119895=119896+1
(120582 minus 119906119895V119895+1)
(119906119895V119895 minus 120582)
(19)
By using Lemma 4 we obtain that
10038171003817100381710038171003817(119866 (119906 V Δ) minus 120582119868)
minus110038171003817100381710038171003817(ℓ1ℓ1)
= sup119899
119899
sum
119896=0
100381610038161003816100381610038161003816((119866 (119906 V Δ) minus 120582119868)
minus1)119899119896
100381610038161003816100381610038161003816
le sup119899
(
10038161003816100381610038161003816100381610038161003816
1
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
+
119899minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
(V119896 minus V119896+1) 119906119899(119906119896V119896 minus 120582) (119906119899V119899 minus 120582)
times
119899minus1
prod
119895=119896+1
(120582 minus 119906119895V119895+1)
(119906119895V119895 minus 120582)
10038161003816100381610038161003816100381610038161003816100381610038161003816
)
International Journal of Analysis 5
= sup119899
10038161003816100381610038161003816100381610038161003816
1
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
(1 +
119899minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
(V119896 minus V119896+1) 119906119899(119906119896V119896 minus 120582)
100381610038161003816100381610038161003816100381610038161003816
times
119899minus1
prod
119895=119896+1
10038161003816100381610038161003816100381610038161003816100381610038161003816
(120582 minus 119906119895V119895+1)
(119906119895V119895 minus 120582)
10038161003816100381610038161003816100381610038161003816100381610038161003816
)
le sup119899
10038161003816100381610038161003816100381610038161003816
119906119899
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
(1
10038161003816100381610038161199061198991003816100381610038161003816
+
119899minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
(V119896 minus V119896+1)(119906119896V119896 minus 120582)
100381610038161003816100381610038161003816100381610038161003816
119899minus1
prod
119895=119896+1
10038161003816100381610038161199031205821003816100381610038161003816)
le sup119899
10038161003816100381610038161003816100381610038161003816
119876120582119906119899
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
(1
10038161003816100381610038161198761205821199061198991003816100381610038161003816
+1
10038161003816100381610038161199031205821003816100381610038161003816
119899minus1
sum
119896=0
10038161003816100381610038161199031205821003816100381610038161003816
119899minus119896minus1)
= sup119899
10038161003816100381610038161003816100381610038161003816
119876120582119906119899
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
[1
10038161003816100381610038161198761205821199061198991003816100381610038161003816
+1
10038161003816100381610038161199031205821003816100381610038161003816
(1 minus
10038161003816100381610038161199031205821003816100381610038161003816
119899minus1
1 minus1003816100381610038161003816119903120582
1003816100381610038161003816
)] lt infin
(20)
For simplicity we write119876120582 = sup119896((V119896minusV119896+1)(119906119896V119896minus120582)) and
119903120582 = sup119895((120582minus119906119895V119895+1)(119906119895V119895 minus120582)) Since (119906119896) and (V119896) are two
bounded sequences of positive reals the quantity119876120582 is finiteAs per the assumption120582 isin C and |(120582minus119906119896V119896+1)(120582minus119906119896V119896)| lt 1
for all 119896 isin N0 Therefore |119903120582| lt 1 Now we consider the fourpossible cases of the sequences (119906119896) and (V119896)
Case 1 If (119906119896) and (V119896) are strictly decreasing sequencesof positive reals then from the above relation we have(119866(119906 V Δ) minus 120582119868)
minus1isin (ℓ1 ℓ1) Thus
120590 (119866 (119906 V Δ) ℓ1) sube 120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
ge 1 (119896 isin N0)
(21)
Conversely consider120582 = 119906119896V119896 and |(120582minus119906119896V119896+1)(120582minus119906119896V119896)| ge 1
for all 119896 isin N0 and clearly (119866(119906 V Δ) minus 120582119868) is a triangle andhence (119866(119906 V Δ)minus120582119868)
minus1 exists but (119866(119906 V Δ)minus120582119868)minus1(ℓ1ℓ1)
is not finite Again for 120582 = 119906119896V119896 the matrix (119866(119906 V Δ) minus
119906119896V119896119868)(119896 isin N0) is not invertible Thus
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
ge 1 (119896 isin N0) sube 120590 (119866 (119906 V Δ) ℓ1)
(22)
Combining (21) and (22) we complete the proof for thiscase
Case 2 If (119906119896) is a strictly decreasing sequence and (V119896) is aconstant sequence of positive reals then
(119866 (119906 V Δ) minus 120582119868)minus1
=
(((((((
(
1
1199060V minus 1205820 0 0 sdot sdot sdot
01
1199061V minus 1205820 0 sdot sdot sdot
0 01
1199062V minus 1205820 sdot sdot sdot
0 0 01
1199063V minus 120582sdot sdot sdot
d
)))))))
)
(23)
Since 119876 = 0 (119866(119906 V Δ) minus 120582119868)minus1(ℓ1ℓ1) = sup
119899|1(119906119899V minus 120582)| lt
infin and the rest part of the proof is similar to that of Case 1
Case 3 If (119906119896) and (V119896) are constant sequences of positivereals then the proof is similar to that of Case 2 and only thedifference is (119866(119906 V Δ) minus 120582119868)
minus1(ℓ1ℓ1) = |1119906V minus 120582| lt infin
Case 4 If (119906119896) is constant sequence and (V119896) is strictlydecreasing sequence of positive reals then the proof is similarto that of Case 1
Theorem8 The point spectrum of the operator119866(119906 V Δ) overℓ1 is given by
120590119901 (119866 (119906 V Δ) ℓ1)=
119906119896V (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
0 (119900119905ℎ119890119903119908119894119904119890)
(24)
Proof Suppose 119909 isin ℓ1 and consider the system of linearequations 119866(119906 V Δ)119909 = 120582119909 for 119909 = 0 = 0 0 0 in ℓ1
1199060V01199090 = 1205821199090
1199061 (V0 minus V1) 1199090 + 1199061V11199091 = 1205821199091
1199062 (V0 minus V1) 1199090 + 1199062 (V1 minus V2) 1199091 + 1199062V21199092 = 1205821199092
119906119899 (V0 minus V1) 1199090 + 119906119899 (V1 minus V2) 1199091 + sdot sdot sdot + 119906119899V119899119909119899 = 120582119909119899
(25)
On solving the system of (25) we obtain that 120582 = 1199060V0whenever 1199090 = 0 and
1199091 =1199061 (V0 minus V1)1199060V0 minus 1199061V1
1199090
1199092 =1199062 (V0 minus V1) (1199060V0 minus 1199061V2)(1199060V0 minus 1199061V1) (1199060V0 minus 1199062V2)
1199090 and in general
119909119899 =119906119899 (V0 minus V1)1199060V0 minus 119906119899V119899
119899minus1
prod
119895=1
1199060V0 minus 119906119895V119895+11199060V0 minus 119906119895V119895
1199090
(26)
Now we have the following cases
Case 1 If (119906119896) and (V119896) are strictly decreasing sequencesof positive reals then we need to show that |(1199060V0 minus
119906119895V119895+1)(1199060V0 minus 119906119895V119895)| ge 1 Suppose for the contrary if|(1199060V0 minus 119906119895V119895+1)(1199060V0 minus 119906119895V119895)| lt 1 then V119895+1 gt V119895 whichcontradicts the fact that (V119896) is strictly decreasing sequenceTherefore 119909 notin ℓ1 and hence 120590119901(119866(119906 V Δ) ℓ1) = 0
Case 2 If (119906119896) is a strictly decreasing and (V119896) is a con-stant sequence of positive reals then for all 119896 isin N0120582 = 119906119896V is an eigen value corresponding to the eigen
6 International Journal of Analysis
vector 119909 = 0 0 0 1 0 whose 119896th entry is 1 Thus120590119901(119866(119906 V Δ) ℓ1) = 119906119896V
Case 3 If (119906119896) and (V119896) are constant sequences of positivereals then the proof is similar to that of Case 2 and120590119901(119866(119906 V Δ) ℓ1) = 119906V
Case 4 If (119906119896) is a constant and (V119896) is a strictly decreasingsequence of positive reals then the proof is similar to that ofCase 1
Theorem 9 The point spectrum of the dual operator(119866(119906 V Δ))
lowast over the sequence space ℓlowast1≊ ℓinfin is given by
120590119901 ((119866 (119906 V Δ))lowast ℓinfin) =
119906119896V (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
gt 1 (119896 isin N0) (119900119905ℎ119890119903119908119894119904119890)
(27)
Proof Consider (119866(119906 V Δ))lowast119891 = 120582119891 and 0 = 119891 isin ℓ1 and then
the system of equations
1199060V01198910 + (V0 minus V1) 11990611198911 + (V0 minus V1) 11990621198912 + sdot sdot sdot = 1205821198910
1199061V11198911 + (V1 minus V2) 11990621198912 + (V1 minus V2) 11990631198913 + sdot sdot sdot = 1205821198911
1199062V21198912 + (V2 minus V3) 11990631198913 + (V2 minus V3) 11990641198914 + sdot sdot sdot = 1205821198912
119906119896V119896119891119896 + (V119896 minus V119896+1) 119906119896+1119891119896+1 + (V119896 minus V119896+1) 119906119896+1119891119896+1
+ sdot sdot sdot = 120582119891119896
(28)
From the system of linear equations (28) we obtain that
119891119896minus1 =V119896minus1 minus V119896
120582 minus 119906119896minus1V119896minus1(119906119896119891119896 + 119906119896+1119891119896+1 + 119906119896+2119891119896+2 + sdot sdot sdot )
(29)
119891119896 =V119896 minus V119896+1120582 minus 119906119896V119896
(119906119896+1119891119896+1 + 119906119896+2119891119896+2 + 119906119896+3119891119896+3 + sdot sdot sdot )
(119896 ge 1)
(30)
Combining (29) and (30) we have
119891119896 =V119896 minus V119896+1120582 minus 119906119896V119896
((120582 minus 119906119896minus1V119896minus1)(V119896minus1 minus V119896)
119891119896minus1 minus 119906119896119891119896) (31)
Therefore by using ratio text it is observed that
lim119896997888rarrinfin
10038161003816100381610038161003816100381610038161003816
119891119896
119891119896minus1
10038161003816100381610038161003816100381610038161003816
= lim119896997888rarrinfin
100381610038161003816100381610038161003816100381610038161003816
(V119896 minus V119896+1) (120582 minus 119906119896minus1V119896minus1)(V119896minus1 minus V119896) (120582 minus 119906119896V119896+1)
100381610038161003816100381610038161003816100381610038161003816
(32)
Now we have the following cases
Case 1 If (119906119896) and (V119896) are strictly decreasing sequencesof positive reals then by using (2) we conclude thatlim119896997888rarrinfin|119891119896119891119896minus1| lt 1 if and only if |(120582 minus 119906119896minus1V119896minus1)(120582 minus
119906119896V119896+1)| lt 1 for all 119896 ge 1 Therefore 120590119901((119866(119906 V Δ))lowast ℓinfin) =
120582 isin C |(120582 minus 119906119896V119896+1)(120582 minus 119906119896V119896)| gt 1 (119896 isin N0)
Case 2 If (119906119896) is a strictly decreasing and (V119896) is a constantsequence of positive reals then from the system of (28) weobserve that for all 119896 isin N0 120582 = 119906119896V is an eigen valuecorresponding to the eigen vector 119891 = 0 0 0 1 0
whose 119896th entry is 1 Thus 120590119901((119866(119906 V Δ))lowast ℓinfin) = 119906119896V
Case 3 If (119906119896) and (V119896) are constant sequences of positivereals then the proof is similar to that of Case 2
Case 4 If (119906119896) is a constant and (V119896) is a strictly decreasingsequence of positive reals then the proof is similar to that ofCase 1
Theorem 10 The residual spectrum of the operator 119866(119906 V Δ)
over the sequence space ℓ1 is given by
120590119903 (119866 (119906 V Δ) ℓ1) =
0 (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
gt 1 (119896 isin N0) (119900119905ℎ119890119903119908119894119904119890)
(33)
Proof Proof follows from Lemma 2 and Theorems 8 and 9 Theorem 11 The continuous spectrum of the operator119866(119906 V Δ) over the sequence space ℓ1 is given by
International Journal of Analysis 7
120590119888 (119866 (119906 V Δ) ℓ1) =
0 (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
= 1 (119896 isin N0) (119900119905ℎ119890119903119908119894119904119890)
(34)
Proof The proof of this theorem follows fromTheorems 7 8and 10 and along with the fact that
120590 (119866 (119906 V Δ) ℓ1)
= 120590119901 (119866 (119906 V Δ) ℓ1) cup 120590119903 (119866 (119906 V Δ) ℓ1)
cup 120590119888 (119866 (119906 V Δ) ℓ1)
(35)
3 Conclusion
In this work the authors have determined the spectrum ofthe generalized weighted mean difference operator 119866(119906 V Δ)
over the Banach space ℓ1
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Polat V Karakaya and N Simsek ldquoDifference sequencespaces derived by using a generalized weighted meanrdquo AppliedMathematics Letters vol 24 no 5 pp 608ndash614 2011
[2] B E Rhoades ldquoThe fine spectra for weighted mean operatorsrdquoPacific Journal of Mathematics vol 104 no 1 pp 219ndash230 1983
[3] B Altay and F Basar ldquoThe fine spectrum and the matrixdomain of the difference operator 998779 on the sequence space 119897119901(0 lt 119901 lt 1)rdquo Communications in Mathematical Analysis vol 2no 2 pp 1ndash11 2007
[4] B Altay and F Basar ldquoOn the fine spectrum of the differenceoperator 998779 on 1198880 and 119888rdquo Information Sciences vol 168 no 1ndash4pp 217ndash224 2004
[5] K Kayaduman and H Furkan ldquoThe fine spectra of thedifference operator 998779 over the sequence spaces 1198971 and 119887VrdquoInternational Mathematical Forum vol 1 no 21ndash24 pp 1153ndash1160 2006
[6] P D Srivastava and S Kumar ldquoFine spectrumof the generalizeddifference operator 998779V on sequence space 1198971rdquo Thai Journal ofMathematics vol 8 no 2 pp 221ndash233 2010
[7] P D Srivastava and S Kumar ldquoOn the fine spectrum of thegeneralized difference operator 998779V over the sequence space 1198880rdquoCommunications in Mathematical Analysis vol 6 no 1 pp 8ndash21 2009
[8] S Dutta and P Baliarsingh ldquoOn the fine spectra of thegeneralized rth difference operator 998779
119903
V on the sequence space1198971rdquo Applied Mathematics and Computation vol 219 no 4 pp1776ndash1784 2012
[9] S Dutta and P Baliarsingh ldquoOn the spectrum of 2-nd ordergeneralized difference operator 9987792 over the sequence space 1198880rdquo
Boletim da Sociedade Paranaense de Matematica 3rd Serie vol31 no 2 pp 235ndash244 2013
[10] S Dutta and P Baliarsingh ldquoOn a spectral classification of theoperator 998779119903V over thesequence space 1198880rdquo Proceeding of NationalAcademy of Sciences India A Physical Science In press
[11] H Bilgic and H Furkan ldquoOn the fine spectrum of the gener-alized difference operator 119861 (119903 119904) over the sequence spaces 119897119901
