Abstract and Applied Analysis

Volume 2012 (2012), Article ID 204031, 11 pages

doi:10.1155/2012/204031

Research Article

More on -Normal Operators in Hilbert Spaces

Rasoul Eskandari, Farzollah Mirzapour, and Ali Morassaei

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

Academic Editor: Gerd Teschke

Copyright © 2012 Rasoul Eskandari et al. 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.

Abstract.

We study some properties of -normal operators and we present various inequalities between the operator norm and the numerical radius of -normal operators on Banach algebra ℬ() of all bounded linear operators , where is Hilbert space.

1. Introduction

Throughout the paper, let denote the algebra of all bounded linear operators acting on a complex Hilbert space , denote the algebra of all self-adjoint operators in , and is the identity operator. In case of , we identify with the full matrix algebra of all matrices with entries in the complex field. An operator is called positive if is valid for any , and then we write . Moreover, by we mean for any . For , we say if . An operator is majorized by , if there exists a constant such that for all or equivalently [1].

For real numbers and with , an operator acting on a Hilbert space is called -normal [2, 3] if

An immediate consequence of above definition is

from which we obtain

for all .

Notice that, according to (1.1), if is -normal operator, then and majorize each other.

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

For fixed and ,

(i)give an example of an -normal operator which is neither normal nor hyponormal;(ii)is there any nice relation between norm, numerical radius, and spectral radius of an -normal operator?

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

The matrix in is an -normal with and .

The numerical radius of an operator on is defined by

Obviously, by (1.4), for any we have

It is well known that is a norm on the Banach algebra of all bounded linear operators. Moreover, we have

For other results and historical comments on the numerical radius see [4].

The antieigenvalue of an operator defined by

The vector which takes is called an antieigenvector of . We refer more study on this matter to [4].

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

2. Some Properties of -Normal Operators

In this section, we establish some properties of -normal operators. It is easy to see that if is an -normal then is -normal. We find numbers such that is -normal where is -normal.

We know by the Cauchy-Schwartz inequality that . Also we can write

We define

We know that if is normal operator then is also normal.

Theorem 2.1. Let be an -normal operator on a Hilbert space such that and . Then is -normal, if provided one of the following conditions holds:

(i),(ii).

Proof. In both of above cases, we show that

By the assumption (i), , we have for every with and , consequently we get , and therefore (2.3) is valid. On the other hand, if (ii) holds and we set then we get for every with and , consequently:

Since , we obtain

and so

Now, by using the last inequality, we have

This shows that (2.3) holds for (ii), too. Thus, for any with we have

and this completes the proof.

Corollary 2.2. Let be an -normal operator. We have the following.

(i)If then is -normal operator for any .(ii)If then is -normal operator for any .

Proof. (i) By the definition of the first antieigenvalue of , for all we have

By using Theorem 2.1(i) we imply that is an -normal.

(ii) If , then

By using Theorem 2.1(i) we imply that is an -normal.

Corollary 2.3. Let be an injective and -normal operator with . Then

(i) is dense,(ii) is injective,(iii)if is surjective then is also -normal.

Proof. Since the inequality (1.3) is valid, we obtain , and therefore , thus is a dense subspace of and is injective. This proves (i) and (ii).

To prove (iii), we note that since is surjective, we imply that is invertible. On the other hand we have . Also we know that if and are two positive and invertible operators with then . Since is -normal, by taking inverse from all sides of (1.1), we get

This means that is -normal, thus is -normal.

Example 2.4. Consider the following matrix in :

is an -normal operator, with parameters and . Then is -normal.

For we call

the spectral radius of , where is the spectrum of and it is known that [5, page 102].

Theorem 2.5. Let be an -normal operator such that is -normal operator for every , too. Then, we have

Proof. For any we have

In particular, if is a self-adjoint operator then . Thus, by the definition of -normal operator, we have

By induction on , we imply that

from which we obtain

Therefore, we get . This completes the proof.

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

Example 2.6. Assume that is a separable Hilbert space and is an orthonormal basis for . We define the operator as follows:

so

and by simple computation we get

Consequently, is -normal operator and also is -normal operator, for any integer . Thus we have and , 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 be an -normal operator.

(i)For positive real numbers and with and we have

(ii)If or , then we have

where . (iii)If and for any with we have

then, we obtain

Proof. (i) We use the following known inequality:

which is valid for any where is a Hilbert space.

Now, if we take and in (3.5), then for any we get

Taking the supremum in (3.6) over with , we get the desired result (3.1).

(ii) We use the following inequality [6, Theorem 8, page 551]:

where and are two vectors in a Hilbert space and or .

Now, if we put and in (3.7), then we obtain

Now, taking the supremum over in (3.8), we get the desired result (3.2).

(iii) We use the following reverse of Schwarz's inequality:

which is valid for and , with (see [7]). We take and in (3.9) to get

Thus, we obtain

Now, taking the supremum over in recent inequality, we get the desired result (3.4).

Theorem 3.2. Assume that is an -normal operator. Then, we have

Proof. By [2, Theorem 3.1], we have

and also

On the other hand, it is known [8] that for we have

By using this inequality we get

If we put in (3.14), we obtain

Thus we get

Now, we take in (3.13) to obtain

This completes the proof.

Theorem 3.3. Assume that is an -normal operator. Then for any real with , we have

Proof. By [9, Theorem 2.6] (see also [10, Theorem 2.4]), we have

where , and . By taking , and in (3.21), we get

thus, we have

Finally, we take supremum over from both sides of

and we use triangle inequality for supremums to complete the proof.

Corollary 3.4. Let be an -normal operator. Then, we have

Proof. By using the inequality (3.21) we get

We take in inequalities (3.20) and (3.26) to imply

Thus, . Now, taking supremum overall with , 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.

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