FACTA UNIVERSITATIS (NIS)

Ser. Math. Inform. Vol. 23 (2008), pp. 39–47

SOME INEQUALITIES FOR (α, β)-NORMAL OPERATORS INHILBERT SPACES

S.S. Dragomir and M.S. Moslehian

Abstract. An operator T acting on a Hilbert space is called (α, β)-normal (0 ≤ α ≤1 ≤ β) if

α2T ∗T ≤ TT ∗ ≤ β2T ∗T.

In this paper we establish various inequalities between the operator norm and thenumerical radius of (α, β)-normal operators in Hilbert spaces. For this purpose, weemploy some classical inequalities for vectors in inner product spaces.

1. Introduction

An operator T acting on a Hilbert space (H; 〈·, ·〉) is called (α, β)-normal (0 ≤α ≤ 1 ≤ β) if

α2T ∗T ≤ TT ∗ ≤ β2T ∗T.

Thenα2〈T ∗Tx, x〉 ≤ 〈TT ∗x, x〉 ≤ β2〈T ∗Tx, x〉 ,

whence

(1.1) α‖Tx‖ ≤ ‖T ∗x‖ ≤ β‖Tx‖,

for all x ∈ H, namely, both T majorizes T ∗ and T ∗ majorizes T . A seminal result ofR.G. Douglas [6] (majorization lemma) says that an operator T ∈ B(H) majorizesan operator S ∈ B(H) if any one of the following equivalent statements holds:

(i) the range space ran(T ) of T is a subset of ran(S);(ii) TT ∗ ≤ λ2SS∗;(iii) there exists an operator R ∈ B(H) such that T = SR.

Furthermore, R is the unique operator satisfying(a) ‖R‖ = inf{λ : TT ∗ ≤ λSS∗};(b) ker(T ) = ker(R);

Received April 29, 2008.2000 Mathematics Subject Classification. Primary 47A12; Secondary 47A30, 47B20.

39

40 S.S. Dragomir and M.S. Moslehian

(c) ran(R) is a subset of the closure ran(S∗)− of ran(S∗).

Analogues of Douglas’ majorization lemma for Banach space operators werestudied by M.R. Embry [14] (see also [3]). A discussion of the duality between theproperties of majorization and range inclusion can be found in [1].

Using the result of Douglas, we observe that T is (α, β)-normal if and only ifran(T ) = ran(T ∗), or, equivalently, ker(T ) = ker(T ∗). It is therefore obvious thatinvertible, normal and hyponormal operators are (α, β)-normal for some appropriatevalues of α and β. The matrix

»1 01 1

–in B(C2) is an (α, β)-normal with α =√

(3−√

5)/2 and β =√

(3 +√

5)/2, which is neither normal nor hyponormal.There are some results which can be applied to our notion in the literature. Forinstance, one can deduce from [4, Lemma 1] that if z is an eigenvalue of T andz belongs to the topological boundary of the numerical range of T , then T − z is(α, β)-normal for some α and β. There are also some interesting questions in linearalgebra concerning (α, β)-normality, see [18].

Another characterization is that T is (α, β)-normal (0 < α ≤ 1 ≤ β) if and onlyif there are operators S1, S2 ∈ B(H) such that T = T ∗S1 and T = S2T

∗. Moreover,S1, S2 can be chosen in such a way that

‖S1‖ = inf{β ≥ 1 : TT ∗ ≤ βT ∗T}, ‖S2‖ = sup{α > 0 : αT ∗T ≤ TT ∗} .

Let T be an (α, β)-normal operator on a (not necessarily finite dimensional) Hilbertspace H. Using the fact that ker(T ∗)⊥ = ran(T )−, we observe that H = ker(T )⊕ran(T )−. Hence T can be represented as a block matrix

»0 00 C

–, where C :

ran(T )− → ran(T )− has zero kernel. We can define the pseudo-inverse of T , denotedby T+, to be the operator on H, which is zero on ran(T )⊥, and is the inverse to Con ran(T )−. It is easy to see that T+ is closed if and only if ran(T ) is closed. Theoperator pseudo-inverse is a powerful tool in applied mathematics; cf. [2].

Let (H; 〈·, ·〉) be a complex Hilbert space. The numerical radius w(T ) of anoperator T on H is given by

(1.2) w(T ) = sup{|〈Tx, x〉|, ‖x‖ = 1}.

Obviously, by (1.2), for any x ∈ H one has

(1.3) |〈Tx, x〉| ≤ w(T )‖x‖2.

It is well known that w(·) is a norm on the Banach algebra B(H) of all boundedlinear operators. Moreover, we have

w(T ) ≤ ‖T‖ ≤ 2w(T ) (T ∈ B(H)).

