More on ([alpha],[beta]) -Normal Operators in Hilbert Spaces

Abstract

We study some properties of (α,β) -normal operators and we present various inequalities between the operator norm and the numerical radius of (α,β) -normal operators on Banach algebra [Bernoulli]( [Hamiltonian (script capital H)] ) of all bounded linear operators T:[Hamiltonian (script capital H)][arrow right][Hamiltonian (script capital H)] , where [Hamiltonian (script capital H)] is Hilbert space.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Recommended by Gerd Teschke

1, Department of Mathematics, Faculty of Sciences, University of Zanjan, Zanjan 45195-313, Iran

Received 18 November 2011; Revised 3 July 2012; Accepted 20 July 2012

This 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.

1. Introduction

Throughout the paper, let [Bernoulli]([Hamiltonian (script capital H)]) denote the algebra of all bounded linear operators acting on a complex Hilbert space ([Hamiltonian (script capital H)],Y9;·,·YA;) , _[Bernoulli]h ([Hamiltonian (script capital H)]) denote the algebra of all self-adjoint operators in [Bernoulli]([Hamiltonian (script capital H)]) , and I is the identity operator. In case of dim [Hamiltonian (script capital H)]=n , we identify [Bernoulli]([Hamiltonian (script capital H)]) with the full matrix algebra _{[physics M-matrix]n} (...) of all n×n matrices with entries in the complex field. An operator A∈_[Bernoulli]h ([Hamiltonian (script capital H)]) is called positive if Y9;Ax,xYA;...5;0 is valid for any x∈[Hamiltonian (script capital H)] , and then we write A...5;0 . Moreover, by A>0 we mean Y9;Ax,xYA;>0 for any x∈[Hamiltonian (script capital H)] . For A,B∈_[Bernoulli]h ([Hamiltonian (script capital H)]) , we say A...4;B if B-A...5;0 . An operator A is majorized by B , if there exists a constant λ such that ||Ax||...4;λ||Bx|| for all x∈[Hamiltonian (script capital H)] or equivalently ^A* A...4;^λ2B* B [1].

For real numbers α and β with 0...4;α...4;1...4;β , an operator T acting on a Hilbert space [Hamiltonian (script capital H)] is called (α,β) -normal [2, 3] if [figure omitted; refer to PDF] An immediate consequence of above definition is [figure omitted; refer to PDF] from which we obtain [figure omitted; refer to PDF] for all x∈[Hamiltonian (script capital H)] .

Notice that, according to (1.1), if T is (α,β) -normal operator, then T and ^T* majorize each other.

In [3], Moslehian posed two problems about (α,β) -normal operators as follows.

For fixed α>0 and β...0;1 ,

(i) give an example of an (α,β) -normal operator which is neither normal nor hyponormal;

(ii) is there any nice relation between norm, numerical radius, and spectral radius of an (α,β) -normal operator?

Dragomir and Moslehian answered these problems in [2], as more as, they propounded a nice example of (α,β) -normal operator that is neither normal nor hyponormal, as follows.

The matrix (1011) in [Bernoulli](^...2 ) is an (α,β) -normal with α=(3-5)/2 and β=(3+5)/2 .

The numerical radius w(T) of an operator T on [Hamiltonian (script capital H)] is defined by [figure omitted; refer to PDF] Obviously, by (1.4), for any x∈[Hamiltonian (script capital H)] we have [figure omitted; refer to PDF] It is well known that w(·) is a norm on the Banach algebra [Bernoulli]([Hamiltonian (script capital H)]) of all bounded linear operators. Moreover, we have [figure omitted; refer to PDF] For other results and historical comments on the numerical radius see [4].

The antieigenvalue of an operator T∈[Bernoulli]([Hamiltonian (script capital H)]) defined by [figure omitted; refer to PDF] The vector x∈[Hamiltonian (script capital H)] which takes _μ1 (T) is called an antieigenvector of T . We refer more study on this matter to [4].

In this paper, we prove some properties of (α,β) -normal operators and state various inequalities between the operator norm and the numerical radius of (α,β) -normal operators in Hilbert spaces.

2. Some Properties of (α,β) -Normal Operators

In this section, we establish some properties of (α,β) -normal operators. It is easy to see that if T is an (α,β) -normal (α>0) then ^T* is (1/β,1/α) -normal. We find numbers z∈... such that z+T is (α,β) -normal where T is (α,β) -normal.

We know by the Cauchy-Schwartz inequality that -1...4;_μ1 (T)...4;1 . Also we can write [figure omitted; refer to PDF] We define [figure omitted; refer to PDF]

We know that if T is normal operator then z+T is also normal.