and 119887V119901 (1 lt 119901 lt infin)rdquo Nonlinear Analysis Theory Methods ampApplications vol 68 no 3 pp 499ndash506 2008
[12] H Bilgic and H Furkan ldquoOn the fine spectrum of the operator119861 (119903 119904 119905) over the sequence spaces 1198971 and 119887VrdquoMathematical andComputer Modelling vol 45 no 7-8 pp 883ndash891 2007
[13] F Basar Summability Theory and Its Applications BenthamScience Publishers e-books Mono-graphs Istanbul Turkey2012
[14] S Dutta and P Baliarsingh ldquoSome spectral aspects of theoperator Δ119903V over the sequence spaces 119897119901 and 119887V119901(1 lt 119901 lt infin)rdquoChinese Journal of Mathematics vol 2013 Article ID 286748 10pages 2013
[15] S Dutta and P Baliarsingh ldquoOn the spectrum of differenceoperator 998779119886119887 over the sequence spaces 119897119901 and 119887V119901(1 lt 119901 lt infin)rdquoMathematical Sciences Letters vol 3 no 2 pp 115ndash120 2014
[16] E Kreyszig Introductory Functional Analysis with ApplicationsJohn Wiley amp Sons New York NY USA 1978
[17] S Goldberg Unbounded Linear Operators Dover New YorkNY USA 1985
[18] A Wilansky Summability through Functional Analysis vol 85of North-Holland Mathematics Studies North-Holland Ams-terdam The Netherlands 1984
[19] S Dutta and P Baliarsingh ldquoOn some Toeplitz matrices andtheir inversionsrdquo Journal of the Egyptian Mathematical SocietyIn press
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 International Journal of Analysis
(119892119899119896) =((
(
1199060V0 0 0 0 0 sdot sdot sdot
1199061 (V0 minus V1) 1199061V1 0 0 0 sdot sdot sdot
1199062 (V0 minus V1) 1199062 (V1 minus V2) 1199062V2 0 0 sdot sdot sdot
1199063 (V0 minus V1) 1199063 (V1 minus V2) 1199063 (V2 minus V3) 1199063V3 0 sdot sdot sdot
d
))
)
(5)
Themain objective of this paper is to determine the spec-trum of the operator 119866(119906 V Δ) over the basic sequence spaceℓ1 The operator 119866(119906 V Δ) has been studied by Polat et al[1] in detail by introducing the difference sequence spacesℓinfin(119906 V Δ) 1198880(119906 V Δ) and 119888(119906 V Δ) In the existing literatureseveral researchers have been actively engaged in findingthe spectrum and fine spectrum of different bounded linearoperators over various sequence spaces The spectrum ofweighted mean operator has been studied by Rhoades [2]whereas that of the difference operator Δ over the sequencespaces ℓ119901 for 0 lt 119901 lt 1 and 119888 1198880 has been studied byAltay andBasar [3 4] Kayaduman and Furkan [5] have determined thefine spectrum of the difference operator Δ over the sequencespaces ℓ1 and 119887V119901 and on generalizing these results Srivastavaand Kumar [6 7] have determined the fine spectrum of theoperator Δ ] over the sequence spaces ℓ1 and 1198880 where (]119896) isa sequence of either constant or strictly deceasing sequence ofreals satisfying certain conditions Dutta and Baliarsingh [8ndash10] have computed the spectrum of the operator Δ119903] (119903 isin N0)and Δ
2 over the sequence spaces ℓ1 1198880 and 1198880 respectivelyThe fine spectrum of the generalized difference operators119861(119903 119904) and 119861(119903 119904 119905) over the sequence spaces ℓ119901 119887V119901 and ℓ1119887V has been studied by Bilgic and Furkan [11 12] respectivelyRecently the spectrum of some particular limitationmatricesin certain sequence spaces has been studied by [13ndash15] andmany others
Let 119883 and 119884 be Banach spaces and 119879 119883 997888rarr 119884 be abounded linear operator ByR(119879) we denote the range of 119879that is
R (119879) = 119910 isin 119884 119910 = 119879119909 119909 isin 119883 (6)
By119861(119883) we denote the set of all bounded linear operatorson 119883 into itself If 119883 is any Banach space and 119879 isin 119861(119883) thenthe adjoint 119879lowast of 119879 is a bounded linear operator on the dual119883lowast of 119883 defined by (119879
lowast120601)(119909) = 120601(119879119909) for all 120601 isin 119883
lowast and119909 isin 119883 with 119879 = 119879
lowast
Let 119883 = 0 be a normed linear space over the complexfield and 119879 119863(119879) 997888rarr 119883 be a linear operator where 119863(119879)
denotes the domain of119879With119879 for a complex number 120582 weassociate an operator 119879120582 = (119879minus120582119868) where 119868 is called identityoperator on 119863(119879) and if 119879120582 has an inverse we denote it by119879minus1
120582 that is
119879minus1
120582= (119879 minus 120582119868)
minus1 (7)
and is called the resolvent operator of 119879 Many propertiesof 119879120582 and 119879
minus1
120582depend on 120582 and the spectral theory is
concerned with those properties We are interested in theset of all 120582rsquos in the complex plane such that 119879minus1
120582exists119879minus1
120582
is boundeddomain of 119879minus1120582
is dense in 119883 Now we state thefollowing results which are essential for our investigation
Definition 1 (see [16 page 371]) Let119883 = 0 be normed linearspace over the complex field and 119879 119863(119879) 997888rarr 119883 be a linearoperator with domain 119863(119879) sube 119883 A regular value of 119879 is acomplex number 120582 such that
(R1) 119879minus1120582
exists
(R2) 119879minus1120582
is bounded
(R3) 119879minus1120582
is defined on a set which is dense in 119883
The resolvent set 120588(119879119883) of 119879 is the set of all regularvalues of 119879 Its complement 120590(119879119883) = C 120588(119879119883) in thecomplex plane C is called the spectrum of 119879 Furthermorethe spectrum 120588(119879119883) is partitioned into three disjoint sets asfollows
(I) Point Spectrum 120590119901(119879119883) It is the set of all 120582 isin C suchthat (R1) does not hold The elements of 120590119901(119879119883) arecalled eigen values of 119879
(II) Continuous Spectrum 120590119888(119879119883) It is the set of all 120582 isin C
such that (R1) holds and satisfies (R3) but does notsatisfy (R2)
(III) Residual Spectrum 120590119903(119879119883) It is the set of all 120582 isin
C such that (R1) holds but does not satisfy (R3)Condition (R2) may or may not hold
Goldbergrsquos Classification of the Operator 119879120582 (see [17] page 58ndash71) Let119883 be a Banach space and119879120582 = (119879minus120582119868) isin 119861(119883) where120582 is a complex number Again let 119877(119879120582) and 119879
minus1
120582denote the
range and inverse of the operator 119879120582 respectively Then thefollowing possibilities may occur
(A) 119877(119879120582) = 119883
(B) 119877(119879120582) = 119877(119879120582) = 119883
(C) 119877(119879120582) =119883
and
(1) 119879120582 is injective and 119879minus1
120582is continuous
(2) 119879120582 is injective and 119879minus1
120582is discontinuous
(3) 119879120582 is not injective
International Journal of Analysis 3
Taking permutations (A) (B) (C) and (1) (2) (3) we getnine different states These are labelled by 11986011198602 1198603 1198611 11986121198613 1198621 1198622 and 1198623 If 120582 is a complex number such that 119879120582 isin
1198601 or 119879120582 isin 1198611 then 120582 is in the resolvent set 120588(119879119883) of 119879on 119883 The other classifications give rise to the fine spectrumof 119879 We use 120582 isin 1198612120590(119879119883) which means the operator 119879120582 isin
1198612 that is 119877(119879120582) = 119877(119879120582) = 119883 and 119879120582 is injective but 119879minus1
120582is
discontinuous similarly are the others
Lemma 2 (see [17 page 59]) A linear operator 119879 has a denserange if and only if the adjoint 119879lowast is one to one
Lemma 3 (see [17 page 60]) The adjoint operator 119879lowast is ontoif and and only if 119879 has a bounded inverse
Let 119875 119876 be two nonempty subsets of the space 119908 of allreal or complex sequences and let 119860 = (119886119899119896) be an infinitematrix of complex numbers 119886119899119896 where 119899 119896 isin N0 For every119909 = (119909119896) isin 119875 we write
119860119899 (119909) = sum
119896
119886119899119896119909119896 (119899 isin N) (8)
The sequence 119860119909 = (119860119899(119909)) if it exists is called thetransformation of 119909 by the matrix 119860 Infinite matrix 119860 isin
(119875 119876) if and only if 119860119909 isin 119876 whenever 119909 isin 119875
Lemma 4 (see [18 page 126]) Thematrix119860 = (119886119899119896) gives riseto a bounded linear operator 119879 isin 119861(ℓ1) from ℓ1 to itself if andonly if the supremum ℓ1 norms of the columns of119860 is bounded
Lemma 5 (see [19] Theorem 2) If 119886119896(0) = 0 for all 119896 isin N0then the inverse of the difference operator 119861(119886[119898]) is given bya lower triangular Toeplitz matrix 119862 = (119888119899119896) as follows
119888119899119896 =
1
119886119899 (0) (119896 = 119899)
(minus1)119899minus119896
prod119899
119895=119896119886119895 (0)
times119863(119896)
119899minus119896(119886 [119898]) (0 le 119896 le 119899 minus 1)
0 (119896 gt 119899)
(119899 119896 isin N0)
(9)
where (119861(119886[119898])119909)119896 = 119886119896(0)119909119896 + 119886119896minus1(1)119909119896minus1 + sdot sdot sdot +
119886119896minus119898(119898)119909119896minus119898 (119896 isin N0) and
119863(119896)
119899(119886 [119898]) =
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119886119896 (1) 119886119896+1 (0) 0 sdot sdot sdot 0 0 sdot sdot sdot 0
119886119896 (2) 119886119896+1 (1) 119886119896+2 (0) sdot sdot sdot 0 0 sdot sdot sdot 0
119886119896 (119898) 119886119896+1 (119898 minus 1) 119886119896+2 (119898 minus 2) sdot sdot sdot 119886119898minus1 (1) 119886119898 (0) sdot sdot sdot 0
0 119886119896+1 (119898) 119886119896+2 (119898 minus 1) sdot sdot sdot 119886119898minus1 (2) 119886119898 (1) sdot sdot sdot 0
d
0 0 sdot sdot sdot 119886119899minus119898 (119898) sdot sdot sdot sdot sdot sdot sdot sdot sdot 119886119899minus1 (1)
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(10)
2 The Spectrum of the Operator 119866(119906 V Δ)
over the Sequence ℓ1
In this sectionwe compute the spectrum the point spectrumthe continuous spectrum and the residual spectrum of thedifference matrix 119866(119906 V Δ) on the sequence space ℓ1
Theorem 6 The operator 119866(119906 V Δ) ℓ1 997888rarr ℓ1 is a linearoperator and
119866(119906 V Δ)(ℓ1ℓ1) = sup119899
10038161003816100381610038161199061198991003816100381610038161003816 (
119899minus1
sum
119896=0
1003816100381610038161003816V119896 minus V119896+11003816100381610038161003816 +
1003816100381610038161003816V1198991003816100381610038161003816)
(11)
Proof Proof of this theorem follows from Lemma 4 and inparticular cases
(i) If (119906119896) is a strictly decreasing sequence and (V119896) is aconstant sequence of positive reals (say V119896 = V for all119896) then
119866 (119906 V Δ)(ℓ1ℓ1) = 1199060V (12)
(ii) If (119906119896) and (V119896) are strictly decreasing sequences ofpositive reals then
119866 (119906 V Δ)(ℓ1ℓ1) = 1199060V0 (13)
(iii) If (119906119896) and (V119896) are constant sequences of positivereals then
119866 (119906 V Δ)(ℓ1ℓ1) = 119906V (14)
(iv) If (119906119896) is a constant sequence and (V119896) is a strictlydecreasing sequence of positive reals then
119866 (119906 V Δ)(ℓ1ℓ1) = 119906V0 (15)
4 International Journal of Analysis
Theorem7 The spectrumof119866(119906 V Δ) over the sequence spaceℓ1 is given by
120590 (119866 (119906 V Δ) ℓ1) = 120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
ge 1 (119896 isin N0)
(16)
Proof The proof is divided into four parts as followsLet (119906119896) and (V119896) be two bounded sequences of positive
reals satisfying (1) and (2) and 120582 isin C such that |(120582 minus
119906119896V119896+1)(120582 minus 119906119896V119896)| lt 1 for all 119896 isin N0 then (119866(119906 V Δ) minus 120582119868)
is a triangle and hence has an inverse that is
(119866 (119906 V Δ) minus 120582119868)minus1
=
(((((((((((((((((
(
1
1199060V0 minus 1205820 0 0 sdot sdot sdot
minus1199061 (V0 minus V1)
prod1
119895=0(119906119895V119895 minus 120582)
1
1199061V1 minus 1205820 0 sdot sdot sdot
1199062 (V0 minus V1) (120582 minus 1199061V2)
prod2
119895=0(119906119895V119895 minus 120582)
minus1199062 (V1 minus V2)
prod2
119895=1(119906119895V119895 minus 120582)
1
1199062V2 minus 1205820 sdot sdot sdot
minus1199063 (V0 minus V1) (120582 minus 1199061V2) (120582 minus 1199062V3)
prod3
119895=0(119906119895V119895 minus 120582)
1199063 (V1 minus V2) (120582 minus 1199062V3)
prod3
119895=1(119906119895V119895 minus 120582)
minus1199063 (V2 minus V3)
prod3
119895=2(119906119895V119895 minus 120582)
1
1199063V3 minus 120582sdot sdot sdot
d
)))))))))))))))))
)
(17)
In general by using Lemma 5 we observe that
119863(119896)
119899(119906 V 120582) =
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119906119896+1 (V119896 minus V119896+1) (119906119896+1V119896+1 minus 120582) 0 sdot sdot sdot 0
119906119896+2 (V119896 minus V119896+1) 119906119896+2 (V119896+1 minus V119896+2) (119906119896+2V119896+2 minus 120582) sdot sdot sdot 0
119906119896+3 (V119896 minus V119896+1) 119906119896+3 (V119896+1 minus V119896+2) 119906119896+3 (V119896+2 minus V119896+3) sdot sdot sdot 0
d
119906119899 (V119896 minus V119896+1) 119906119899 (V119896+1 minus V119896+2) 119906119899 (V119896+2 minus V119896+3) sdot sdot sdot 119906119899 (V119899minus1 minus V119899)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= (V119896 minus V119896+1) (V119896+1 minus V119896+2) sdot sdot sdot 119906119899 (V119899minus1 minus V119899) times (119906119896+1 minus(119906119896+1V119896+1 minus 120582)
(V119896+1 minus V119896+2))
times (119906119896+2 minus(119906119896+2V119896+2 minus 120582)
(V119896+2 minus V119896+3)) times sdot sdot sdot times (119906119899minus1 minus
(119906119899minus1V119899minus1 minus 120582)
(V119899minus1 minus V119899))
(18)
Therefore for all 119899 = 119896 ((119866(119906 V Δ)minus120582119868)minus1)119899119896 = 1(119906119899V119899minus120582)
for all 119896 gt 119899 ((119866(119906 V Δ) minus 120582119868)minus1)119899119896 = 0 and for all 0 le 119896 lt 119899
one can deduce that
((119866 (119906 V Δ) minus 120582119868)minus1)119899119896
=(V119896 minus V119896+1) 119906119899
(119906119896V119896 minus 120582) (119906119899V119899 minus 120582)