For other results and historical comments on the numerical radius see [16].In this paper, we establish various inequalities between the operator norm and

the numerical radius of (α, β)-normal operators in Hilbert spaces. For this purpose,we employ some classical inequalities for vectors in inner product spaces due toBuzano, Dunkl–Williams, Dragomir–Sandor, Goldstein–Ryff–Clarke and Dragomir.

Some Inequalities for (α, β)-Normal Operators in Hilbert Spaces 41

2. Inequalities Involving Numerical Radius

In this section we study some inequalities concerning the numerical radius andnorm of (α, β)-normal operators. Our first result reads as follows, see also [11]:

Theorem 2.1. Let T ∈ B(H) be an (α, β)-normal operator. Then

(2.1) (α2r + β2r)‖T‖2 ≤

2βrw(T 2) + r2β2r−2‖βT − T ∗‖2, if r ≥ 1,

2βrw(T 2) + ‖βT − T ∗‖2, if r < 1.

Proof. We use the following inequality for vectors in inner product spaces due toGoldstein, Ryff and Clarke [15]:

(2.2) ‖a‖2r + ‖b‖2r − 2‖a‖r‖b‖r · Re〈a, b〉‖a‖ ‖b‖

≤

r2‖a‖2r−2‖a− b‖2 if r ≥ 1,

‖b‖2r−2‖a− b‖2 if r < 1,

provided r ∈ R and a, b ∈ H with ‖a‖ ≥ ‖b‖.Suppose that r ≥ 1. Let x ∈ H with ‖x‖ = 1. Noting to (1.1) and applying

(2.2) for the choices a = βTx, b = T ∗x we get

(2.3) ‖βTx‖2r + ‖T ∗x‖2r − 2‖βTx‖r−1‖ ‖T ∗x‖r−1 Re〈βTx, T ∗x〉≤ r2‖βTx‖2r−2‖βTx− T ∗x‖2

for any x ∈ H, ‖x‖ = 1 and r ≥ 1. Using (1.1) and (2.3) we get

(2.4) (α2r + β2r)‖Tx‖2r

≤ 2βr‖Tx‖r−1‖T ∗x‖r−1|〈T 2x, x〉|+ r2β2r−2‖Tx‖2r−2‖βTx− T ∗x‖2.

Taking the supremum in (2.4) over x ∈ H, ‖x‖ = 1, we deduce

(α2r + β2r)‖T‖2r ≤ 2βr‖T‖2r−2‖T ∗‖r−1w(T 2) + r2β2r−2‖T‖2r−2‖βT − T ∗‖2,

which is the first inequality in (2.1). If r < 1, then one can similarly prove thesecond inequality in (2.1).

Theorem 2.2. Let T ∈ B(H) be an (α, β)-normal operator. Then

(2.5) w(T )2 ≤ 12[β‖T‖2 + w(T 2)

].

Proof. The following inequality is known in the literature as the Buzano inequality[5]:

(2.6) |〈a, e〉〈e, b〉| ≤ 12

(‖a‖ ‖b‖+ |〈a, b〉|),

42 S.S. Dragomir and M.S. Moslehian

for any a, b, e in H with ‖e‖ = 1.Let x ∈ H with ‖x|| = 1. Put e = x, a = Tx, b = T ∗x in (2.6) to get

|〈Tx, x〉〈x, T ∗x〉| ≤ 12

(‖Tx‖ ‖T ∗x‖+ |〈Tx, T ∗x〉|) ≤ 12

(β‖Tx‖2 + |〈T 2x, x〉|).

Taking the supremum over x ∈ H, ‖x‖ = 1, we obtain (2.5).

Theorem 2.3. Let T ∈ B(H) be an (α, β)-normal operator and λ ∈ C. Then

(2.7) α‖T‖2 ≤ w(T 2) +2β‖T − λT ∗‖2

(1 + |λ|α)2.

Proof. Using the Dunkl–Williams inequality [13]

12

(‖a‖+ ‖b‖)∥∥∥∥ a

‖a‖− b

‖b‖

∥∥∥∥ ≤ ‖a− b‖ (a, b ∈ H \ {0})

we get

2− 2 · Re〈a, b〉‖a‖‖b‖

=∥∥∥∥ a

‖a‖− b

‖b‖

∥∥∥∥2

≤ 4‖a− b‖2

(‖a||+ ‖b‖)2(a, b ∈ H \ {0})

whence

‖a‖‖b‖ ≤ 2‖a‖ ‖b‖ ‖a− b‖2

(‖a||+ ‖b‖)2+ |〈a, b〉| (a, b ∈ H \ {0}).

