On p α,β -Normal Operators - m-hikari.com€¦ · ⊆ p−(α,β) normal ⊆ M - hyponormal ⊆...

Applied Mathematical Sciences, Vol. 8, 2014, no. 41, 2041 - 2052HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2014.42140

On p-(α, β)-Normal Operators

D. Senthilkumar

Post Graduate and Research Department of MathematicsGovernment Arts College (Autonomous)Coimbatore - 641 018,Tamil Nadu, India

R. Santhi1

Department of MathematicsSri Ramakrishna Engineering College

Vattamalaipalayam, Coimbatore - 641 022Tamil Nadu, India

Copyright c© 2014 D. Senthilkumar and R. Santhi. This is an open access article dis-tributed under the Creative Commons Attribution License, which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An operator T ∈ B(H) is said to be p−(α, β) normal operators for 0 < p ≤ 1if α2(T ∗T )p ≤ (TT ∗)p ≤ β2(T ∗T )p, 0 ≤ α ≤ 1 ≤ β. In this paper, we provethat if T is p−(α, β) - normal operator then T n is p

n−(α, β) - normal operator

for all positive integer n. Moreover, we prove that if T = U |T | is p − (α, β)

- normal operator for 0 < p < 1, the Aluthge transform T̃ = |T | 12 U |T | 12 is(p + 1

2) - (α, β) - normal operator. In this paper, some of the properties and

structure of p − (α, β) normal operators are discussed.

Mathematics Subject Classification: Primary 47B33; Secondary 47B37

Keywords: p- (α, β) - normal operators, (α, β) - normal operators, AluthgeTransformation

1. Introduction and Preliminaries

Let H be a separable complex Hilbert space and B(H) denote the algebraof bounded linear operators on an infinite dimensional separable Hilbert space

1Corresponding author

2042 D. Senthilkumar and R. Santhi

H . An operator T is said to be normal if TT ∗ = T ∗T and hyponormal ifTT ∗ ≤ T ∗T . An operator T is said to be dominant if ran (T − λI) ⊆ ran(T − λI)∗ for all λ ∈ C or equivalently there exists a real number Mλ foreach λ ∈ C such that ‖(T − λI)∗x‖ ≤ Mλ ‖(T − λI)x‖ for each x ∈ H . Ifthere exists a constant M such that Mλ ⊆ M for all λ ∈ C, then T is calledM - hyponormal and if M = 1, T is hyponormal. The class of Hyponormaloperators has been studied by many authors. In recent years this class hasbeen generalized, in some sense,to the larger sets of so called p - hyponormal,log hyponormal, Posinormal, etc [7],[8], [5], [1] and [2]. An operator T ∈ B(H)is said to be-p-hyponormal for 0 < p < 1 iff (TT ∗)p ≤ (T ∗T )p,-p-posinormal if (TT ∗)p ≤ c2(T ∗T )p,-(α, β)-nomal operators if α2T ∗T ≤ TT ∗ ≤ β2T ∗T, 0 ≤ α ≤ 1 ≤ β [6].

The example of an M - hyponormal operator given by Wadhwa [12], theweighted shift operator defined by Te1 = e2, T e2 = 2e3 and Tei = ei+1 fori ≥ 0, is not an p − (α, β) normal, which is neither normal nor hyponormal.So it is clear that the class of p − (α, β) normal lies between hyponormal andM - hyponormal operators. Now the inclusion relation becomes

Normal ⊆ Hyponormal ⊆ (α, β) normal⊆ p − (α, β) normal ⊆ M - hyponormal ⊆ Dominant

S.S Dragomir and M.S.Moslehian [3] and [6] has given various inequalitiesbetween the operator norm and numerical radius of (α, β) - normal operators. Weyl type theorems and composition operators of (α, β) have been studiedby D.SenthilKumar and Sherin Joy.S.M [10, 11]. As a generalisation of (α, β)- normal operators, we introduce p − (α, β) - normal operators. When p = 1,this coincide with (α, β)-normal operators.

2. on p − (α, β) normal operators

Proposition 2.1. If T ∈ B(H) ,then the following statements are equivalent:(i) T ∈ p − (α, β) normal operators.(ii)Range |T |p= Range |T |∗p

, kerT p=kerT ∗p

(iii)There exist S1, S2 ∈ B(H) such that |T ∗|p = |T |pS1, |T |p = |T ∗|pS2

(iv)There exist positive operators P1, P2 such that|T ∗|2p = |T |pP1|T |p, |T |2p = |T |∗p

P2|T |∗p.