119899minus1
prod
119895=119896+1
(120582 minus 119906119895V119895+1)
(119906119895V119895 minus 120582)
(19)
By using Lemma 4 we obtain that
10038171003817100381710038171003817(119866 (119906 V Δ) minus 120582119868)
minus110038171003817100381710038171003817(ℓ1ℓ1)
= sup119899
119899
sum
119896=0
100381610038161003816100381610038161003816((119866 (119906 V Δ) minus 120582119868)
minus1)119899119896
100381610038161003816100381610038161003816
le sup119899
(
10038161003816100381610038161003816100381610038161003816
1
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
+
119899minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
(V119896 minus V119896+1) 119906119899(119906119896V119896 minus 120582) (119906119899V119899 minus 120582)
times
119899minus1
prod
119895=119896+1
(120582 minus 119906119895V119895+1)
(119906119895V119895 minus 120582)
10038161003816100381610038161003816100381610038161003816100381610038161003816
)
International Journal of Analysis 5
= sup119899
10038161003816100381610038161003816100381610038161003816
1
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
(1 +
119899minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
(V119896 minus V119896+1) 119906119899(119906119896V119896 minus 120582)
100381610038161003816100381610038161003816100381610038161003816
times
119899minus1
prod
119895=119896+1
10038161003816100381610038161003816100381610038161003816100381610038161003816
(120582 minus 119906119895V119895+1)
(119906119895V119895 minus 120582)
10038161003816100381610038161003816100381610038161003816100381610038161003816
)
le sup119899
10038161003816100381610038161003816100381610038161003816
119906119899
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
(1
10038161003816100381610038161199061198991003816100381610038161003816
+
119899minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
(V119896 minus V119896+1)(119906119896V119896 minus 120582)
100381610038161003816100381610038161003816100381610038161003816
119899minus1
prod
119895=119896+1
10038161003816100381610038161199031205821003816100381610038161003816)
le sup119899
10038161003816100381610038161003816100381610038161003816
119876120582119906119899
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
(1
10038161003816100381610038161198761205821199061198991003816100381610038161003816
+1
10038161003816100381610038161199031205821003816100381610038161003816
119899minus1
sum
119896=0
10038161003816100381610038161199031205821003816100381610038161003816
119899minus119896minus1)
= sup119899
10038161003816100381610038161003816100381610038161003816
119876120582119906119899
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
[1
10038161003816100381610038161198761205821199061198991003816100381610038161003816
+1
10038161003816100381610038161199031205821003816100381610038161003816
(1 minus
10038161003816100381610038161199031205821003816100381610038161003816
119899minus1
1 minus1003816100381610038161003816119903120582
1003816100381610038161003816
)] lt infin
(20)
For simplicity we write119876120582 = sup119896((V119896minusV119896+1)(119906119896V119896minus120582)) and
119903120582 = sup119895((120582minus119906119895V119895+1)(119906119895V119895 minus120582)) Since (119906119896) and (V119896) are two
bounded sequences of positive reals the quantity119876120582 is finiteAs per the assumption120582 isin C and |(120582minus119906119896V119896+1)(120582minus119906119896V119896)| lt 1
for all 119896 isin N0 Therefore |119903120582| lt 1 Now we consider the fourpossible cases of the sequences (119906119896) and (V119896)
Case 1 If (119906119896) and (V119896) are strictly decreasing sequencesof positive reals then from the above relation we have(119866(119906 V Δ) minus 120582119868)
minus1isin (ℓ1 ℓ1) Thus
120590 (119866 (119906 V Δ) ℓ1) sube 120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
ge 1 (119896 isin N0)
(21)
Conversely consider120582 = 119906119896V119896 and |(120582minus119906119896V119896+1)(120582minus119906119896V119896)| ge 1
for all 119896 isin N0 and clearly (119866(119906 V Δ) minus 120582119868) is a triangle andhence (119866(119906 V Δ)minus120582119868)
minus1 exists but (119866(119906 V Δ)minus120582119868)minus1(ℓ1ℓ1)
is not finite Again for 120582 = 119906119896V119896 the matrix (119866(119906 V Δ) minus
119906119896V119896119868)(119896 isin N0) is not invertible Thus
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
ge 1 (119896 isin N0) sube 120590 (119866 (119906 V Δ) ℓ1)
(22)
Combining (21) and (22) we complete the proof for thiscase
Case 2 If (119906119896) is a strictly decreasing sequence and (V119896) is aconstant sequence of positive reals then
(119866 (119906 V Δ) minus 120582119868)minus1
=
(((((((
(
1
1199060V minus 1205820 0 0 sdot sdot sdot
01
1199061V minus 1205820 0 sdot sdot sdot
0 01
1199062V minus 1205820 sdot sdot sdot
0 0 01
1199063V minus 120582sdot sdot sdot
d
)))))))
)
(23)
Since 119876 = 0 (119866(119906 V Δ) minus 120582119868)minus1(ℓ1ℓ1) = sup
119899|1(119906119899V minus 120582)| lt
infin and the rest part of the proof is similar to that of Case 1
Case 3 If (119906119896) and (V119896) are constant sequences of positivereals then the proof is similar to that of Case 2 and only thedifference is (119866(119906 V Δ) minus 120582119868)
minus1(ℓ1ℓ1) = |1119906V minus 120582| lt infin
Case 4 If (119906119896) is constant sequence and (V119896) is strictlydecreasing sequence of positive reals then the proof is similarto that of Case 1
Theorem8 The point spectrum of the operator119866(119906 V Δ) overℓ1 is given by
120590119901 (119866 (119906 V Δ) ℓ1)=
119906119896V (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
0 (119900119905ℎ119890119903119908119894119904119890)
(24)
Proof Suppose 119909 isin ℓ1 and consider the system of linearequations 119866(119906 V Δ)119909 = 120582119909 for 119909 = 0 = 0 0 0 in ℓ1
1199060V01199090 = 1205821199090
1199061 (V0 minus V1) 1199090 + 1199061V11199091 = 1205821199091
1199062 (V0 minus V1) 1199090 + 1199062 (V1 minus V2) 1199091 + 1199062V21199092 = 1205821199092
119906119899 (V0 minus V1) 1199090 + 119906119899 (V1 minus V2) 1199091 + sdot sdot sdot + 119906119899V119899119909119899 = 120582119909119899
(25)
On solving the system of (25) we obtain that 120582 = 1199060V0whenever 1199090 = 0 and
1199091 =1199061 (V0 minus V1)1199060V0 minus 1199061V1
1199090
1199092 =1199062 (V0 minus V1) (1199060V0 minus 1199061V2)(1199060V0 minus 1199061V1) (1199060V0 minus 1199062V2)
1199090 and in general
119909119899 =119906119899 (V0 minus V1)1199060V0 minus 119906119899V119899
119899minus1
prod
119895=1
1199060V0 minus 119906119895V119895+11199060V0 minus 119906119895V119895
1199090
(26)
Now we have the following cases
Case 1 If (119906119896) and (V119896) are strictly decreasing sequencesof positive reals then we need to show that |(1199060V0 minus
119906119895V119895+1)(1199060V0 minus 119906119895V119895)| ge 1 Suppose for the contrary if|(1199060V0 minus 119906119895V119895+1)(1199060V0 minus 119906119895V119895)| lt 1 then V119895+1 gt V119895 whichcontradicts the fact that (V119896) is strictly decreasing sequenceTherefore 119909 notin ℓ1 and hence 120590119901(119866(119906 V Δ) ℓ1) = 0
Case 2 If (119906119896) is a strictly decreasing and (V119896) is a con-stant sequence of positive reals then for all 119896 isin N0120582 = 119906119896V is an eigen value corresponding to the eigen
6 International Journal of Analysis
vector 119909 = 0 0 0 1 0 whose 119896th entry is 1 Thus120590119901(119866(119906 V Δ) ℓ1) = 119906119896V
Case 3 If (119906119896) and (V119896) are constant sequences of positivereals then the proof is similar to that of Case 2 and120590119901(119866(119906 V Δ) ℓ1) = 119906V
Case 4 If (119906119896) is a constant and (V119896) is a strictly decreasingsequence of positive reals then the proof is similar to that ofCase 1
Theorem 9 The point spectrum of the dual operator(119866(119906 V Δ))
lowast over the sequence space ℓlowast1≊ ℓinfin is given by
120590119901 ((119866 (119906 V Δ))lowast ℓinfin) =
119906119896V (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
gt 1 (119896 isin N0) (119900119905ℎ119890119903119908119894119904119890)
(27)
Proof Consider (119866(119906 V Δ))lowast119891 = 120582119891 and 0 = 119891 isin ℓ1 and then
the system of equations
1199060V01198910 + (V0 minus V1) 11990611198911 + (V0 minus V1) 11990621198912 + sdot sdot sdot = 1205821198910
1199061V11198911 + (V1 minus V2) 11990621198912 + (V1 minus V2) 11990631198913 + sdot sdot sdot = 1205821198911
1199062V21198912 + (V2 minus V3) 11990631198913 + (V2 minus V3) 11990641198914 + sdot sdot sdot = 1205821198912
119906119896V119896119891119896 + (V119896 minus V119896+1) 119906119896+1119891119896+1 + (V119896 minus V119896+1) 119906119896+1119891119896+1
+ sdot sdot sdot = 120582119891119896
(28)
From the system of linear equations (28) we obtain that
119891119896minus1 =V119896minus1 minus V119896
120582 minus 119906119896minus1V119896minus1(119906119896119891119896 + 119906119896+1119891119896+1 + 119906119896+2119891119896+2 + sdot sdot sdot )
(29)
119891119896 =V119896 minus V119896+1120582 minus 119906119896V119896
(119906119896+1119891119896+1 + 119906119896+2119891119896+2 + 119906119896+3119891119896+3 + sdot sdot sdot )
(119896 ge 1)
(30)
Combining (29) and (30) we have
119891119896 =V119896 minus V119896+1120582 minus 119906119896V119896
((120582 minus 119906119896minus1V119896minus1)(V119896minus1 minus V119896)
119891119896minus1 minus 119906119896119891119896) (31)
Therefore by using ratio text it is observed that
lim119896997888rarrinfin
10038161003816100381610038161003816100381610038161003816
119891119896
119891119896minus1
10038161003816100381610038161003816100381610038161003816
= lim119896997888rarrinfin
100381610038161003816100381610038161003816100381610038161003816
(V119896 minus V119896+1) (120582 minus 119906119896minus1V119896minus1)(V119896minus1 minus V119896) (120582 minus 119906119896V119896+1)
100381610038161003816100381610038161003816100381610038161003816
(32)
Now we have the following cases
Case 1 If (119906119896) and (V119896) are strictly decreasing sequencesof positive reals then by using (2) we conclude thatlim119896997888rarrinfin|119891119896119891119896minus1| lt 1 if and only if |(120582 minus 119906119896minus1V119896minus1)(120582 minus
119906119896V119896+1)| lt 1 for all 119896 ge 1 Therefore 120590119901((119866(119906 V Δ))lowast ℓinfin) =
120582 isin C |(120582 minus 119906119896V119896+1)(120582 minus 119906119896V119896)| gt 1 (119896 isin N0)
Case 2 If (119906119896) is a strictly decreasing and (V119896) is a constantsequence of positive reals then from the system of (28) weobserve that for all 119896 isin N0 120582 = 119906119896V is an eigen valuecorresponding to the eigen vector 119891 = 0 0 0 1 0
whose 119896th entry is 1 Thus 120590119901((119866(119906 V Δ))lowast ℓinfin) = 119906119896V
Case 3 If (119906119896) and (V119896) are constant sequences of positivereals then the proof is similar to that of Case 2
Case 4 If (119906119896) is a constant and (V119896) is a strictly decreasingsequence of positive reals then the proof is similar to that ofCase 1
Theorem 10 The residual spectrum of the operator 119866(119906 V Δ)
over the sequence space ℓ1 is given by
120590119903 (119866 (119906 V Δ) ℓ1) =
0 (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
gt 1 (119896 isin N0) (119900119905ℎ119890119903119908119894119904119890)
(33)
Proof Proof follows from Lemma 2 and Theorems 8 and 9 Theorem 11 The continuous spectrum of the operator119866(119906 V Δ) over the sequence space ℓ1 is given by
International Journal of Analysis 7
120590119888 (119866 (119906 V Δ) ℓ1) =
0 (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
= 1 (119896 isin N0) (119900119905ℎ119890119903119908119894119904119890)
(34)
Proof The proof of this theorem follows fromTheorems 7 8and 10 and along with the fact that
120590 (119866 (119906 V Δ) ℓ1)
= 120590119901 (119866 (119906 V Δ) ℓ1) cup 120590119903 (119866 (119906 V Δ) ℓ1)
cup 120590119888 (119866 (119906 V Δ) ℓ1)
(35)
3 Conclusion
In this work the authors have determined the spectrum ofthe generalized weighted mean difference operator 119866(119906 V Δ)
over the Banach space ℓ1
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Polat V Karakaya and N Simsek ldquoDifference sequencespaces derived by using a generalized weighted meanrdquo AppliedMathematics Letters vol 24 no 5 pp 608ndash614 2011
[2] B E Rhoades ldquoThe fine spectra for weighted mean operatorsrdquoPacific Journal of Mathematics vol 104 no 1 pp 219ndash230 1983
[3] B Altay and F Basar ldquoThe fine spectrum and the matrixdomain of the difference operator 998779 on the sequence space 119897119901(0 lt 119901 lt 1)rdquo Communications in Mathematical Analysis vol 2no 2 pp 1ndash11 2007
[4] B Altay and F Basar ldquoOn the fine spectrum of the differenceoperator 998779 on 1198880 and 119888rdquo Information Sciences vol 168 no 1ndash4pp 217ndash224 2004
[5] K Kayaduman and H Furkan ldquoThe fine spectra of thedifference operator 998779 over the sequence spaces 1198971 and 119887VrdquoInternational Mathematical Forum vol 1 no 21ndash24 pp 1153ndash1160 2006
[6] P D Srivastava and S Kumar ldquoFine spectrumof the generalizeddifference operator 998779V on sequence space 1198971rdquo Thai Journal ofMathematics vol 8 no 2 pp 221ndash233 2010
[7] P D Srivastava and S Kumar ldquoOn the fine spectrum of thegeneralized difference operator 998779V over the sequence space 1198880rdquoCommunications in Mathematical Analysis vol 6 no 1 pp 8ndash21 2009
[8] S Dutta and P Baliarsingh ldquoOn the fine spectra of thegeneralized rth difference operator 998779
119903
V on the sequence space1198971rdquo Applied Mathematics and Computation vol 219 no 4 pp1776ndash1784 2012
[9] S Dutta and P Baliarsingh ldquoOn the spectrum of 2-nd ordergeneralized difference operator 9987792 over the sequence space 1198880rdquo
Boletim da Sociedade Paranaense de Matematica 3rd Serie vol31 no 2 pp 235ndash244 2013
[10] S Dutta and P Baliarsingh ldquoOn a spectral classification of theoperator 998779119903V over thesequence space 1198880rdquo Proceeding of NationalAcademy of Sciences India A Physical Science In press