Put a = Tx and b = λT ∗ to get

‖Tx‖ ‖T ∗x‖ ≤ |〈T 2x, x〉|+ 2‖Tx‖ ‖T ∗x‖ ‖Tx− λT ∗x‖2

(‖Tx‖+ |λ| ‖T ∗x‖)2

so that

(2.8) α‖Tx‖2 ≤ |〈T 2x, x〉|+ 2β‖Tx‖2‖Tx− λT ∗x‖2

(‖Tx‖+ |λ|α ‖Tx‖)2

≤ |〈T 2x, x〉|+ 2β‖(T − λT ∗)x‖2

(1 + |λ|α)2.

Taking the supremum in (2.8) over x ∈ H, ‖x‖ = 1, we get the desired result(2.7).

Theorem 2.4. Let T ∈ B(H) be an (α, β)-normal operator and λ ∈ C\{0}. Then

(2.9)

[α2 −

(1|λ|

+ β

)2]‖T‖4 ≤ w(T 2).

Some Inequalities for (α, β)-Normal Operators in Hilbert Spaces 43

Proof. We apply the following reverse of the quadratic Schwarz inequality obtainedby Dragomir in [10]

(2.10) (0 ≤)‖a‖2‖b‖2 − |〈a, b〉|2 ≤ 1|λ|2‖a‖2‖a− λb‖2

provided a, b ∈ H and λ ∈ C\{0}.Set a = Tx, b = T ∗x in (2.10), to get

α2‖Tx‖4 ≤ |〈Tx, T ∗x〉|2 +1|λ|2‖Tx‖2‖Tx− λT ∗x‖2

≤ |〈T 2x, x〉|2 +1|λ|2‖Tx‖2(1 + |λ|β)2‖Tx‖2

whence

(2.11)

[α2 −

(1|λ|

+ β

)2]‖Tx‖4 ≤ |〈T 2x, x〉|2.

Taking the supremum in (2.11) over x ∈ H, ‖x‖ = 1, we get the desired result(2.9).

Theorem 2.5. Let T ∈ B(H) be an (α, β)-normal operator, r ≥ 0 and λ ∈C� {0}. If ‖λT ∗ − T‖ ≤ r and r

|λ| ≤ inf{‖T ∗x‖ : ‖x‖ = 1}, then

(2.12) α2‖T‖4 ≤ w(T 2)2 +r2

|λ|2‖T‖2.

Proof. We use the following reverse of the Schwarz inequality obtained by Dragomirin [8] (see also [9, p. 20]):

(2.13) (0 ≤) ‖y‖2 ‖a‖2 − [Re〈y, a〉]2 ≤ r2‖y‖2,

provided ‖y − a‖ ≤ r ≤ ‖a‖.By the assumption of theorem ‖Tx− λT ∗x‖ ≤ r ≤ ‖λT ∗x‖. Setting a = λT ∗x

and y = Tx, with ‖x‖ = 1 in (2.13) we get

‖Tx‖2 ‖λT ∗x‖2 ≤ [Re〈Tx, λT ∗x〉]2 + r2‖Tx‖2

whence

(2.14) α2|λ|2‖Tx‖4 ≤ |λ|2|〈T 2x, x〉|2 + r2‖Tx‖2.

Taking the supremum in (2.14) over x ∈ H, ‖x‖ = 1, we get the desired result(2.12).

Finally, the following result that is less restrictive for the involved parameters rand λ (from the above theorem) may be stated as well:

44 S.S. Dragomir and M.S. Moslehian

Theorem 2.6. Let T ∈ B(H) be an (α, β)-normal operator, r ≥ 0 and λ ∈C� {0}. If ‖λT ∗ − T‖ ≤ r, then

(2.15) α‖T‖2 ≤ w(T 2) +r2

2|λ|.

Proof. We use the following reverse of the Schwarz inequality obtained by Dragomirin [7] (see also [9, p. 27]):

(2.16) (0 ≤) ‖y‖ ‖a‖ − Re〈y, a〉 ≤ 12r2,

provided ‖y − a‖ ≤ r.Setting a = λT ∗x and y = Tx, with ‖x‖ = 1 in (2.16) we get

‖Tx‖ ‖λT ∗x‖ ≤ |〈Tx, λT ∗x〉|+ 12r2

which gives

α‖Tx‖2 ≤ |〈T 2x, x〉|+ 12|λ|

r2.

Now, taking the supremum over ‖x‖ = 1 in this inequality, we get the desired result(2.15)

3. Inequalities Involving Norms

Our first result in this section reads as follows.

Theorem 3.1. Let T ∈ B(H) be an (α, β)-normal operator. If p ≥ 2, then

(3.1) 2(1 + αp)‖T‖p ≤ 12

(‖T + T ∗‖p + ‖T − T ∗‖p).