Theorem 2.2. Let T be an p − (α, β) normal operator then(i)λT is p − (α, β) normal operator(ii)the translate T − λ need not be p − (α, β) normal operator.

On p-(α, β)-normal operators 2043

Proof. (i)Let T be an p − (α, β) normal operator, then

α2(T ∗T )p ≤ (TT ∗)p ≤ β2(T ∗T )p

α2((λT )∗λT )p ≤ (λT (λT )∗)p ≤ β2((λT )∗λT )p

α2(λ∗)pλp(T ∗T )p ≤ (λ∗)pλp(TT ∗)p ≤ β2(λ∗)pλp(T ∗T )p

α2(T ∗T )p ≤ (TT ∗)p ≤ β2(T ∗T )p.

Therefore λT is p − (α, β) normal operator.(ii) The operator T = U∗ − 2, where U is the unilateral shift since 2 /∈ σ(U∗)[5] , T is p − (α, β) normal operator but U∗ is not p − (α, β) normal, U isp − (α, β) normal.

Theorem 2.3. Let T = U |T | ∈ B(H) be the polar decomposition of T. ThenT ∈ p− (α, β) normal operator iff there exist a positive number α, β such thatα‖|T |px‖‖|T |py‖ ≤ |(U |T |2px, y)| ≤ β‖|T |px‖‖|T |py‖.Proof. By assumption T is p − (α, β) normal operator

|(U |T |2px, y)| = |(|T |2px, U∗y)|= ‖|T |px‖‖|T |pU∗y‖= ‖|T |px‖‖|T ∗|py‖≤ ‖|T |px‖β‖|T |py‖.

Similarily

|(U |T |2px, y)| = |(|T |2px, U∗y)|= ‖|T |px‖‖|T ∗|py‖≥ ‖|T |px‖α‖|T |py‖.

Thereforeα‖|T |px‖‖|T |py‖ ≤ |(U |T |2px, y)| ≤ β‖|T |px‖‖|T |py‖.

Theorem 2.4 (Mc Carthy [2]). Let A ≥ 0. Then(i) (Ax, x)r ≤ ‖x‖2(r−1)(Arx, x) if r ≥ 1(ii) (Ax, x)r ≥ ‖x‖2(r−1)(Arx, x) if 0 ≤ r ≤ 1.

Theorem 2.5. If T is p− (α, β) normal operator then T is M - paranormal.

Proof. Let T = U |T | be the polar decomposition of T. Since T is p − (α, β)normal operator,

α2|T |2p ≤ U |T |2pU∗ ≤ β2|T |2p

⇒ U |T |2pU∗ ≤ β2|T |2p

⇒ |T |2p ≤ β2U∗|T |2pU.

Hence by Theorem 2.4, for all x ∈ H with ‖x‖ = 1

β2‖T 2x‖2 = β2(T ∗TTx, Tx)


= β2((|T |2p)(1/p) Tx

‖Tx‖ ,Tx

‖Tx‖)

,

0 ≤ p ≤ 1

≥ β2 (|T |2pTx, Tx)1/p‖Tx‖2

‖Tx‖2/p

≥ β2(U∗|T |2p|T |x, |T |x)1/p ‖Tx‖2

‖Tx‖2/p

≥ (|T |2p+2x, x)1/p ‖Tx‖2

‖Tx‖2/p

for all

β ≥ 1

≥ ((|T |2)p+1x, x)1/p ‖Tx‖2

‖Tx‖2/p

≥ (|T |2x, x)(1/p)+1 ‖Tx‖2

‖Tx‖2/p

≥ ‖Tx‖4

‖Tx‖4 ≤ β2‖T 2x‖2

Therefore , ‖Tx‖2 ≤ β‖T 2x‖ T is M -paranormal.

Corollary 2.6. If T is p − (α, β) normal operator and V is isometry, thenV TV ∗ is also p − (α, β) normal operator.

Proof. Let T ∈ B(H) , there exists some positive operators K1 and K2 suchthat

|T ∗|2p = |T |pK1|T |p|T |2p = |T ∗|pK2|T ∗|p

Let

|T ∗0 |2p = V |T ∗|2pV ∗

= V |T |pK1|T |pV ∗

= |T0|pV K1V∗|T0|p( since V is isometry ).