[11] H Bilgic and H Furkan ldquoOn the fine spectrum of the gener-alized difference operator 119861 (119903 119904) over the sequence spaces 119897119901
and 119887V119901 (1 lt 119901 lt infin)rdquo Nonlinear Analysis Theory Methods ampApplications vol 68 no 3 pp 499ndash506 2008
[12] H Bilgic and H Furkan ldquoOn the fine spectrum of the operator119861 (119903 119904 119905) over the sequence spaces 1198971 and 119887VrdquoMathematical andComputer Modelling vol 45 no 7-8 pp 883ndash891 2007
[13] F Basar Summability Theory and Its Applications BenthamScience Publishers e-books Mono-graphs Istanbul Turkey2012
[14] S Dutta and P Baliarsingh ldquoSome spectral aspects of theoperator Δ119903V over the sequence spaces 119897119901 and 119887V119901(1 lt 119901 lt infin)rdquoChinese Journal of Mathematics vol 2013 Article ID 286748 10pages 2013
[15] S Dutta and P Baliarsingh ldquoOn the spectrum of differenceoperator 998779119886119887 over the sequence spaces 119897119901 and 119887V119901(1 lt 119901 lt infin)rdquoMathematical Sciences Letters vol 3 no 2 pp 115ndash120 2014
[16] E Kreyszig Introductory Functional Analysis with ApplicationsJohn Wiley amp Sons New York NY USA 1978
[17] S Goldberg Unbounded Linear Operators Dover New YorkNY USA 1985
[18] A Wilansky Summability through Functional Analysis vol 85of North-Holland Mathematics Studies North-Holland Ams-terdam The Netherlands 1984
[19] S Dutta and P Baliarsingh ldquoOn some Toeplitz matrices andtheir inversionsrdquo Journal of the Egyptian Mathematical SocietyIn press
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Analysis 3
Taking permutations (A) (B) (C) and (1) (2) (3) we getnine different states These are labelled by 11986011198602 1198603 1198611 11986121198613 1198621 1198622 and 1198623 If 120582 is a complex number such that 119879120582 isin
1198601 or 119879120582 isin 1198611 then 120582 is in the resolvent set 120588(119879119883) of 119879on 119883 The other classifications give rise to the fine spectrumof 119879 We use 120582 isin 1198612120590(119879119883) which means the operator 119879120582 isin
1198612 that is 119877(119879120582) = 119877(119879120582) = 119883 and 119879120582 is injective but 119879minus1
120582is
discontinuous similarly are the others
Lemma 2 (see [17 page 59]) A linear operator 119879 has a denserange if and only if the adjoint 119879lowast is one to one
Lemma 3 (see [17 page 60]) The adjoint operator 119879lowast is ontoif and and only if 119879 has a bounded inverse
Let 119875 119876 be two nonempty subsets of the space 119908 of allreal or complex sequences and let 119860 = (119886119899119896) be an infinitematrix of complex numbers 119886119899119896 where 119899 119896 isin N0 For every119909 = (119909119896) isin 119875 we write
119860119899 (119909) = sum
119896
119886119899119896119909119896 (119899 isin N) (8)
The sequence 119860119909 = (119860119899(119909)) if it exists is called thetransformation of 119909 by the matrix 119860 Infinite matrix 119860 isin
(119875 119876) if and only if 119860119909 isin 119876 whenever 119909 isin 119875
Lemma 4 (see [18 page 126]) Thematrix119860 = (119886119899119896) gives riseto a bounded linear operator 119879 isin 119861(ℓ1) from ℓ1 to itself if andonly if the supremum ℓ1 norms of the columns of119860 is bounded
Lemma 5 (see [19] Theorem 2) If 119886119896(0) = 0 for all 119896 isin N0then the inverse of the difference operator 119861(119886[119898]) is given bya lower triangular Toeplitz matrix 119862 = (119888119899119896) as follows
119888119899119896 =
1
119886119899 (0) (119896 = 119899)
(minus1)119899minus119896
prod119899
119895=119896119886119895 (0)
times119863(119896)
119899minus119896(119886 [119898]) (0 le 119896 le 119899 minus 1)
0 (119896 gt 119899)
(119899 119896 isin N0)
(9)
where (119861(119886[119898])119909)119896 = 119886119896(0)119909119896 + 119886119896minus1(1)119909119896minus1 + sdot sdot sdot +
119886119896minus119898(119898)119909119896minus119898 (119896 isin N0) and
119863(119896)
119899(119886 [119898]) =
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119886119896 (1) 119886119896+1 (0) 0 sdot sdot sdot 0 0 sdot sdot sdot 0
119886119896 (2) 119886119896+1 (1) 119886119896+2 (0) sdot sdot sdot 0 0 sdot sdot sdot 0
119886119896 (119898) 119886119896+1 (119898 minus 1) 119886119896+2 (119898 minus 2) sdot sdot sdot 119886119898minus1 (1) 119886119898 (0) sdot sdot sdot 0
0 119886119896+1 (119898) 119886119896+2 (119898 minus 1) sdot sdot sdot 119886119898minus1 (2) 119886119898 (1) sdot sdot sdot 0
d
0 0 sdot sdot sdot 119886119899minus119898 (119898) sdot sdot sdot sdot sdot sdot sdot sdot sdot 119886119899minus1 (1)
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(10)
2 The Spectrum of the Operator 119866(119906 V Δ)
over the Sequence ℓ1
In this sectionwe compute the spectrum the point spectrumthe continuous spectrum and the residual spectrum of thedifference matrix 119866(119906 V Δ) on the sequence space ℓ1
Theorem 6 The operator 119866(119906 V Δ) ℓ1 997888rarr ℓ1 is a linearoperator and
119866(119906 V Δ)(ℓ1ℓ1) = sup119899
10038161003816100381610038161199061198991003816100381610038161003816 (
119899minus1
sum
119896=0
1003816100381610038161003816V119896 minus V119896+11003816100381610038161003816 +
1003816100381610038161003816V1198991003816100381610038161003816)
(11)
Proof Proof of this theorem follows from Lemma 4 and inparticular cases
(i) If (119906119896) is a strictly decreasing sequence and (V119896) is aconstant sequence of positive reals (say V119896 = V for all119896) then
119866 (119906 V Δ)(ℓ1ℓ1) = 1199060V (12)
(ii) If (119906119896) and (V119896) are strictly decreasing sequences ofpositive reals then
119866 (119906 V Δ)(ℓ1ℓ1) = 1199060V0 (13)
(iii) If (119906119896) and (V119896) are constant sequences of positivereals then
119866 (119906 V Δ)(ℓ1ℓ1) = 119906V (14)
(iv) If (119906119896) is a constant sequence and (V119896) is a strictlydecreasing sequence of positive reals then
119866 (119906 V Δ)(ℓ1ℓ1) = 119906V0 (15)
4 International Journal of Analysis
Theorem7 The spectrumof119866(119906 V Δ) over the sequence spaceℓ1 is given by
120590 (119866 (119906 V Δ) ℓ1) = 120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
ge 1 (119896 isin N0)
(16)
Proof The proof is divided into four parts as followsLet (119906119896) and (V119896) be two bounded sequences of positive
reals satisfying (1) and (2) and 120582 isin C such that |(120582 minus
119906119896V119896+1)(120582 minus 119906119896V119896)| lt 1 for all 119896 isin N0 then (119866(119906 V Δ) minus 120582119868)
is a triangle and hence has an inverse that is
(119866 (119906 V Δ) minus 120582119868)minus1
=
(((((((((((((((((
(
1
1199060V0 minus 1205820 0 0 sdot sdot sdot
minus1199061 (V0 minus V1)
prod1
119895=0(119906119895V119895 minus 120582)
1
1199061V1 minus 1205820 0 sdot sdot sdot
1199062 (V0 minus V1) (120582 minus 1199061V2)
prod2
119895=0(119906119895V119895 minus 120582)
minus1199062 (V1 minus V2)
prod2
119895=1(119906119895V119895 minus 120582)
1
1199062V2 minus 1205820 sdot sdot sdot
minus1199063 (V0 minus V1) (120582 minus 1199061V2) (120582 minus 1199062V3)
prod3
119895=0(119906119895V119895 minus 120582)
1199063 (V1 minus V2) (120582 minus 1199062V3)
prod3
119895=1(119906119895V119895 minus 120582)
minus1199063 (V2 minus V3)
prod3
119895=2(119906119895V119895 minus 120582)
1
1199063V3 minus 120582sdot sdot sdot
d
)))))))))))))))))
)
(17)
In general by using Lemma 5 we observe that
119863(119896)
119899(119906 V 120582) =
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119906119896+1 (V119896 minus V119896+1) (119906119896+1V119896+1 minus 120582) 0 sdot sdot sdot 0
119906119896+2 (V119896 minus V119896+1) 119906119896+2 (V119896+1 minus V119896+2) (119906119896+2V119896+2 minus 120582) sdot sdot sdot 0
119906119896+3 (V119896 minus V119896+1) 119906119896+3 (V119896+1 minus V119896+2) 119906119896+3 (V119896+2 minus V119896+3) sdot sdot sdot 0
d
119906119899 (V119896 minus V119896+1) 119906119899 (V119896+1 minus V119896+2) 119906119899 (V119896+2 minus V119896+3) sdot sdot sdot 119906119899 (V119899minus1 minus V119899)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= (V119896 minus V119896+1) (V119896+1 minus V119896+2) sdot sdot sdot 119906119899 (V119899minus1 minus V119899) times (119906119896+1 minus(119906119896+1V119896+1 minus 120582)
(V119896+1 minus V119896+2))
times (119906119896+2 minus(119906119896+2V119896+2 minus 120582)
(V119896+2 minus V119896+3)) times sdot sdot sdot times (119906119899minus1 minus
(119906119899minus1V119899minus1 minus 120582)
(V119899minus1 minus V119899))
(18)
Therefore for all 119899 = 119896 ((119866(119906 V Δ)minus120582119868)minus1)119899119896 = 1(119906119899V119899minus120582)
for all 119896 gt 119899 ((119866(119906 V Δ) minus 120582119868)minus1)119899119896 = 0 and for all 0 le 119896 lt 119899
one can deduce that
((119866 (119906 V Δ) minus 120582119868)minus1)119899119896
=(V119896 minus V119896+1) 119906119899
(119906119896V119896 minus 120582) (119906119899V119899 minus 120582)
119899minus1
prod
119895=119896+1
(120582 minus 119906119895V119895+1)
(119906119895V119895 minus 120582)
(19)
By using Lemma 4 we obtain that
10038171003817100381710038171003817(119866 (119906 V Δ) minus 120582119868)
minus110038171003817100381710038171003817(ℓ1ℓ1)
= sup119899
119899
sum
119896=0
100381610038161003816100381610038161003816((119866 (119906 V Δ) minus 120582119868)
minus1)119899119896
100381610038161003816100381610038161003816
le sup119899
(
10038161003816100381610038161003816100381610038161003816
1
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
+
119899minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
(V119896 minus V119896+1) 119906119899(119906119896V119896 minus 120582) (119906119899V119899 minus 120582)
times
119899minus1
prod
119895=119896+1
(120582 minus 119906119895V119895+1)
(119906119895V119895 minus 120582)
10038161003816100381610038161003816100381610038161003816100381610038161003816
)
International Journal of Analysis 5
= sup119899
10038161003816100381610038161003816100381610038161003816
1
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
(1 +
119899minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
(V119896 minus V119896+1) 119906119899(119906119896V119896 minus 120582)
100381610038161003816100381610038161003816100381610038161003816
times
119899minus1
prod
119895=119896+1
10038161003816100381610038161003816100381610038161003816100381610038161003816
(120582 minus 119906119895V119895+1)
(119906119895V119895 minus 120582)
10038161003816100381610038161003816100381610038161003816100381610038161003816
)
le sup119899
10038161003816100381610038161003816100381610038161003816
119906119899
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
(1
10038161003816100381610038161199061198991003816100381610038161003816
+
119899minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
(V119896 minus V119896+1)(119906119896V119896 minus 120582)
100381610038161003816100381610038161003816100381610038161003816
119899minus1
prod
119895=119896+1
10038161003816100381610038161199031205821003816100381610038161003816)
le sup119899
10038161003816100381610038161003816100381610038161003816
119876120582119906119899
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
(1
10038161003816100381610038161198761205821199061198991003816100381610038161003816
+1
10038161003816100381610038161199031205821003816100381610038161003816
119899minus1
sum
119896=0
10038161003816100381610038161199031205821003816100381610038161003816
119899minus119896minus1)
= sup119899
10038161003816100381610038161003816100381610038161003816
119876120582119906119899
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
[1
10038161003816100381610038161198761205821199061198991003816100381610038161003816
+1
10038161003816100381610038161199031205821003816100381610038161003816
(1 minus
10038161003816100381610038161199031205821003816100381610038161003816
119899minus1
1 minus1003816100381610038161003816119903120582
1003816100381610038161003816
)] lt infin
(20)
For simplicity we write119876120582 = sup119896((V119896minusV119896+1)(119906119896V119896minus120582)) and
119903120582 = sup119895((120582minus119906119895V119895+1)(119906119895V119895 minus120582)) Since (119906119896) and (V119896) are two
bounded sequences of positive reals the quantity119876120582 is finiteAs per the assumption120582 isin C and |(120582minus119906119896V119896+1)(120582minus119906119896V119896)| lt 1
for all 119896 isin N0 Therefore |119903120582| lt 1 Now we consider the fourpossible cases of the sequences (119906119896) and (V119896)
Case 1 If (119906119896) and (V119896) are strictly decreasing sequencesof positive reals then from the above relation we have(119866(119906 V Δ) minus 120582119868)
minus1isin (ℓ1 ℓ1) Thus
120590 (119866 (119906 V Δ) ℓ1) sube 120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
ge 1 (119896 isin N0)
(21)
Conversely consider120582 = 119906119896V119896 and |(120582minus119906119896V119896+1)(120582minus119906119896V119896)| ge 1
for all 119896 isin N0 and clearly (119866(119906 V Δ) minus 120582119868) is a triangle andhence (119866(119906 V Δ)minus120582119868)
minus1 exists but (119866(119906 V Δ)minus120582119868)minus1(ℓ1ℓ1)
is not finite Again for 120582 = 119906119896V119896 the matrix (119866(119906 V Δ) minus
119906119896V119896119868)(119896 isin N0) is not invertible Thus
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
ge 1 (119896 isin N0) sube 120590 (119866 (119906 V Δ) ℓ1)
(22)
Combining (21) and (22) we complete the proof for thiscase
Case 2 If (119906119896) is a strictly decreasing sequence and (V119896) is aconstant sequence of positive reals then
(119866 (119906 V Δ) minus 120582119868)minus1
=
(((((((
(
1
1199060V minus 1205820 0 0 sdot sdot sdot
01
1199061V minus 1205820 0 sdot sdot sdot
0 01
1199062V minus 1205820 sdot sdot sdot
0 0 01
1199063V minus 120582sdot sdot sdot
d
)))))))
)
(23)
Since 119876 = 0 (119866(119906 V Δ) minus 120582119868)minus1(ℓ1ℓ1) = sup
119899|1(119906119899V minus 120582)| lt
infin and the rest part of the proof is similar to that of Case 1
Case 3 If (119906119896) and (V119896) are constant sequences of positivereals then the proof is similar to that of Case 2 and only thedifference is (119866(119906 V Δ) minus 120582119868)
minus1(ℓ1ℓ1) = |1119906V minus 120582| lt infin