Theorem 2.1.

Let T be an (α,β) -normal operator on a Hilbert space such that 0...4;α<1<β and z∈... . Then z+T is (α,β) -normal, if provided one of the following conditions holds:

(i) _μ1 (z¯T)...5;0 ,

(ii) _μ1 (z¯T)<0,^|z|2 ...5;-2|z|||T||_μ1 (z¯T) .

Proof.

In both of above cases, we show that [figure omitted; refer to PDF] By the assumption (i), _μ1 (z¯T)...5;0 , we have ReY9;z¯Tx,xYA;/|z|||Tx||...5;0 for every x∈[Hamiltonian (script capital H)] with ||x||=1 and Tx...0;0 , consequently we get ReY9;z¯Tx,xYA;...5;0 , and therefore (2.3) is valid. On the other hand, if (ii) holds and we set B...=_μ1 (z¯T) then we get B...4;ReY9;z¯Tx,xYA;/|z|||Tx|| for every x∈[Hamiltonian (script capital H)] with ||x||=1 and Tx...0;0 , consequently: [figure omitted; refer to PDF] Since B<0 , we obtain [figure omitted; refer to PDF] and so [figure omitted; refer to PDF] Now, by using the last inequality, we have [figure omitted; refer to PDF] This shows that (2.3) holds for (ii), too. Thus, for any x∈[Hamiltonian (script capital H)] with ||x||=1 we have [figure omitted; refer to PDF] and this completes the proof.

Corollary 2.2.

Let T be an (α,β) -normal operator. We have the following.

(i) If _μ1 (T)...5;0 then z+T is (α,β) -normal operator for any z>0 .

(ii) If _μ2 (T)...4;0 then z+T is (α,β) -normal operator for any z<0 .

Proof.

(i) By the definition of the first antieigenvalue of T , for all z>0 we have [figure omitted; refer to PDF] By using Theorem 2.1(i) we imply that z+T is an (α,β) -normal.

(ii) If z<0 , then [figure omitted; refer to PDF] By using Theorem 2.1(i) we imply that z+T is an (α,β) -normal.

Corollary 2.3.

Let T be an injective and (α,β) -normal operator with α>0 . Then

(i) ...(T) is dense,

(ii) ^T* is injective,

(iii): if T is surjective then ^T-1 is also (α,β) -normal.

Proof.

Since the inequality (1.3) is valid, we obtain ...A9;(^T* )=...A9;(T) , and therefore ...(T^{)[perpendicular]} =...A9;(^T* )=...A9;(T)=0 , thus ...(T) is a dense subspace of [Hamiltonian (script capital H)] and ^T* is injective. This proves (i) and (ii).

To prove (iii), we note that since T is surjective, we imply that T is invertible. On the other hand we have (^T*)-1 =(^T-1)* . Also we know that if A and B are two positive and invertible operators with 0<A...4;B then ^B-1 ...4;^A-1 . Since T is (α,β) -normal, by taking inverse from all sides of (1.1), we get [figure omitted; refer to PDF] This means that ^{(T-1 )*} is (1/β,1/α) -normal, thus ^T-1 is (α,β) -normal.

Example 2.4.

Consider the following matrix T in [Bernoulli](^...2 ) : [figure omitted; refer to PDF] T is an (α,β) -normal operator, with parameters α=(3-5)/2 and β=(3+5)/2 . Then ^T-1 =(10-11) is (α,β) -normal.

For T∈[Bernoulli]([Hamiltonian (script capital H)]) we call [figure omitted; refer to PDF] the spectral radius of T , where σ(T) is the spectrum of T and it is known that r(T)=_{lim n[arrow right]∞}^{||Tn ||1/n} [5, page 102].

Theorem 2.5.

Let T be an (α,β) -normal operator such that ^T2n is (α,β) -normal operator for every n∈... , too. Then, we have [figure omitted; refer to PDF]

Proof.

For any T∈[Bernoulli]([Hamiltonian (script capital H)]) we have [figure omitted; refer to PDF] In particular, if T is a self-adjoint operator then ||^T2 ||=^||T||2 . Thus, by the definition of (α,β) -normal operator, we have [figure omitted; refer to PDF] By induction on n , we imply that [figure omitted; refer to PDF] from which we obtain [figure omitted; refer to PDF] Therefore, we get (1/β)||T||...4;r(T)...4;||T|| . This completes the proof.

Below, we give an example of (α,β) -normal operator such that it satisfies in Theorem 2.5.

Example 2.6.