Let

|T0|2p = V |T |2pV ∗

= V |T ∗|pK2|T ∗|pV ∗

= |T ∗0 |pV K2V

∗|T ∗0 |p( since V is isometry )

V TV ∗ is p − (α, β) normal operator .


Theorem 2.7. Let T be p − (α, β) normal operator , then there exist M > 0such that

‖T kx‖2 ≤ Mp‖T k−1x‖‖T k+1x‖ for all unit vectors x ∈ H.

Proof. T is p − (α, β) normal operator

‖T k+1x‖2 = (T k+1x, T k+1x)

= (T ∗TT kx, T kx)

= ((|T |2p)1/pT kx, T kx)

≥ (|T |2pT kx, T kx)1/p‖T kx‖2(1−1/p)

≥ β−2/p(|T ∗|2kT kx, T kx)1/p‖T kx‖2(1−1/p)

≥ β−2/p(|T |2(p+1)T k−1x, T k−1x)1/p‖T kx‖2(1−1/p)

≥ β−2/p(|T |2T k−1x, T k−1x)p+1/p‖T kx‖2(1−1/p)‖T k−1x‖2(1−p−1/p)

≥ β−2/p‖T kx‖4‖T k−1x‖−2

‖T kx‖2 ≤ β1/p‖T k−1x‖‖T k+1x‖

Theorem 2.8. Let T be p − (α, β) normal operator and S is a self adjointoperator on H. If TS is a contraction,then α ≤ ‖TS‖p ≤ β.

Proof. By p − (α, β) normal operator, we have

α‖TSx‖p ≤ ‖T ∗S∗x‖p ≤ β‖TSx‖p

α‖x‖p ≤ ‖(ST )∗x‖p ≤ β‖x‖p

α ≤ ‖ST‖p ≤ β

α ≤ ‖TS‖p ≤ β.

Lemma 2.9. If T is p− (α, β) normal operator such that αβ = 1, then T ∗ isalso p − (α, β) normal operator.

Proof. From the definition of p − (α, β) normal operator

α2(T ∗T )p ≤ (TT ∗)p ≤ β2(T ∗T )p

α4(T ∗T )p ≤ α2(TT ∗)p ≤ α2β2(T ∗T )p (2.1)

and

α2β2(T ∗T )p ≤ β2(TT ∗)p ≤ β4(T ∗T )p (2.2)


From above two equations,α2(T ∗T )p ≤ α2β2(T ∗T )p ≤ β2(TT ∗)p.

Therefore,α2(TT ∗)p ≤ (T ∗T )p ≤ β2(TT ∗)p

T ∗ is also p − (α, β) normal operator if αβ = 1.

Theorem 2.10. Tf T is p − (α, β) normal operator and S is an unitary op-erator such that TS =ST then C= TS is also p − (α, β) normal operator .

Proof.

α2(C∗C)p ≤ CC∗p ≤ β2(C∗C)p

α2((TS)∗TS)p ≤ TS(TS)∗p ≤ β2((TS)∗TS)p

α2(S∗T ∗TS)p ≤ TSS∗T ∗p ≤ β2(S∗T ∗TS)p

α2(T ∗S∗ST )p ≤ (TT ∗)p ≤ β2(T ∗S∗ST )p

α2(T ∗T )p ≤ (TT ∗)p ≤ β2(T ∗T )p( since S is unitary ).

Hence, TS is p − (α, β) normal operator.

3. Aluthge Transformation on powers of p − (α, β) - normaloperators

An operator T can be decomposed into T = U |T | where U is partial isometryand |T | is square root of T ∗T with N(U) = N(|T |) and this kernal conditionN(U) = N(|T |) uniquely determines U and |T | in the polar decomposition ofT . In this section T = U |T | denotes the polar decomposition satisfying thekernal condition N(U) = N(|T |).

In this section, we consider new properties as an extension of p - hyponormaloperators using the generalized Aluthge transform. In this section, we provethat if T is p−(α, β) - normal operator then T n is p

n−(α, β) - normal operator

for all positive integer n. Moreover, we prove that if T = U |T | is p − (α, β)

- normal operator for 0 < p < 1, the Aluthge transform T̃ = |T | 12 U |T | 12 is(p + 1

2) - (α, β) - normal operator.

For an operator T = U |T | defines T̃ as follows :

T̃s,t = |T |sU |T |tfor s, t > 0 which is called the generalized Aluthge transform of T . In thissection, we will study p − (α, β) - normal operators using their generalizedAluthge transform.