Case 4 If (119906119896) is constant sequence and (V119896) is strictlydecreasing sequence of positive reals then the proof is similarto that of Case 1
Theorem8 The point spectrum of the operator119866(119906 V Δ) overℓ1 is given by
120590119901 (119866 (119906 V Δ) ℓ1)=
119906119896V (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
0 (119900119905ℎ119890119903119908119894119904119890)
(24)
Proof Suppose 119909 isin ℓ1 and consider the system of linearequations 119866(119906 V Δ)119909 = 120582119909 for 119909 = 0 = 0 0 0 in ℓ1
1199060V01199090 = 1205821199090
1199061 (V0 minus V1) 1199090 + 1199061V11199091 = 1205821199091
1199062 (V0 minus V1) 1199090 + 1199062 (V1 minus V2) 1199091 + 1199062V21199092 = 1205821199092
119906119899 (V0 minus V1) 1199090 + 119906119899 (V1 minus V2) 1199091 + sdot sdot sdot + 119906119899V119899119909119899 = 120582119909119899
(25)
On solving the system of (25) we obtain that 120582 = 1199060V0whenever 1199090 = 0 and
1199091 =1199061 (V0 minus V1)1199060V0 minus 1199061V1
1199090
1199092 =1199062 (V0 minus V1) (1199060V0 minus 1199061V2)(1199060V0 minus 1199061V1) (1199060V0 minus 1199062V2)
1199090 and in general
119909119899 =119906119899 (V0 minus V1)1199060V0 minus 119906119899V119899
119899minus1
prod
119895=1
1199060V0 minus 119906119895V119895+11199060V0 minus 119906119895V119895
1199090
(26)
Now we have the following cases
Case 1 If (119906119896) and (V119896) are strictly decreasing sequencesof positive reals then we need to show that |(1199060V0 minus
119906119895V119895+1)(1199060V0 minus 119906119895V119895)| ge 1 Suppose for the contrary if|(1199060V0 minus 119906119895V119895+1)(1199060V0 minus 119906119895V119895)| lt 1 then V119895+1 gt V119895 whichcontradicts the fact that (V119896) is strictly decreasing sequenceTherefore 119909 notin ℓ1 and hence 120590119901(119866(119906 V Δ) ℓ1) = 0
Case 2 If (119906119896) is a strictly decreasing and (V119896) is a con-stant sequence of positive reals then for all 119896 isin N0120582 = 119906119896V is an eigen value corresponding to the eigen
6 International Journal of Analysis
vector 119909 = 0 0 0 1 0 whose 119896th entry is 1 Thus120590119901(119866(119906 V Δ) ℓ1) = 119906119896V
Case 3 If (119906119896) and (V119896) are constant sequences of positivereals then the proof is similar to that of Case 2 and120590119901(119866(119906 V Δ) ℓ1) = 119906V
Case 4 If (119906119896) is a constant and (V119896) is a strictly decreasingsequence of positive reals then the proof is similar to that ofCase 1
Theorem 9 The point spectrum of the dual operator(119866(119906 V Δ))
lowast over the sequence space ℓlowast1≊ ℓinfin is given by
120590119901 ((119866 (119906 V Δ))lowast ℓinfin) =
119906119896V (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
gt 1 (119896 isin N0) (119900119905ℎ119890119903119908119894119904119890)
(27)
Proof Consider (119866(119906 V Δ))lowast119891 = 120582119891 and 0 = 119891 isin ℓ1 and then
the system of equations
1199060V01198910 + (V0 minus V1) 11990611198911 + (V0 minus V1) 11990621198912 + sdot sdot sdot = 1205821198910
1199061V11198911 + (V1 minus V2) 11990621198912 + (V1 minus V2) 11990631198913 + sdot sdot sdot = 1205821198911
1199062V21198912 + (V2 minus V3) 11990631198913 + (V2 minus V3) 11990641198914 + sdot sdot sdot = 1205821198912
119906119896V119896119891119896 + (V119896 minus V119896+1) 119906119896+1119891119896+1 + (V119896 minus V119896+1) 119906119896+1119891119896+1
+ sdot sdot sdot = 120582119891119896
(28)
From the system of linear equations (28) we obtain that
119891119896minus1 =V119896minus1 minus V119896
120582 minus 119906119896minus1V119896minus1(119906119896119891119896 + 119906119896+1119891119896+1 + 119906119896+2119891119896+2 + sdot sdot sdot )
(29)
119891119896 =V119896 minus V119896+1120582 minus 119906119896V119896
(119906119896+1119891119896+1 + 119906119896+2119891119896+2 + 119906119896+3119891119896+3 + sdot sdot sdot )
(119896 ge 1)
(30)
Combining (29) and (30) we have
119891119896 =V119896 minus V119896+1120582 minus 119906119896V119896
((120582 minus 119906119896minus1V119896minus1)(V119896minus1 minus V119896)
119891119896minus1 minus 119906119896119891119896) (31)
Therefore by using ratio text it is observed that
lim119896997888rarrinfin
10038161003816100381610038161003816100381610038161003816
119891119896
119891119896minus1
10038161003816100381610038161003816100381610038161003816
= lim119896997888rarrinfin
100381610038161003816100381610038161003816100381610038161003816
(V119896 minus V119896+1) (120582 minus 119906119896minus1V119896minus1)(V119896minus1 minus V119896) (120582 minus 119906119896V119896+1)
100381610038161003816100381610038161003816100381610038161003816
(32)
Now we have the following cases
Case 1 If (119906119896) and (V119896) are strictly decreasing sequencesof positive reals then by using (2) we conclude thatlim119896997888rarrinfin|119891119896119891119896minus1| lt 1 if and only if |(120582 minus 119906119896minus1V119896minus1)(120582 minus
119906119896V119896+1)| lt 1 for all 119896 ge 1 Therefore 120590119901((119866(119906 V Δ))lowast ℓinfin) =
120582 isin C |(120582 minus 119906119896V119896+1)(120582 minus 119906119896V119896)| gt 1 (119896 isin N0)
Case 2 If (119906119896) is a strictly decreasing and (V119896) is a constantsequence of positive reals then from the system of (28) weobserve that for all 119896 isin N0 120582 = 119906119896V is an eigen valuecorresponding to the eigen vector 119891 = 0 0 0 1 0
whose 119896th entry is 1 Thus 120590119901((119866(119906 V Δ))lowast ℓinfin) = 119906119896V
Case 3 If (119906119896) and (V119896) are constant sequences of positivereals then the proof is similar to that of Case 2
Case 4 If (119906119896) is a constant and (V119896) is a strictly decreasingsequence of positive reals then the proof is similar to that ofCase 1
Theorem 10 The residual spectrum of the operator 119866(119906 V Δ)
over the sequence space ℓ1 is given by
120590119903 (119866 (119906 V Δ) ℓ1) =
0 (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
gt 1 (119896 isin N0) (119900119905ℎ119890119903119908119894119904119890)
(33)
Proof Proof follows from Lemma 2 and Theorems 8 and 9 Theorem 11 The continuous spectrum of the operator119866(119906 V Δ) over the sequence space ℓ1 is given by
International Journal of Analysis 7
120590119888 (119866 (119906 V Δ) ℓ1) =
0 (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
= 1 (119896 isin N0) (119900119905ℎ119890119903119908119894119904119890)
(34)
Proof The proof of this theorem follows fromTheorems 7 8and 10 and along with the fact that
120590 (119866 (119906 V Δ) ℓ1)
= 120590119901 (119866 (119906 V Δ) ℓ1) cup 120590119903 (119866 (119906 V Δ) ℓ1)
cup 120590119888 (119866 (119906 V Δ) ℓ1)
(35)
3 Conclusion
In this work the authors have determined the spectrum ofthe generalized weighted mean difference operator 119866(119906 V Δ)
over the Banach space ℓ1
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Polat V Karakaya and N Simsek ldquoDifference sequencespaces derived by using a generalized weighted meanrdquo AppliedMathematics Letters vol 24 no 5 pp 608ndash614 2011
[2] B E Rhoades ldquoThe fine spectra for weighted mean operatorsrdquoPacific Journal of Mathematics vol 104 no 1 pp 219ndash230 1983
[3] B Altay and F Basar ldquoThe fine spectrum and the matrixdomain of the difference operator 998779 on the sequence space 119897119901(0 lt 119901 lt 1)rdquo Communications in Mathematical Analysis vol 2no 2 pp 1ndash11 2007
[4] B Altay and F Basar ldquoOn the fine spectrum of the differenceoperator 998779 on 1198880 and 119888rdquo Information Sciences vol 168 no 1ndash4pp 217ndash224 2004
[5] K Kayaduman and H Furkan ldquoThe fine spectra of thedifference operator 998779 over the sequence spaces 1198971 and 119887VrdquoInternational Mathematical Forum vol 1 no 21ndash24 pp 1153ndash1160 2006
[6] P D Srivastava and S Kumar ldquoFine spectrumof the generalizeddifference operator 998779V on sequence space 1198971rdquo Thai Journal ofMathematics vol 8 no 2 pp 221ndash233 2010
[7] P D Srivastava and S Kumar ldquoOn the fine spectrum of thegeneralized difference operator 998779V over the sequence space 1198880rdquoCommunications in Mathematical Analysis vol 6 no 1 pp 8ndash21 2009
[8] S Dutta and P Baliarsingh ldquoOn the fine spectra of thegeneralized rth difference operator 998779
119903
V on the sequence space1198971rdquo Applied Mathematics and Computation vol 219 no 4 pp1776ndash1784 2012
[9] S Dutta and P Baliarsingh ldquoOn the spectrum of 2-nd ordergeneralized difference operator 9987792 over the sequence space 1198880rdquo
Boletim da Sociedade Paranaense de Matematica 3rd Serie vol31 no 2 pp 235ndash244 2013
[10] S Dutta and P Baliarsingh ldquoOn a spectral classification of theoperator 998779119903V over thesequence space 1198880rdquo Proceeding of NationalAcademy of Sciences India A Physical Science In press
[11] H Bilgic and H Furkan ldquoOn the fine spectrum of the gener-alized difference operator 119861 (119903 119904) over the sequence spaces 119897119901
and 119887V119901 (1 lt 119901 lt infin)rdquo Nonlinear Analysis Theory Methods ampApplications vol 68 no 3 pp 499ndash506 2008
[12] H Bilgic and H Furkan ldquoOn the fine spectrum of the operator119861 (119903 119904 119905) over the sequence spaces 1198971 and 119887VrdquoMathematical andComputer Modelling vol 45 no 7-8 pp 883ndash891 2007
[13] F Basar Summability Theory and Its Applications BenthamScience Publishers e-books Mono-graphs Istanbul Turkey2012
[14] S Dutta and P Baliarsingh ldquoSome spectral aspects of theoperator Δ119903V over the sequence spaces 119897119901 and 119887V119901(1 lt 119901 lt infin)rdquoChinese Journal of Mathematics vol 2013 Article ID 286748 10pages 2013
[15] S Dutta and P Baliarsingh ldquoOn the spectrum of differenceoperator 998779119886119887 over the sequence spaces 119897119901 and 119887V119901(1 lt 119901 lt infin)rdquoMathematical Sciences Letters vol 3 no 2 pp 115ndash120 2014
[16] E Kreyszig Introductory Functional Analysis with ApplicationsJohn Wiley amp Sons New York NY USA 1978
[17] S Goldberg Unbounded Linear Operators Dover New YorkNY USA 1985
[18] A Wilansky Summability through Functional Analysis vol 85of North-Holland Mathematics Studies North-Holland Ams-terdam The Netherlands 1984
[19] S Dutta and P Baliarsingh ldquoOn some Toeplitz matrices andtheir inversionsrdquo Journal of the Egyptian Mathematical SocietyIn press
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 International Journal of Analysis
Theorem7 The spectrumof119866(119906 V Δ) over the sequence spaceℓ1 is given by
120590 (119866 (119906 V Δ) ℓ1) = 120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
ge 1 (119896 isin N0)
(16)
Proof The proof is divided into four parts as followsLet (119906119896) and (V119896) be two bounded sequences of positive
reals satisfying (1) and (2) and 120582 isin C such that |(120582 minus
119906119896V119896+1)(120582 minus 119906119896V119896)| lt 1 for all 119896 isin N0 then (119866(119906 V Δ) minus 120582119868)
is a triangle and hence has an inverse that is
(119866 (119906 V Δ) minus 120582119868)minus1
=
(((((((((((((((((
(
1
1199060V0 minus 1205820 0 0 sdot sdot sdot
minus1199061 (V0 minus V1)
prod1
119895=0(119906119895V119895 minus 120582)
1
1199061V1 minus 1205820 0 sdot sdot sdot
1199062 (V0 minus V1) (120582 minus 1199061V2)
prod2
119895=0(119906119895V119895 minus 120582)
minus1199062 (V1 minus V2)
prod2
119895=1(119906119895V119895 minus 120582)
1
1199062V2 minus 1205820 sdot sdot sdot
minus1199063 (V0 minus V1) (120582 minus 1199061V2) (120582 minus 1199062V3)
prod3
119895=0(119906119895V119895 minus 120582)
1199063 (V1 minus V2) (120582 minus 1199062V3)
prod3
119895=1(119906119895V119895 minus 120582)
minus1199063 (V2 minus V3)
prod3
119895=2(119906119895V119895 minus 120582)
1
1199063V3 minus 120582sdot sdot sdot
d
)))))))))))))))))
)
(17)
In general by using Lemma 5 we observe that
119863(119896)
119899(119906 V 120582) =
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119906119896+1 (V119896 minus V119896+1) (119906119896+1V119896+1 minus 120582) 0 sdot sdot sdot 0
119906119896+2 (V119896 minus V119896+1) 119906119896+2 (V119896+1 minus V119896+2) (119906119896+2V119896+2 minus 120582) sdot sdot sdot 0
119906119896+3 (V119896 minus V119896+1) 119906119896+3 (V119896+1 minus V119896+2) 119906119896+3 (V119896+2 minus V119896+3) sdot sdot sdot 0
d
119906119899 (V119896 minus V119896+1) 119906119899 (V119896+1 minus V119896+2) 119906119899 (V119896+2 minus V119896+3) sdot sdot sdot 119906119899 (V119899minus1 minus V119899)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= (V119896 minus V119896+1) (V119896+1 minus V119896+2) sdot sdot sdot 119906119899 (V119899minus1 minus V119899) times (119906119896+1 minus(119906119896+1V119896+1 minus 120582)
(V119896+1 minus V119896+2))
times (119906119896+2 minus(119906119896+2V119896+2 minus 120582)
(V119896+2 minus V119896+3)) times sdot sdot sdot times (119906119899minus1 minus
(119906119899minus1V119899minus1 minus 120582)
(V119899minus1 minus V119899))
(18)
Therefore for all 119899 = 119896 ((119866(119906 V Δ)minus120582119868)minus1)119899119896 = 1(119906119899V119899minus120582)
for all 119896 gt 119899 ((119866(119906 V Δ) minus 120582119868)minus1)119899119896 = 0 and for all 0 le 119896 lt 119899
one can deduce that
((119866 (119906 V Δ) minus 120582119868)minus1)119899119896
=(V119896 minus V119896+1) 119906119899
(119906119896V119896 minus 120582) (119906119899V119899 minus 120582)
119899minus1
prod
119895=119896+1
(120582 minus 119906119895V119895+1)
(119906119895V119895 minus 120582)
(19)
By using Lemma 4 we obtain that
10038171003817100381710038171003817(119866 (119906 V Δ) minus 120582119868)
minus110038171003817100381710038171003817(ℓ1ℓ1)
= sup119899
119899
sum
119896=0
100381610038161003816100381610038161003816((119866 (119906 V Δ) minus 120582119868)
minus1)119899119896
100381610038161003816100381610038161003816
le sup119899
(
10038161003816100381610038161003816100381610038161003816
1
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
+
119899minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
(V119896 minus V119896+1) 119906119899(119906119896V119896 minus 120582) (119906119899V119899 minus 120582)
times
119899minus1
prod
119895=119896+1