Assume that [Hamiltonian (script capital H)] is a separable Hilbert space and {_en :n∈...} is an orthonormal basis for [Hamiltonian (script capital H)] . We define the operator T∈[Bernoulli]([Hamiltonian (script capital H)]) as follows: [figure omitted; refer to PDF] so [figure omitted; refer to PDF] and by simple computation we get [figure omitted; refer to PDF] Consequently, T is (1/4,4) -normal operator and also ^Tn is (1/4,4) -normal operator, for any integer n...5;0 . Thus we have ||T||=2 and r(T)=1 , hence (2.14) is valid.

3. Inequalities Involving Norms and Numerical Radius

In this section we state some inequalities involving norms and numerical radius.

Theorem 3.1.

Let T∈[Bernoulli]([Hamiltonian (script capital H)]) be an (α,β) -normal operator.

(i) For positive real numbers p and q with p...5;2 and (1/p)+(1/q)=1 we have [figure omitted; refer to PDF]

(ii) If 0...4;p...4;1 or p...5;2 , then we have [figure omitted; refer to PDF] where [straight phi](α,p)=^2p [^{(1+αp )2} +(^2p -²² )^αp ] .

(iii): If ...A9;(T)=0 and for any x∈[Hamiltonian (script capital H)] with ||x||=1 we have [figure omitted; refer to PDF] then, we obtain [figure omitted; refer to PDF]

Proof.

(i) We use the following known inequality: [figure omitted; refer to PDF] which is valid for any a,b∈[Hamiltonian (script capital H)] where [Hamiltonian (script capital H)] is a Hilbert space.

Now, if we take a=Tx and b=^T* x in (3.5), then for any x∈[Hamiltonian (script capital H)] we get [figure omitted; refer to PDF] Taking the supremum in (3.6) over x∈[Hamiltonian (script capital H)] with ||x||=1 , we get the desired result (3.1).

(ii) We use the following inequality [6, Theorem 8, page 551]: [figure omitted; refer to PDF] where a and b are two vectors in a Hilbert space and 0...4;p...4;1 or p...5;2 .

Now, if we put a=Tx and b=^T* x in (3.7), then we obtain [figure omitted; refer to PDF] Now, taking the supremum over ||x||=1 in (3.8), we get the desired result (3.2).

(iii) We use the following reverse of Schwarz's inequality: [figure omitted; refer to PDF] which is valid for a,b∈[Hamiltonian (script capital H)]\{0} and ρ>0 , with ||(a/||b||)-(b/||a||)||...4;ρ (see [7]). We take a=Tx and b=^T* x in (3.9) to get [figure omitted; refer to PDF] Thus, we obtain [figure omitted; refer to PDF] Now, taking the supremum over ||x||=1 in recent inequality, we get the desired result (3.4).

Theorem 3.2.

Assume that T is an (α,β) -normal operator. Then, we have [figure omitted; refer to PDF]

Proof.

By [2, Theorem 3.1], we have [figure omitted; refer to PDF] and also [figure omitted; refer to PDF] On the other hand, it is known [8] that for A,B∈[Bernoulli]([Hamiltonian (script capital H)]) we have [figure omitted; refer to PDF] By using this inequality we get [figure omitted; refer to PDF] If we put p=2 in (3.14), we obtain [figure omitted; refer to PDF] Thus we get [figure omitted; refer to PDF] Now, we take p=2 in (3.13) to obtain [figure omitted; refer to PDF] This completes the proof.

Theorem 3.3.

Assume that T is an (α,β) -normal operator. Then for any real s with 0...4;s...4;1 , we have [figure omitted; refer to PDF]

Proof.

By [9, Theorem 2.6] (see also [10, Theorem 2.4]), we have [figure omitted; refer to PDF] where 0...4;s...4;1 , t∈... and a,b∈[Hamiltonian (script capital H)] . By taking t=1,a=Tx , and b=^T* x in (3.21), we get [figure omitted; refer to PDF] thus, we have [figure omitted; refer to PDF] Finally, we take supremum over ||x||=1 from both sides of [figure omitted; refer to PDF] and we use triangle inequality for supremums to complete the proof.

Corollary 3.4.

Let T be an (α,β) -normal operator. Then, we have [figure omitted; refer to PDF]

Proof.

By using the inequality (3.21) we get [figure omitted; refer to PDF] We take s=0 in inequalities (3.20) and (3.26) to imply [figure omitted; refer to PDF] Thus, max {1/β,α}^||Tx||2 ...4;||Tx||||Tx-^T* x||+ω(^T2 ) . Now, taking supremum overall x with ||x||=1 , the desired inequality is obtained.

Acknowledgments

The authors would like to sincerely thank the anonymous referee for several useful comments improving the paper and also Professor Mehdi Hassani for a useful discussion.

References

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