Theorem 3.1. Furuta Inequality [4] Let A ≥ B ≥ 0. Then for all r > 0,

(1) (Br2 ApB

r2 )

1q ≥ (B

r2 BpB

r2 )

1q

(2) (Ar2 ApA

r2 )

1q ≥ (A

r2 BpA

r2 )

1q

for p ≥ 0, q ≥ 1 with (1 + r)q ≥ p + r.


Theorem 3.2. Let T = U |T | be the polar decomposition of p−(α, β) - normaloperator for 0 < p ≤ 1 then

1) T̃s,t = |T |sU |T |t is p+min(s,t)s+t

−(α, β) - normal for s, t > 0 such that max(s, t) ≥p.2) T̃s,t is (α, β) - normal for 0 < s, t ≤ p.

Proof. Let T be p − (α, β) - normal operator thenα2(T ∗T )p ≤ (TT ∗)p ≤ β2(T ∗T )p

α2|T |2p ≤ |T ∗|2p ≤ β2|T |2p

(1) Assume A = α2|T |2p , B = |T ∗|2p and C = β2|T |2p. Then,

(T̃ ∗s,tT̃s,t)

p+min(s,t)s+t = (|T |tU∗|T |2sU |T |t) p+min(s,t)

s+t

= U∗(|T ∗|t|T |2s|T ∗|t) p+min(s,t)s+t U

= U∗(β−2s/pBt/2pCs/pBt/2p)p+min(s,t)

s+t U

≥ β−2s/pp+min(s,t)

s+t U∗(Bt/2pBs/pBt/2p)p+min(s,t)

s+t U

≥ β−2s/p(p+min(s,t)

s+t)U∗B

p+min(s,t)p U

≥ β−2s/p( p+min(s,t)s+t

)|T |2(p+min(s,t))

(T̃ ∗s,tT̃s,t)

p+min(s,t)s+t = (|T |tU∗|T |2sU |T |t) p+min(s,t)

s+t

= U∗(|T ∗|t|T |2s|T ∗|t) p+min(s,t)s+t U

= U∗(α−2s/pBt/2pAs/pBt/2p)p+min(s,t)

s+t U

≤ α−2s/p p+min(s,t)s+t U∗(Bt/2pBs/pBt/2p)

p+min(s,t)s+t U

≤ α−2s/p(p+min(s,t)

s+t)U∗B

p+min(s,t)p U

≤ α−2s/p(p+min(s,t)

s+t)|T |2(p+min(s,t))

So, we have,

|T |2(p+min(s,t)) ≥α2s/p(T̃ ∗s,tT̃s,t)

p+min(s,t)s+t (3.1)

and

|T |2(p+min(s,t)) ≤β2s/p(T̃ ∗s,tT̃s,t)

p+min(s,t)s+t (3.2)


(T̃s,tT̃∗s,t)

p+min(s,t)s+t = (|T |sU |T |2tU∗|T |s) p+min(s,t)

s+t

= ((C

β2)s/2p|T ∗|2t(

C

β2)s/2p)

p+min(s,t)s+t

= ((C

β2)s/2pB

tp (

C

β2)s/2p)

p+min(s,t)s+t

≤ β−2s/pp+min(s,t)

s+t Cp+min(s,t)

p

≤ β−2s/p p+min(s,t)s+t |T |2(p+min(s,t))(β2)

p+min(s,t)p

≤ (β2)p+min(s,t)

p (T̃ ∗s,tT̃s,t)

p+min(s,t)s+t

(T̃s,tT̃∗s,t)

p+min(s,t)s+t = (|T |sU |T |2tU∗|T |s) p+min(s,t)

s+t

= ((A

α2)s/2p|T ∗|2t(

A

α2)s/2p)

p+min(s,t)s+t

== ((A

α2)s/2pB

tp (

A

α2)s/2p)

p+min(s,t)s+t

≥ α−2s/p p+min(s,t)s+t A

p+min(s,t)p

≥ α−2s/p p+min(s,t)s+t |T |2(p+min(s,t))(α2)

p+min(s,t)p

≥ (α2)p+min(s,t)


p+min(s,t)s+t

So, we have

(T̃s,tT̃∗s,t)

p+min(s,t)s+t ≤ (β2)

p+min(s,t)p (T̃ ∗

s,tT̃s,t)p+min(s,t)

s+t (3.3)

and

(T̃s,tT̃∗s,t)

p+min(s,t)s+t ≥ (α2)

p+min(s,t)p (T̃ ∗

s,tT̃s,t)p+min(s,t)

s+t (3.4)