(120582 minus 119906119895V119895+1)
(119906119895V119895 minus 120582)
10038161003816100381610038161003816100381610038161003816100381610038161003816
)
International Journal of Analysis 5
= sup119899
10038161003816100381610038161003816100381610038161003816
1
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
(1 +
119899minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
(V119896 minus V119896+1) 119906119899(119906119896V119896 minus 120582)
100381610038161003816100381610038161003816100381610038161003816
times
119899minus1
prod
119895=119896+1
10038161003816100381610038161003816100381610038161003816100381610038161003816
(120582 minus 119906119895V119895+1)
(119906119895V119895 minus 120582)
10038161003816100381610038161003816100381610038161003816100381610038161003816
)
le sup119899
10038161003816100381610038161003816100381610038161003816
119906119899
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
(1
10038161003816100381610038161199061198991003816100381610038161003816
+
119899minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
(V119896 minus V119896+1)(119906119896V119896 minus 120582)
100381610038161003816100381610038161003816100381610038161003816
119899minus1
prod
119895=119896+1
10038161003816100381610038161199031205821003816100381610038161003816)
le sup119899
10038161003816100381610038161003816100381610038161003816
119876120582119906119899
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
(1
10038161003816100381610038161198761205821199061198991003816100381610038161003816
+1
10038161003816100381610038161199031205821003816100381610038161003816
119899minus1
sum
119896=0
10038161003816100381610038161199031205821003816100381610038161003816
119899minus119896minus1)
= sup119899
10038161003816100381610038161003816100381610038161003816
119876120582119906119899
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
[1
10038161003816100381610038161198761205821199061198991003816100381610038161003816
+1
10038161003816100381610038161199031205821003816100381610038161003816
(1 minus
10038161003816100381610038161199031205821003816100381610038161003816
119899minus1
1 minus1003816100381610038161003816119903120582
1003816100381610038161003816
)] lt infin
(20)
For simplicity we write119876120582 = sup119896((V119896minusV119896+1)(119906119896V119896minus120582)) and
119903120582 = sup119895((120582minus119906119895V119895+1)(119906119895V119895 minus120582)) Since (119906119896) and (V119896) are two
bounded sequences of positive reals the quantity119876120582 is finiteAs per the assumption120582 isin C and |(120582minus119906119896V119896+1)(120582minus119906119896V119896)| lt 1
for all 119896 isin N0 Therefore |119903120582| lt 1 Now we consider the fourpossible cases of the sequences (119906119896) and (V119896)
Case 1 If (119906119896) and (V119896) are strictly decreasing sequencesof positive reals then from the above relation we have(119866(119906 V Δ) minus 120582119868)
minus1isin (ℓ1 ℓ1) Thus
120590 (119866 (119906 V Δ) ℓ1) sube 120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
ge 1 (119896 isin N0)
(21)
Conversely consider120582 = 119906119896V119896 and |(120582minus119906119896V119896+1)(120582minus119906119896V119896)| ge 1
for all 119896 isin N0 and clearly (119866(119906 V Δ) minus 120582119868) is a triangle andhence (119866(119906 V Δ)minus120582119868)
minus1 exists but (119866(119906 V Δ)minus120582119868)minus1(ℓ1ℓ1)
is not finite Again for 120582 = 119906119896V119896 the matrix (119866(119906 V Δ) minus
119906119896V119896119868)(119896 isin N0) is not invertible Thus
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
ge 1 (119896 isin N0) sube 120590 (119866 (119906 V Δ) ℓ1)
(22)
Combining (21) and (22) we complete the proof for thiscase
Case 2 If (119906119896) is a strictly decreasing sequence and (V119896) is aconstant sequence of positive reals then
(119866 (119906 V Δ) minus 120582119868)minus1
=
(((((((
(
1
1199060V minus 1205820 0 0 sdot sdot sdot
01
1199061V minus 1205820 0 sdot sdot sdot
0 01
1199062V minus 1205820 sdot sdot sdot
0 0 01
1199063V minus 120582sdot sdot sdot
d
)))))))
)
(23)
Since 119876 = 0 (119866(119906 V Δ) minus 120582119868)minus1(ℓ1ℓ1) = sup
119899|1(119906119899V minus 120582)| lt
infin and the rest part of the proof is similar to that of Case 1
Case 3 If (119906119896) and (V119896) are constant sequences of positivereals then the proof is similar to that of Case 2 and only thedifference is (119866(119906 V Δ) minus 120582119868)
minus1(ℓ1ℓ1) = |1119906V minus 120582| lt infin
Case 4 If (119906119896) is constant sequence and (V119896) is strictlydecreasing sequence of positive reals then the proof is similarto that of Case 1
Theorem8 The point spectrum of the operator119866(119906 V Δ) overℓ1 is given by
120590119901 (119866 (119906 V Δ) ℓ1)=
119906119896V (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
0 (119900119905ℎ119890119903119908119894119904119890)
(24)
Proof Suppose 119909 isin ℓ1 and consider the system of linearequations 119866(119906 V Δ)119909 = 120582119909 for 119909 = 0 = 0 0 0 in ℓ1
1199060V01199090 = 1205821199090
1199061 (V0 minus V1) 1199090 + 1199061V11199091 = 1205821199091
1199062 (V0 minus V1) 1199090 + 1199062 (V1 minus V2) 1199091 + 1199062V21199092 = 1205821199092
119906119899 (V0 minus V1) 1199090 + 119906119899 (V1 minus V2) 1199091 + sdot sdot sdot + 119906119899V119899119909119899 = 120582119909119899
(25)
On solving the system of (25) we obtain that 120582 = 1199060V0whenever 1199090 = 0 and
1199091 =1199061 (V0 minus V1)1199060V0 minus 1199061V1
1199090
1199092 =1199062 (V0 minus V1) (1199060V0 minus 1199061V2)(1199060V0 minus 1199061V1) (1199060V0 minus 1199062V2)
1199090 and in general
119909119899 =119906119899 (V0 minus V1)1199060V0 minus 119906119899V119899
119899minus1
prod
119895=1
1199060V0 minus 119906119895V119895+11199060V0 minus 119906119895V119895
1199090
(26)
Now we have the following cases
Case 1 If (119906119896) and (V119896) are strictly decreasing sequencesof positive reals then we need to show that |(1199060V0 minus
119906119895V119895+1)(1199060V0 minus 119906119895V119895)| ge 1 Suppose for the contrary if|(1199060V0 minus 119906119895V119895+1)(1199060V0 minus 119906119895V119895)| lt 1 then V119895+1 gt V119895 whichcontradicts the fact that (V119896) is strictly decreasing sequenceTherefore 119909 notin ℓ1 and hence 120590119901(119866(119906 V Δ) ℓ1) = 0
Case 2 If (119906119896) is a strictly decreasing and (V119896) is a con-stant sequence of positive reals then for all 119896 isin N0120582 = 119906119896V is an eigen value corresponding to the eigen
6 International Journal of Analysis
vector 119909 = 0 0 0 1 0 whose 119896th entry is 1 Thus120590119901(119866(119906 V Δ) ℓ1) = 119906119896V
Case 3 If (119906119896) and (V119896) are constant sequences of positivereals then the proof is similar to that of Case 2 and120590119901(119866(119906 V Δ) ℓ1) = 119906V
Case 4 If (119906119896) is a constant and (V119896) is a strictly decreasingsequence of positive reals then the proof is similar to that ofCase 1
Theorem 9 The point spectrum of the dual operator(119866(119906 V Δ))
lowast over the sequence space ℓlowast1≊ ℓinfin is given by
120590119901 ((119866 (119906 V Δ))lowast ℓinfin) =
119906119896V (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
gt 1 (119896 isin N0) (119900119905ℎ119890119903119908119894119904119890)
(27)
Proof Consider (119866(119906 V Δ))lowast119891 = 120582119891 and 0 = 119891 isin ℓ1 and then
the system of equations
1199060V01198910 + (V0 minus V1) 11990611198911 + (V0 minus V1) 11990621198912 + sdot sdot sdot = 1205821198910
1199061V11198911 + (V1 minus V2) 11990621198912 + (V1 minus V2) 11990631198913 + sdot sdot sdot = 1205821198911
1199062V21198912 + (V2 minus V3) 11990631198913 + (V2 minus V3) 11990641198914 + sdot sdot sdot = 1205821198912
119906119896V119896119891119896 + (V119896 minus V119896+1) 119906119896+1119891119896+1 + (V119896 minus V119896+1) 119906119896+1119891119896+1
+ sdot sdot sdot = 120582119891119896
(28)
From the system of linear equations (28) we obtain that
119891119896minus1 =V119896minus1 minus V119896
120582 minus 119906119896minus1V119896minus1(119906119896119891119896 + 119906119896+1119891119896+1 + 119906119896+2119891119896+2 + sdot sdot sdot )
(29)
119891119896 =V119896 minus V119896+1120582 minus 119906119896V119896
(119906119896+1119891119896+1 + 119906119896+2119891119896+2 + 119906119896+3119891119896+3 + sdot sdot sdot )
(119896 ge 1)
(30)
Combining (29) and (30) we have
119891119896 =V119896 minus V119896+1120582 minus 119906119896V119896
((120582 minus 119906119896minus1V119896minus1)(V119896minus1 minus V119896)
119891119896minus1 minus 119906119896119891119896) (31)
Therefore by using ratio text it is observed that
lim119896997888rarrinfin
10038161003816100381610038161003816100381610038161003816
119891119896
119891119896minus1
10038161003816100381610038161003816100381610038161003816
= lim119896997888rarrinfin
100381610038161003816100381610038161003816100381610038161003816
(V119896 minus V119896+1) (120582 minus 119906119896minus1V119896minus1)(V119896minus1 minus V119896) (120582 minus 119906119896V119896+1)
100381610038161003816100381610038161003816100381610038161003816
(32)
Now we have the following cases
Case 1 If (119906119896) and (V119896) are strictly decreasing sequencesof positive reals then by using (2) we conclude thatlim119896997888rarrinfin|119891119896119891119896minus1| lt 1 if and only if |(120582 minus 119906119896minus1V119896minus1)(120582 minus
119906119896V119896+1)| lt 1 for all 119896 ge 1 Therefore 120590119901((119866(119906 V Δ))lowast ℓinfin) =
120582 isin C |(120582 minus 119906119896V119896+1)(120582 minus 119906119896V119896)| gt 1 (119896 isin N0)
Case 2 If (119906119896) is a strictly decreasing and (V119896) is a constantsequence of positive reals then from the system of (28) weobserve that for all 119896 isin N0 120582 = 119906119896V is an eigen valuecorresponding to the eigen vector 119891 = 0 0 0 1 0
whose 119896th entry is 1 Thus 120590119901((119866(119906 V Δ))lowast ℓinfin) = 119906119896V
Case 3 If (119906119896) and (V119896) are constant sequences of positivereals then the proof is similar to that of Case 2
Case 4 If (119906119896) is a constant and (V119896) is a strictly decreasingsequence of positive reals then the proof is similar to that ofCase 1
Theorem 10 The residual spectrum of the operator 119866(119906 V Δ)
over the sequence space ℓ1 is given by
120590119903 (119866 (119906 V Δ) ℓ1) =
0 (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
gt 1 (119896 isin N0) (119900119905ℎ119890119903119908119894119904119890)
(33)
Proof Proof follows from Lemma 2 and Theorems 8 and 9 Theorem 11 The continuous spectrum of the operator119866(119906 V Δ) over the sequence space ℓ1 is given by
International Journal of Analysis 7
120590119888 (119866 (119906 V Δ) ℓ1) =
0 (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
= 1 (119896 isin N0) (119900119905ℎ119890119903119908119894119904119890)
(34)
Proof The proof of this theorem follows fromTheorems 7 8and 10 and along with the fact that
120590 (119866 (119906 V Δ) ℓ1)
= 120590119901 (119866 (119906 V Δ) ℓ1) cup 120590119903 (119866 (119906 V Δ) ℓ1)
cup 120590119888 (119866 (119906 V Δ) ℓ1)
(35)
3 Conclusion
In this work the authors have determined the spectrum ofthe generalized weighted mean difference operator 119866(119906 V Δ)
over the Banach space ℓ1
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Polat V Karakaya and N Simsek ldquoDifference sequencespaces derived by using a generalized weighted meanrdquo AppliedMathematics Letters vol 24 no 5 pp 608ndash614 2011
[2] B E Rhoades ldquoThe fine spectra for weighted mean operatorsrdquoPacific Journal of Mathematics vol 104 no 1 pp 219ndash230 1983
[3] B Altay and F Basar ldquoThe fine spectrum and the matrixdomain of the difference operator 998779 on the sequence space 119897119901(0 lt 119901 lt 1)rdquo Communications in Mathematical Analysis vol 2no 2 pp 1ndash11 2007
[4] B Altay and F Basar ldquoOn the fine spectrum of the differenceoperator 998779 on 1198880 and 119888rdquo Information Sciences vol 168 no 1ndash4pp 217ndash224 2004
[5] K Kayaduman and H Furkan ldquoThe fine spectra of thedifference operator 998779 over the sequence spaces 1198971 and 119887VrdquoInternational Mathematical Forum vol 1 no 21ndash24 pp 1153ndash1160 2006
[6] P D Srivastava and S Kumar ldquoFine spectrumof the generalizeddifference operator 998779V on sequence space 1198971rdquo Thai Journal ofMathematics vol 8 no 2 pp 221ndash233 2010
[7] P D Srivastava and S Kumar ldquoOn the fine spectrum of thegeneralized difference operator 998779V over the sequence space 1198880rdquoCommunications in Mathematical Analysis vol 6 no 1 pp 8ndash21 2009
[8] S Dutta and P Baliarsingh ldquoOn the fine spectra of thegeneralized rth difference operator 998779
119903
V on the sequence space1198971rdquo Applied Mathematics and Computation vol 219 no 4 pp1776ndash1784 2012
[9] S Dutta and P Baliarsingh ldquoOn the spectrum of 2-nd ordergeneralized difference operator 9987792 over the sequence space 1198880rdquo
Boletim da Sociedade Paranaense de Matematica 3rd Serie vol31 no 2 pp 235ndash244 2013
[10] S Dutta and P Baliarsingh ldquoOn a spectral classification of theoperator 998779119903V over thesequence space 1198880rdquo Proceeding of NationalAcademy of Sciences India A Physical Science In press
[11] H Bilgic and H Furkan ldquoOn the fine spectrum of the gener-alized difference operator 119861 (119903 119904) over the sequence spaces 119897119901
and 119887V119901 (1 lt 119901 lt infin)rdquo Nonlinear Analysis Theory Methods ampApplications vol 68 no 3 pp 499ndash506 2008
[12] H Bilgic and H Furkan ldquoOn the fine spectrum of the operator119861 (119903 119904 119905) over the sequence spaces 1198971 and 119887VrdquoMathematical andComputer Modelling vol 45 no 7-8 pp 883ndash891 2007
[13] F Basar Summability Theory and Its Applications BenthamScience Publishers e-books Mono-graphs Istanbul Turkey2012
[14] S Dutta and P Baliarsingh ldquoSome spectral aspects of theoperator Δ119903V over the sequence spaces 119897119901 and 119887V119901(1 lt 119901 lt infin)rdquoChinese Journal of Mathematics vol 2013 Article ID 286748 10pages 2013