In general, for each T ∈ B(H) and p ≥ 2 we have

(3.2)∥∥∥∥T ∗T + TT ∗

2

∥∥∥∥p/2 ≤ 14

(‖T + T ∗‖p + ‖T − T ∗‖p).

Proof. We use the following inequality obtained by Dragomir and Sandor in [12](see also [17, p. 544]):

(3.3) ‖a+ b‖p + ‖a− b‖p ≥ 2(‖a‖p + ‖b‖p)

for any a, b ∈ H and p ≥ 2.Now, if we choose a = Tx, b = T ∗x in (3.3), then we get

(3.4) ‖Tx+ T ∗x‖p + ‖Tx− T ∗x‖p ≥ 2(‖Tx‖p + ‖T ∗x‖p),

Some Inequalities for (α, β)-Normal Operators in Hilbert Spaces 45

whence

(3.5) ‖Tx+ T ∗x‖p + ‖Tx− T ∗x‖p ≥ 2(‖Tx‖p + αp‖Tx‖p),

for any x ∈ H, ‖x‖ = 1.Taking the supremum in (3.5) over x ∈ H, ‖x‖ = 1, we get the desired result

(3.1).Now for the general case T ∈ B(H), observe that

(3.6) ‖Tx‖p + ‖T ∗x‖p = (‖Tx‖2)p2 + (‖T ∗x‖2)

p2

and by applying the elementary inequality:

aq + bq

2≥(a+ b

2

)q, a, b ≥ 0 and q ≥ 1

we have

(3.7) (‖Tx‖2)p2 + (‖T ∗x‖2)

p2 ≥ 21− p

2 (‖Tx‖2 + ‖T ∗x‖2)p2

= 21− p2 [〈Tx, Tx〉+ 〈T ∗x, T ∗x〉]

p2

= 21− p2 [〈(T ∗T + TT ∗)x, x〉]

p2 .

Combining (3.4) with (3.7) and (3.6) we get

(3.8)14

[‖Tx− T ∗x‖p + ‖Tx+ T ∗x‖p] ≥∣∣∣∣⟨(T ∗T + TT ∗

2

)x, x

⟩∣∣∣∣p/2for any x ∈ H, ‖x‖ = 1. Taking the supremum over x ∈ H, ‖x‖ = 1, and takinginto account that

w

(T ∗T + TT ∗

2

)=∥∥∥∥T ∗T + TT ∗

2

∥∥∥∥ ,we deduce the desired result (3.2).

Theorem 3.2. Let T ∈ B(H) be an (α, β)-normal operator. If p ∈ (1, 2) andλ, µ ∈ C, then

(3.9) [(|λ|+ β|µ|)p + max{|λ| − |µ|β, α|µ| − |λ|}] ‖T‖p

≤ ‖λT + µT ∗‖p + ‖λT − µT ∗‖p.

Proof. We use the following inequality obtained by Dragomir and Sandor in [12](see also [17, p. 544])

(3.10) (‖a‖+ ‖b‖)p + | ‖a‖ − ‖b‖ |p ≤ ||a+ b‖p + ‖a− b‖p,

for any a, b ∈ H and p ∈ (1, 2).

46 S.S. Dragomir and M.S. Moslehian

Put a = λTx, b = µT ∗x in (3.10) to obtain

(‖λTx‖+ ‖µT ∗x‖)p + |‖λTx‖ − ‖µT ∗x‖|p

≤ ‖λTx+ µT ∗x‖p + ‖λTx− µT ∗x‖p,

whence

(3.11) (|λ|+ |µ|α)p‖Tx‖p + (max{|λ| − |µ|β, α|µ| − |λ|}) ‖Tx‖p

≤ ‖λTx+ µT ∗x‖p + ‖λTx− µT ∗x‖p,

for any x ∈ H, ‖x‖ = 1.

Taking the supremum in (3.11) over x ∈ H, ‖x‖ = 1, we get the desired result(3.9).

Acknowledgement. The authors would like to thank Professor Leiba Rodmanfor his useful suggestions and for bringing the majorization lemma of R.G. Douglasinto their attention. The authors would also like to express their gratitude toProfessor M. Mirzavaziri for giving an example of an (α, β)-normal operator whichis neither normal nor hyponormal.

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School of Computer Science and Mathematics

Victoria University

P. O. Box 14428, Melbourne City

Victoria 8001, Australia

sever.dragomir@vu.edu.au

Department of Pure Mathematics

Ferdowsi University of Mashhad

P.O. Box 1159

Mashhad 91775, Iran

Center of Excellence in Analysis on Algebraic Structures (CEAAS)

Ferdowsi University of Mashhad, Iran

moslehian@ferdowsi.um.ac.ir and moslehian@ams.org