From (3.1), (3.2), (3.3) and (3.4), we have

(α2)p+min(s,t)


p+min(s,t)s+t ≤ (T̃s,tT̃

∗s,t)

p+min(s,t)s+t

≤ (β2)p+min(s,t)


p+min(s,t)s+t .Since

s+tp+min(s,t)

≥ 1, (1 + t/p)( s+tp+min(s,t)

) ≥ (s/p) + (t/p), (1 + s/p)( s+tp+min(s,t)

) ≥(s/p) + (t/p).

Therefore T̃s,t is p+min(s,t)s+t

p − (α, β) - normal.

(2) Applying Lowner - Heinz Inequality, we have

α2s/p|T |2s ≤ |T ∗|2s ≤ β2s/p|T |2s


If 0 < s, t ≤ p, we have

T̃ ∗s,tT̃s,t = |T |tU∗|T |2sU |T |t

≥ |T |tU∗|T ∗|2sU |T |tβ

2sp

≥ β−2s

p |T |tU∗|T ∗|2sU |T |t

≥ β−2s

p |T |2(s+t)

T̃ ∗s,tT̃s,t ≥ β

−2sp |T |2(s+t). (3.5)

Similarily, we have

T̃ ∗s,tT̃s,t ≤ α

−2sp |T |2(s+t). (3.6)

T̃s,tT̃∗s,t = |T |sU |T |2tU∗|T |s

= |T |s|T ∗|2t|T |s

Since |T ∗| ≥ |T |α 1p , |T ∗| ≤ β

1p |T |.

T̃s,tT̃∗s,t ≤ β

2tp |T |2(s+t). (3.7)

T̃s,tT̃∗s,t ≥ α

2tp |T |2(s+t). (3.8)

From (3.5), (3.6), (3.7) and (3.8), we have

α2(s+t)

p T̃ ∗s,tT̃s,t ≤ T̃s,tT̃

∗s,t ≤ β

2(s+t)p T̃ ∗

s,tT̃s,t. Therefore T̃s,t is (α, β)- normal.

Corollary 3.3. Let T =U |T | be p − (α, β) - normal operator for 0 < p < 1,then(1) T̃ = |T | 12 U |T | 12 is (p + 1

2) - (α, β) - normal for 0 < p < 1/2.

(2) T̃ is (α, β) - normal operator for 1/2 ≤ p < 1.

Theorem 3.4. Let T =U |T | be p − (α, β) - normal operator for 0 < p < 1,then

(1) β−(p+1)

p(n−1)(T ∗T )p+1 ≤ (T n∗

T n)p+1

n ≤ α−(p+1)

p(n−1)(T ∗T )p+1

(2) β−(p+1)

p(n−1)(T nT n∗

)p+1

n ≤ (TT ∗)p+1 ≤ α−(p+1)

p(n−1)(T nT n∗

)p+1

n holds for allpositive integer n.

Proof. (1)Let An = (T n∗T n)

pn = |T n| 2p

n and Bn = (T nT n∗)pn = |T n∗| 2p

n for allpositive integer n.

By induction β−(p+1)

p(n−1)(T ∗T )p+1 ≤ (T n∗

T n)p+1

n ≤ α−(p+1)

p(n−1)(T ∗T )p+1

holds for n = k. Since Ak = (T ∗kT k)p/k ≥ β−(k−1)(T ∗T )p ≥ β−(k+1)B1.


Ak = (T ∗kT k)p/k ≤ α−(k−1)(T ∗T )p ≤ α−(k+1)B1.

It follows that

(T (k+1)∗T k+1)p+1k+1 = (U∗|T ∗|T k∗

T k|T ∗|U)p+1k+1

= U∗(|T ∗|T k∗T k|T ∗|) p+1

k+1 U

= U∗(B12p

1 Akp

k B12p

1 )p+1k+1 U

≥ β−k(p+1)p(k+1)

(k+1)U∗(B12p

1 Bkp

1 B12p

1 )p+1k+1 U by Furuta Inequality

≥ β−k(p+1)

p (T ∗T )p+1.

Similarily

(T (k+1)∗T k+1)p+1k+1 ≤ α

−k(p+1)p (T ∗T )p+1.