[15] S Dutta and P Baliarsingh ldquoOn the spectrum of differenceoperator 998779119886119887 over the sequence spaces 119897119901 and 119887V119901(1 lt 119901 lt infin)rdquoMathematical Sciences Letters vol 3 no 2 pp 115ndash120 2014
[16] E Kreyszig Introductory Functional Analysis with ApplicationsJohn Wiley amp Sons New York NY USA 1978
[17] S Goldberg Unbounded Linear Operators Dover New YorkNY USA 1985
[18] A Wilansky Summability through Functional Analysis vol 85of North-Holland Mathematics Studies North-Holland Ams-terdam The Netherlands 1984
[19] S Dutta and P Baliarsingh ldquoOn some Toeplitz matrices andtheir inversionsrdquo Journal of the Egyptian Mathematical SocietyIn press
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Analysis 5
= sup119899
10038161003816100381610038161003816100381610038161003816
1
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
(1 +
119899minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
(V119896 minus V119896+1) 119906119899(119906119896V119896 minus 120582)
100381610038161003816100381610038161003816100381610038161003816
times
119899minus1
prod
119895=119896+1
10038161003816100381610038161003816100381610038161003816100381610038161003816
(120582 minus 119906119895V119895+1)
(119906119895V119895 minus 120582)
10038161003816100381610038161003816100381610038161003816100381610038161003816
)
le sup119899
10038161003816100381610038161003816100381610038161003816
119906119899
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
(1
10038161003816100381610038161199061198991003816100381610038161003816
+
119899minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
(V119896 minus V119896+1)(119906119896V119896 minus 120582)
100381610038161003816100381610038161003816100381610038161003816
119899minus1
prod
119895=119896+1
10038161003816100381610038161199031205821003816100381610038161003816)
le sup119899
10038161003816100381610038161003816100381610038161003816
119876120582119906119899
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
(1
10038161003816100381610038161198761205821199061198991003816100381610038161003816
+1
10038161003816100381610038161199031205821003816100381610038161003816
119899minus1
sum
119896=0
10038161003816100381610038161199031205821003816100381610038161003816
119899minus119896minus1)
= sup119899
10038161003816100381610038161003816100381610038161003816
119876120582119906119899
119906119899V119899 minus 120582
10038161003816100381610038161003816100381610038161003816
[1
10038161003816100381610038161198761205821199061198991003816100381610038161003816
+1
10038161003816100381610038161199031205821003816100381610038161003816
(1 minus
10038161003816100381610038161199031205821003816100381610038161003816
119899minus1
1 minus1003816100381610038161003816119903120582
1003816100381610038161003816
)] lt infin
(20)
For simplicity we write119876120582 = sup119896((V119896minusV119896+1)(119906119896V119896minus120582)) and
119903120582 = sup119895((120582minus119906119895V119895+1)(119906119895V119895 minus120582)) Since (119906119896) and (V119896) are two
bounded sequences of positive reals the quantity119876120582 is finiteAs per the assumption120582 isin C and |(120582minus119906119896V119896+1)(120582minus119906119896V119896)| lt 1
for all 119896 isin N0 Therefore |119903120582| lt 1 Now we consider the fourpossible cases of the sequences (119906119896) and (V119896)
Case 1 If (119906119896) and (V119896) are strictly decreasing sequencesof positive reals then from the above relation we have(119866(119906 V Δ) minus 120582119868)
minus1isin (ℓ1 ℓ1) Thus
120590 (119866 (119906 V Δ) ℓ1) sube 120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
ge 1 (119896 isin N0)
(21)
Conversely consider120582 = 119906119896V119896 and |(120582minus119906119896V119896+1)(120582minus119906119896V119896)| ge 1
for all 119896 isin N0 and clearly (119866(119906 V Δ) minus 120582119868) is a triangle andhence (119866(119906 V Δ)minus120582119868)
minus1 exists but (119866(119906 V Δ)minus120582119868)minus1(ℓ1ℓ1)
is not finite Again for 120582 = 119906119896V119896 the matrix (119866(119906 V Δ) minus
119906119896V119896119868)(119896 isin N0) is not invertible Thus
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
ge 1 (119896 isin N0) sube 120590 (119866 (119906 V Δ) ℓ1)
(22)
Combining (21) and (22) we complete the proof for thiscase
Case 2 If (119906119896) is a strictly decreasing sequence and (V119896) is aconstant sequence of positive reals then
(119866 (119906 V Δ) minus 120582119868)minus1
=
(((((((
(
1
1199060V minus 1205820 0 0 sdot sdot sdot
01
1199061V minus 1205820 0 sdot sdot sdot
0 01
1199062V minus 1205820 sdot sdot sdot
0 0 01
1199063V minus 120582sdot sdot sdot
d
)))))))
)
(23)
Since 119876 = 0 (119866(119906 V Δ) minus 120582119868)minus1(ℓ1ℓ1) = sup
119899|1(119906119899V minus 120582)| lt
infin and the rest part of the proof is similar to that of Case 1
Case 3 If (119906119896) and (V119896) are constant sequences of positivereals then the proof is similar to that of Case 2 and only thedifference is (119866(119906 V Δ) minus 120582119868)
minus1(ℓ1ℓ1) = |1119906V minus 120582| lt infin
Case 4 If (119906119896) is constant sequence and (V119896) is strictlydecreasing sequence of positive reals then the proof is similarto that of Case 1
Theorem8 The point spectrum of the operator119866(119906 V Δ) overℓ1 is given by
120590119901 (119866 (119906 V Δ) ℓ1)=
119906119896V (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
0 (119900119905ℎ119890119903119908119894119904119890)
(24)
Proof Suppose 119909 isin ℓ1 and consider the system of linearequations 119866(119906 V Δ)119909 = 120582119909 for 119909 = 0 = 0 0 0 in ℓ1
1199060V01199090 = 1205821199090
1199061 (V0 minus V1) 1199090 + 1199061V11199091 = 1205821199091
1199062 (V0 minus V1) 1199090 + 1199062 (V1 minus V2) 1199091 + 1199062V21199092 = 1205821199092
119906119899 (V0 minus V1) 1199090 + 119906119899 (V1 minus V2) 1199091 + sdot sdot sdot + 119906119899V119899119909119899 = 120582119909119899
(25)
On solving the system of (25) we obtain that 120582 = 1199060V0whenever 1199090 = 0 and
1199091 =1199061 (V0 minus V1)1199060V0 minus 1199061V1
1199090
1199092 =1199062 (V0 minus V1) (1199060V0 minus 1199061V2)(1199060V0 minus 1199061V1) (1199060V0 minus 1199062V2)
1199090 and in general
119909119899 =119906119899 (V0 minus V1)1199060V0 minus 119906119899V119899
119899minus1
prod
119895=1
1199060V0 minus 119906119895V119895+11199060V0 minus 119906119895V119895
1199090
(26)
Now we have the following cases
Case 1 If (119906119896) and (V119896) are strictly decreasing sequencesof positive reals then we need to show that |(1199060V0 minus
119906119895V119895+1)(1199060V0 minus 119906119895V119895)| ge 1 Suppose for the contrary if|(1199060V0 minus 119906119895V119895+1)(1199060V0 minus 119906119895V119895)| lt 1 then V119895+1 gt V119895 whichcontradicts the fact that (V119896) is strictly decreasing sequenceTherefore 119909 notin ℓ1 and hence 120590119901(119866(119906 V Δ) ℓ1) = 0
Case 2 If (119906119896) is a strictly decreasing and (V119896) is a con-stant sequence of positive reals then for all 119896 isin N0120582 = 119906119896V is an eigen value corresponding to the eigen
6 International Journal of Analysis
vector 119909 = 0 0 0 1 0 whose 119896th entry is 1 Thus120590119901(119866(119906 V Δ) ℓ1) = 119906119896V
Case 3 If (119906119896) and (V119896) are constant sequences of positivereals then the proof is similar to that of Case 2 and120590119901(119866(119906 V Δ) ℓ1) = 119906V
Case 4 If (119906119896) is a constant and (V119896) is a strictly decreasingsequence of positive reals then the proof is similar to that ofCase 1
Theorem 9 The point spectrum of the dual operator(119866(119906 V Δ))
lowast over the sequence space ℓlowast1≊ ℓinfin is given by
120590119901 ((119866 (119906 V Δ))lowast ℓinfin) =
119906119896V (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
gt 1 (119896 isin N0) (119900119905ℎ119890119903119908119894119904119890)
(27)
Proof Consider (119866(119906 V Δ))lowast119891 = 120582119891 and 0 = 119891 isin ℓ1 and then
the system of equations
1199060V01198910 + (V0 minus V1) 11990611198911 + (V0 minus V1) 11990621198912 + sdot sdot sdot = 1205821198910
1199061V11198911 + (V1 minus V2) 11990621198912 + (V1 minus V2) 11990631198913 + sdot sdot sdot = 1205821198911
1199062V21198912 + (V2 minus V3) 11990631198913 + (V2 minus V3) 11990641198914 + sdot sdot sdot = 1205821198912
119906119896V119896119891119896 + (V119896 minus V119896+1) 119906119896+1119891119896+1 + (V119896 minus V119896+1) 119906119896+1119891119896+1
+ sdot sdot sdot = 120582119891119896
(28)
From the system of linear equations (28) we obtain that
119891119896minus1 =V119896minus1 minus V119896
120582 minus 119906119896minus1V119896minus1(119906119896119891119896 + 119906119896+1119891119896+1 + 119906119896+2119891119896+2 + sdot sdot sdot )
(29)
119891119896 =V119896 minus V119896+1120582 minus 119906119896V119896
(119906119896+1119891119896+1 + 119906119896+2119891119896+2 + 119906119896+3119891119896+3 + sdot sdot sdot )
(119896 ge 1)
(30)
Combining (29) and (30) we have
119891119896 =V119896 minus V119896+1120582 minus 119906119896V119896
((120582 minus 119906119896minus1V119896minus1)(V119896minus1 minus V119896)
119891119896minus1 minus 119906119896119891119896) (31)
Therefore by using ratio text it is observed that
lim119896997888rarrinfin
10038161003816100381610038161003816100381610038161003816
119891119896
119891119896minus1
10038161003816100381610038161003816100381610038161003816
= lim119896997888rarrinfin
100381610038161003816100381610038161003816100381610038161003816
(V119896 minus V119896+1) (120582 minus 119906119896minus1V119896minus1)(V119896minus1 minus V119896) (120582 minus 119906119896V119896+1)
100381610038161003816100381610038161003816100381610038161003816
(32)
Now we have the following cases
Case 1 If (119906119896) and (V119896) are strictly decreasing sequencesof positive reals then by using (2) we conclude thatlim119896997888rarrinfin|119891119896119891119896minus1| lt 1 if and only if |(120582 minus 119906119896minus1V119896minus1)(120582 minus
119906119896V119896+1)| lt 1 for all 119896 ge 1 Therefore 120590119901((119866(119906 V Δ))lowast ℓinfin) =
120582 isin C |(120582 minus 119906119896V119896+1)(120582 minus 119906119896V119896)| gt 1 (119896 isin N0)
Case 2 If (119906119896) is a strictly decreasing and (V119896) is a constantsequence of positive reals then from the system of (28) weobserve that for all 119896 isin N0 120582 = 119906119896V is an eigen valuecorresponding to the eigen vector 119891 = 0 0 0 1 0
whose 119896th entry is 1 Thus 120590119901((119866(119906 V Δ))lowast ℓinfin) = 119906119896V
Case 3 If (119906119896) and (V119896) are constant sequences of positivereals then the proof is similar to that of Case 2
Case 4 If (119906119896) is a constant and (V119896) is a strictly decreasingsequence of positive reals then the proof is similar to that ofCase 1
Theorem 10 The residual spectrum of the operator 119866(119906 V Δ)
over the sequence space ℓ1 is given by
120590119903 (119866 (119906 V Δ) ℓ1) =
0 (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
gt 1 (119896 isin N0) (119900119905ℎ119890119903119908119894119904119890)
(33)
Proof Proof follows from Lemma 2 and Theorems 8 and 9 Theorem 11 The continuous spectrum of the operator119866(119906 V Δ) over the sequence space ℓ1 is given by
International Journal of Analysis 7
120590119888 (119866 (119906 V Δ) ℓ1) =
0 (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
= 1 (119896 isin N0) (119900119905ℎ119890119903119908119894119904119890)
(34)
Proof The proof of this theorem follows fromTheorems 7 8and 10 and along with the fact that
120590 (119866 (119906 V Δ) ℓ1)
= 120590119901 (119866 (119906 V Δ) ℓ1) cup 120590119903 (119866 (119906 V Δ) ℓ1)
cup 120590119888 (119866 (119906 V Δ) ℓ1)
(35)
3 Conclusion
In this work the authors have determined the spectrum ofthe generalized weighted mean difference operator 119866(119906 V Δ)
over the Banach space ℓ1
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Polat V Karakaya and N Simsek ldquoDifference sequencespaces derived by using a generalized weighted meanrdquo AppliedMathematics Letters vol 24 no 5 pp 608ndash614 2011
[2] B E Rhoades ldquoThe fine spectra for weighted mean operatorsrdquoPacific Journal of Mathematics vol 104 no 1 pp 219ndash230 1983
[3] B Altay and F Basar ldquoThe fine spectrum and the matrixdomain of the difference operator 998779 on the sequence space 119897119901(0 lt 119901 lt 1)rdquo Communications in Mathematical Analysis vol 2no 2 pp 1ndash11 2007
[4] B Altay and F Basar ldquoOn the fine spectrum of the differenceoperator 998779 on 1198880 and 119888rdquo Information Sciences vol 168 no 1ndash4pp 217ndash224 2004
[5] K Kayaduman and H Furkan ldquoThe fine spectra of thedifference operator 998779 over the sequence spaces 1198971 and 119887VrdquoInternational Mathematical Forum vol 1 no 21ndash24 pp 1153ndash1160 2006
[6] P D Srivastava and S Kumar ldquoFine spectrumof the generalizeddifference operator 998779V on sequence space 1198971rdquo Thai Journal ofMathematics vol 8 no 2 pp 221ndash233 2010
[7] P D Srivastava and S Kumar ldquoOn the fine spectrum of thegeneralized difference operator 998779V over the sequence space 1198880rdquoCommunications in Mathematical Analysis vol 6 no 1 pp 8ndash21 2009
[8] S Dutta and P Baliarsingh ldquoOn the fine spectra of thegeneralized rth difference operator 998779
119903
V on the sequence space1198971rdquo Applied Mathematics and Computation vol 219 no 4 pp1776ndash1784 2012
[9] S Dutta and P Baliarsingh ldquoOn the spectrum of 2-nd ordergeneralized difference operator 9987792 over the sequence space 1198880rdquo
Boletim da Sociedade Paranaense de Matematica 3rd Serie vol31 no 2 pp 235ndash244 2013
[10] S Dutta and P Baliarsingh ldquoOn a spectral classification of theoperator 998779119903V over thesequence space 1198880rdquo Proceeding of NationalAcademy of Sciences India A Physical Science In press
[11] H Bilgic and H Furkan ldquoOn the fine spectrum of the gener-alized difference operator 119861 (119903 119904) over the sequence spaces 119897119901
and 119887V119901 (1 lt 119901 lt infin)rdquo Nonlinear Analysis Theory Methods ampApplications vol 68 no 3 pp 499ndash506 2008