Hence β−(p+1)

p(n−1)(T ∗T )p+1 ≤ (T n∗

T n)p+1

n ≤ α−(p+1)

p(n−1)(T ∗T )p+1.

(2) Assume that β−(p+1)

p(n−1)(T nT n∗

)p+1

n ≤ (TT ∗)p+1 ≤ α−(p+1)

p(n−1)(T nT n∗

)p+1

n

holds for n = k. ThenA1 = (T ∗T )p ≥ β−2(TT ∗)p ≥ β−(k+1)(T kTK∗

)p/k ≥ β−(k+1)Bk.Similarily A1 ≤ α−(k+1)Bk. Hence we have

(T (k+1)T (k+1)∗)p+1k+1 = (U |T |T kT k∗|T |U∗)

p+1k+1

= U(|T ∗|T kT k∗|T |) p+1k+1 U∗

= U(A12p

1 Bkp

k A12p

1 )p+1k+1 U∗

≤ βk(p+1)p(k+1)

(k+1)U(A12p

1 Akp

1 A12p

1 )p+1k+1 U∗ by Furuta Inequality

≤ βk(p+1)

p |T ∗|2(p+1).

Similarily, (T(k+1)T (k+1)∗)p+1k+1 ≥ α

k(p+1)p |T ∗|2(p+1).

Therefore,β−(p+1)

p(n−1)(T nT n∗

)p+1

n ≤ (TT ∗)p+1 ≤ α−(p+1)

p(n−1)(T nT n∗

)p+1

n holdsfor all positive integer n.

Corollary 3.5. If T is p − (α, β) - normal operator then T n is pn−(α, β) -

normal operator for all positive integer n

Proof. Let T be p − (α, β) - normal operator, then by Theorem 3.4, we have

So, T n is pn

- (α, β) - normal operator.


4. Properties of p − (α, β) - normal operators

Definition 4.1. An operator T is called totally p − (α, β)- normal, if thetranslate T − λ is p − (α, β)- normal for all λ ∈ C

Definition 4.2. An operator T is called algebraically p − (α, β)- normal, ifthere exists a non constant polynomial p such that p(T ) is p− (α, β)- normal.

Definition 4.3. Let T ∈ B(H). The reducing point spectrum of T , σpr(T )is the set of scalars λ for which there exists x = 0 such that Tx = λx andT ∗x = λx.

Definition 4.4. An operator T ∈ B(H) is called finite if ‖TX −XT −I‖ ≥ 1for every X ∈ B(H).

Equivalently if σpr(T ) = 0 then T is a finite operator.

Theorem 4.5. Every totally p − (α, β) - normal operator is finite.

Proof. Since ker(T − λ)p = ker((T − λ)∗)p and since T − λ is p - (α,β)-normalfor every λ, we get σp(T ) = σpr(T ). And since ∂σ(T ) ⊂ σp(T ), we get σpr(T ) =0.

Definition 4.6. Let M2 = {AB : A, B ∈ M} where M denote the class ofall self adjoint operators. This class is defined and its various properties arestudied in [9]

Theorem 4.7. An p− (α, β)-normal operator T ∈ M2 is necessarily normal.

Proof. Let T = AB with A = A∗ and B = B∗. Consider the following twocases:

Case 1: A is injective, Then AT ∗ = TA and hence T is normal [9]Theorem3.a.

Case 2: A has a non-trivial null space N.

Let A =

(A1 00 0

)and B =

(B1 B2

B3 B4

)with respect to N⊥ ⊕ N where A1 is an injective self adjoint operator and

B1 is self adjoint. Then T =

(A1B1 A1B2

0 0

)Since A1B1 is the restriction of T to an invariant subspace, A1B1 is p− (α, β)- normal and hence by case 1 it is normal. By [9](Theorem 4) N is a reducinginvariant subspace of T and hence A1B2 = 0. This implies T is normal.

Theorem 4.8. Every p−(α, β) - normal operator can be written as T = A⊕Swhere A is normal and S is p − (α, β) - normal with σw(S) = σ(S).

Proof. By Weyl’s theorem σ(T )\σw(T ) = π00(T ) . Let N be a closed linearsubspace of H generated by λi ∈ π00(T )

⋃N(T − λi) Then N is reduced by

T . The decomposition H = N ⊕ N⊥ gives T = A ⊕ S where A is normal andS is p − (α, β) - normal and σw(S) = σ(S).


References

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1q ≥ B

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Received: February 21, 2014

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