[12] H Bilgic and H Furkan ldquoOn the fine spectrum of the operator119861 (119903 119904 119905) over the sequence spaces 1198971 and 119887VrdquoMathematical andComputer Modelling vol 45 no 7-8 pp 883ndash891 2007
[13] F Basar Summability Theory and Its Applications BenthamScience Publishers e-books Mono-graphs Istanbul Turkey2012
[14] S Dutta and P Baliarsingh ldquoSome spectral aspects of theoperator Δ119903V over the sequence spaces 119897119901 and 119887V119901(1 lt 119901 lt infin)rdquoChinese Journal of Mathematics vol 2013 Article ID 286748 10pages 2013
[15] S Dutta and P Baliarsingh ldquoOn the spectrum of differenceoperator 998779119886119887 over the sequence spaces 119897119901 and 119887V119901(1 lt 119901 lt infin)rdquoMathematical Sciences Letters vol 3 no 2 pp 115ndash120 2014
[16] E Kreyszig Introductory Functional Analysis with ApplicationsJohn Wiley amp Sons New York NY USA 1978
[17] S Goldberg Unbounded Linear Operators Dover New YorkNY USA 1985
[18] A Wilansky Summability through Functional Analysis vol 85of North-Holland Mathematics Studies North-Holland Ams-terdam The Netherlands 1984
[19] S Dutta and P Baliarsingh ldquoOn some Toeplitz matrices andtheir inversionsrdquo Journal of the Egyptian Mathematical SocietyIn press
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 International Journal of Analysis
vector 119909 = 0 0 0 1 0 whose 119896th entry is 1 Thus120590119901(119866(119906 V Δ) ℓ1) = 119906119896V
Case 3 If (119906119896) and (V119896) are constant sequences of positivereals then the proof is similar to that of Case 2 and120590119901(119866(119906 V Δ) ℓ1) = 119906V
Case 4 If (119906119896) is a constant and (V119896) is a strictly decreasingsequence of positive reals then the proof is similar to that ofCase 1
Theorem 9 The point spectrum of the dual operator(119866(119906 V Δ))
lowast over the sequence space ℓlowast1≊ ℓinfin is given by
120590119901 ((119866 (119906 V Δ))lowast ℓinfin) =
119906119896V (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
gt 1 (119896 isin N0) (119900119905ℎ119890119903119908119894119904119890)
(27)
Proof Consider (119866(119906 V Δ))lowast119891 = 120582119891 and 0 = 119891 isin ℓ1 and then
the system of equations
1199060V01198910 + (V0 minus V1) 11990611198911 + (V0 minus V1) 11990621198912 + sdot sdot sdot = 1205821198910
1199061V11198911 + (V1 minus V2) 11990621198912 + (V1 minus V2) 11990631198913 + sdot sdot sdot = 1205821198911
1199062V21198912 + (V2 minus V3) 11990631198913 + (V2 minus V3) 11990641198914 + sdot sdot sdot = 1205821198912
119906119896V119896119891119896 + (V119896 minus V119896+1) 119906119896+1119891119896+1 + (V119896 minus V119896+1) 119906119896+1119891119896+1
+ sdot sdot sdot = 120582119891119896
(28)
From the system of linear equations (28) we obtain that
119891119896minus1 =V119896minus1 minus V119896
120582 minus 119906119896minus1V119896minus1(119906119896119891119896 + 119906119896+1119891119896+1 + 119906119896+2119891119896+2 + sdot sdot sdot )
(29)
119891119896 =V119896 minus V119896+1120582 minus 119906119896V119896
(119906119896+1119891119896+1 + 119906119896+2119891119896+2 + 119906119896+3119891119896+3 + sdot sdot sdot )
(119896 ge 1)
(30)
Combining (29) and (30) we have
119891119896 =V119896 minus V119896+1120582 minus 119906119896V119896
((120582 minus 119906119896minus1V119896minus1)(V119896minus1 minus V119896)
119891119896minus1 minus 119906119896119891119896) (31)
Therefore by using ratio text it is observed that
lim119896997888rarrinfin
10038161003816100381610038161003816100381610038161003816
119891119896
119891119896minus1
10038161003816100381610038161003816100381610038161003816
= lim119896997888rarrinfin
100381610038161003816100381610038161003816100381610038161003816
(V119896 minus V119896+1) (120582 minus 119906119896minus1V119896minus1)(V119896minus1 minus V119896) (120582 minus 119906119896V119896+1)
100381610038161003816100381610038161003816100381610038161003816
(32)
Now we have the following cases
Case 1 If (119906119896) and (V119896) are strictly decreasing sequencesof positive reals then by using (2) we conclude thatlim119896997888rarrinfin|119891119896119891119896minus1| lt 1 if and only if |(120582 minus 119906119896minus1V119896minus1)(120582 minus
119906119896V119896+1)| lt 1 for all 119896 ge 1 Therefore 120590119901((119866(119906 V Δ))lowast ℓinfin) =
120582 isin C |(120582 minus 119906119896V119896+1)(120582 minus 119906119896V119896)| gt 1 (119896 isin N0)
Case 2 If (119906119896) is a strictly decreasing and (V119896) is a constantsequence of positive reals then from the system of (28) weobserve that for all 119896 isin N0 120582 = 119906119896V is an eigen valuecorresponding to the eigen vector 119891 = 0 0 0 1 0
whose 119896th entry is 1 Thus 120590119901((119866(119906 V Δ))lowast ℓinfin) = 119906119896V
Case 3 If (119906119896) and (V119896) are constant sequences of positivereals then the proof is similar to that of Case 2
Case 4 If (119906119896) is a constant and (V119896) is a strictly decreasingsequence of positive reals then the proof is similar to that ofCase 1
Theorem 10 The residual spectrum of the operator 119866(119906 V Δ)
over the sequence space ℓ1 is given by
120590119903 (119866 (119906 V Δ) ℓ1) =
0 (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
gt 1 (119896 isin N0) (119900119905ℎ119890119903119908119894119904119890)
(33)
Proof Proof follows from Lemma 2 and Theorems 8 and 9 Theorem 11 The continuous spectrum of the operator119866(119906 V Δ) over the sequence space ℓ1 is given by
International Journal of Analysis 7
120590119888 (119866 (119906 V Δ) ℓ1) =
0 (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
= 1 (119896 isin N0) (119900119905ℎ119890119903119908119894119904119890)
(34)
Proof The proof of this theorem follows fromTheorems 7 8and 10 and along with the fact that
120590 (119866 (119906 V Δ) ℓ1)
= 120590119901 (119866 (119906 V Δ) ℓ1) cup 120590119903 (119866 (119906 V Δ) ℓ1)
cup 120590119888 (119866 (119906 V Δ) ℓ1)
(35)
3 Conclusion
In this work the authors have determined the spectrum ofthe generalized weighted mean difference operator 119866(119906 V Δ)
over the Banach space ℓ1
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Polat V Karakaya and N Simsek ldquoDifference sequencespaces derived by using a generalized weighted meanrdquo AppliedMathematics Letters vol 24 no 5 pp 608ndash614 2011
[2] B E Rhoades ldquoThe fine spectra for weighted mean operatorsrdquoPacific Journal of Mathematics vol 104 no 1 pp 219ndash230 1983
[3] B Altay and F Basar ldquoThe fine spectrum and the matrixdomain of the difference operator 998779 on the sequence space 119897119901(0 lt 119901 lt 1)rdquo Communications in Mathematical Analysis vol 2no 2 pp 1ndash11 2007
[4] B Altay and F Basar ldquoOn the fine spectrum of the differenceoperator 998779 on 1198880 and 119888rdquo Information Sciences vol 168 no 1ndash4pp 217ndash224 2004
[5] K Kayaduman and H Furkan ldquoThe fine spectra of thedifference operator 998779 over the sequence spaces 1198971 and 119887VrdquoInternational Mathematical Forum vol 1 no 21ndash24 pp 1153ndash1160 2006
[6] P D Srivastava and S Kumar ldquoFine spectrumof the generalizeddifference operator 998779V on sequence space 1198971rdquo Thai Journal ofMathematics vol 8 no 2 pp 221ndash233 2010
[7] P D Srivastava and S Kumar ldquoOn the fine spectrum of thegeneralized difference operator 998779V over the sequence space 1198880rdquoCommunications in Mathematical Analysis vol 6 no 1 pp 8ndash21 2009
[8] S Dutta and P Baliarsingh ldquoOn the fine spectra of thegeneralized rth difference operator 998779
119903
V on the sequence space1198971rdquo Applied Mathematics and Computation vol 219 no 4 pp1776ndash1784 2012
[9] S Dutta and P Baliarsingh ldquoOn the spectrum of 2-nd ordergeneralized difference operator 9987792 over the sequence space 1198880rdquo
Boletim da Sociedade Paranaense de Matematica 3rd Serie vol31 no 2 pp 235ndash244 2013
[10] S Dutta and P Baliarsingh ldquoOn a spectral classification of theoperator 998779119903V over thesequence space 1198880rdquo Proceeding of NationalAcademy of Sciences India A Physical Science In press
[11] H Bilgic and H Furkan ldquoOn the fine spectrum of the gener-alized difference operator 119861 (119903 119904) over the sequence spaces 119897119901
and 119887V119901 (1 lt 119901 lt infin)rdquo Nonlinear Analysis Theory Methods ampApplications vol 68 no 3 pp 499ndash506 2008
[12] H Bilgic and H Furkan ldquoOn the fine spectrum of the operator119861 (119903 119904 119905) over the sequence spaces 1198971 and 119887VrdquoMathematical andComputer Modelling vol 45 no 7-8 pp 883ndash891 2007
[13] F Basar Summability Theory and Its Applications BenthamScience Publishers e-books Mono-graphs Istanbul Turkey2012
[14] S Dutta and P Baliarsingh ldquoSome spectral aspects of theoperator Δ119903V over the sequence spaces 119897119901 and 119887V119901(1 lt 119901 lt infin)rdquoChinese Journal of Mathematics vol 2013 Article ID 286748 10pages 2013
[15] S Dutta and P Baliarsingh ldquoOn the spectrum of differenceoperator 998779119886119887 over the sequence spaces 119897119901 and 119887V119901(1 lt 119901 lt infin)rdquoMathematical Sciences Letters vol 3 no 2 pp 115ndash120 2014
[16] E Kreyszig Introductory Functional Analysis with ApplicationsJohn Wiley amp Sons New York NY USA 1978
[17] S Goldberg Unbounded Linear Operators Dover New YorkNY USA 1985
[18] A Wilansky Summability through Functional Analysis vol 85of North-Holland Mathematics Studies North-Holland Ams-terdam The Netherlands 1984
[19] S Dutta and P Baliarsingh ldquoOn some Toeplitz matrices andtheir inversionsrdquo Journal of the Egyptian Mathematical SocietyIn press
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Analysis 7
120590119888 (119866 (119906 V Δ) ℓ1) =
0 (119894119891 (V119896) 119894119904 119886 119888119900119899119904119905119886119899119905 119904119890119902119906119890119899119888119890)
120582 isin C
10038161003816100381610038161003816100381610038161003816
120582 minus 119906119896V119896+1120582 minus 119906119896V119896
10038161003816100381610038161003816100381610038161003816
= 1 (119896 isin N0) (119900119905ℎ119890119903119908119894119904119890)
(34)
Proof The proof of this theorem follows fromTheorems 7 8and 10 and along with the fact that
120590 (119866 (119906 V Δ) ℓ1)
= 120590119901 (119866 (119906 V Δ) ℓ1) cup 120590119903 (119866 (119906 V Δ) ℓ1)
cup 120590119888 (119866 (119906 V Δ) ℓ1)
(35)
3 Conclusion
In this work the authors have determined the spectrum ofthe generalized weighted mean difference operator 119866(119906 V Δ)
over the Banach space ℓ1
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Polat V Karakaya and N Simsek ldquoDifference sequencespaces derived by using a generalized weighted meanrdquo AppliedMathematics Letters vol 24 no 5 pp 608ndash614 2011
[2] B E Rhoades ldquoThe fine spectra for weighted mean operatorsrdquoPacific Journal of Mathematics vol 104 no 1 pp 219ndash230 1983
[3] B Altay and F Basar ldquoThe fine spectrum and the matrixdomain of the difference operator 998779 on the sequence space 119897119901(0 lt 119901 lt 1)rdquo Communications in Mathematical Analysis vol 2no 2 pp 1ndash11 2007
[4] B Altay and F Basar ldquoOn the fine spectrum of the differenceoperator 998779 on 1198880 and 119888rdquo Information Sciences vol 168 no 1ndash4pp 217ndash224 2004
[5] K Kayaduman and H Furkan ldquoThe fine spectra of thedifference operator 998779 over the sequence spaces 1198971 and 119887VrdquoInternational Mathematical Forum vol 1 no 21ndash24 pp 1153ndash1160 2006
[6] P D Srivastava and S Kumar ldquoFine spectrumof the generalizeddifference operator 998779V on sequence space 1198971rdquo Thai Journal ofMathematics vol 8 no 2 pp 221ndash233 2010
[7] P D Srivastava and S Kumar ldquoOn the fine spectrum of thegeneralized difference operator 998779V over the sequence space 1198880rdquoCommunications in Mathematical Analysis vol 6 no 1 pp 8ndash21 2009
[8] S Dutta and P Baliarsingh ldquoOn the fine spectra of thegeneralized rth difference operator 998779
119903
V on the sequence space1198971rdquo Applied Mathematics and Computation vol 219 no 4 pp1776ndash1784 2012
[9] S Dutta and P Baliarsingh ldquoOn the spectrum of 2-nd ordergeneralized difference operator 9987792 over the sequence space 1198880rdquo
Boletim da Sociedade Paranaense de Matematica 3rd Serie vol31 no 2 pp 235ndash244 2013
[10] S Dutta and P Baliarsingh ldquoOn a spectral classification of theoperator 998779119903V over thesequence space 1198880rdquo Proceeding of NationalAcademy of Sciences India A Physical Science In press
[11] H Bilgic and H Furkan ldquoOn the fine spectrum of the gener-alized difference operator 119861 (119903 119904) over the sequence spaces 119897119901
and 119887V119901 (1 lt 119901 lt infin)rdquo Nonlinear Analysis Theory Methods ampApplications vol 68 no 3 pp 499ndash506 2008
[12] H Bilgic and H Furkan ldquoOn the fine spectrum of the operator119861 (119903 119904 119905) over the sequence spaces 1198971 and 119887VrdquoMathematical andComputer Modelling vol 45 no 7-8 pp 883ndash891 2007
[13] F Basar Summability Theory and Its Applications BenthamScience Publishers e-books Mono-graphs Istanbul Turkey2012
[14] S Dutta and P Baliarsingh ldquoSome spectral aspects of theoperator Δ119903V over the sequence spaces 119897119901 and 119887V119901(1 lt 119901 lt infin)rdquoChinese Journal of Mathematics vol 2013 Article ID 286748 10pages 2013
[15] S Dutta and P Baliarsingh ldquoOn the spectrum of differenceoperator 998779119886119887 over the sequence spaces 119897119901 and 119887V119901(1 lt 119901 lt infin)rdquoMathematical Sciences Letters vol 3 no 2 pp 115ndash120 2014
[16] E Kreyszig Introductory Functional Analysis with ApplicationsJohn Wiley amp Sons New York NY USA 1978
[17] S Goldberg Unbounded Linear Operators Dover New YorkNY USA 1985
[18] A Wilansky Summability through Functional Analysis vol 85of North-Holland Mathematics Studies North-Holland Ams-terdam The Netherlands 1984
[19] S Dutta and P Baliarsingh ldquoOn some Toeplitz matrices andtheir inversionsrdquo Journal of the Egyptian Mathematical SocietyIn press
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MathematicsJournal of
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Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
