# Some Inequalities of the Grüss Type for the Numerical Radius of Bounded Linear Operators in Hilbert Spaces

- S. S. Dragomir
^{1}Email author

**2008**:763102

**DOI: **10.1155/2008/763102

© The Author(s). 2008

**Received: **27 May 2008

**Accepted: **4 August 2008

**Published: **12 August 2008

## Abstract

Some inequalities of the Grüss type for the numerical radius of bounded linear operators in Hilbert spaces are established.

## 1. Introduction

*numerical range*of an operator is the subset of the complex numbers given by [1, page 1]:

It is well known that is a norm on the Banach algebra of all bounded linear operators This norm is equivalent to the operator norm. In fact, the following more precise result holds [1, page 9].

Theorem 1.1 (equivalent norm).

For other results on numerical radius (see [2, Chapter 11]).

We recall some classical results involving the numerical radius of two linear operators

The following general result for the product of two operators holds [1, page 37].

Theorem 1.2.

The following results are also well known [1, page 38].

Theorem 1.3.

If is an isometry and then (1.6) also holds true.

We say that
and
*double commute*, if
and

The following result holds [1, page 38].

Theorem 1.4 (double commute).

As a consequence of the above, one has [1, page 39] the following.

Corollary 1.5.

For other results and historical comments on the above (see [1, pages 39–41]). For more results on the numerical radius, see [2].

In the recent survey paper [3], we provided other inequalities for the numerical radius of the product of two operators. We list here some of the results.

Theorem 1.6.

respectively.

If more information regarding one of the operators is available, then the following results may be stated as well.

Theorem 1.7.

respectively.

where and are specified and desirably simple constants (depending on the given operators and

*superunitary*quantities

are also given.

## 2. Numerical Radius Inequalities of Grüss Type

where by we denote the adjoint of .

We list some properties of the transform that are useful in the following.

- (ii)
The operator is normal, if and only if for each .

We recall that a bounded linear operator
on the complex Hilbert space
is called *accretive,* if
, for any

that holds for any scalars and any vector with we can give a simple characterization result that is useful in the following.

Lemma 2.1.

For and the following statements are equivalent.

(i)*The transform*
*is accretive*.

(ii)*The transform*
*is accretive*.

*One has the norm inequality*

*or, equivalently,*

Remark 2.2.

In order to give examples of operators and numbers such that the transform is accretive, it suffices to select a bounded linear operator and the complex numbers with the property that , and by choosing and we observe that satisfies (2.4), that is, is accretive.

The following results compare the quantities and provided that some information about the transforms and are available, where .

Theorem 2.3.

Proof.

for any

Now, we make use of the following Grüss type inequality for vectors in inner product spaces obtained by the author in [4] (see also [5] or [6, page 43]).

for any which is an inequality of interest in itself.

for any On taking the supremum over in (2.13), we deduce the desired result (2.6).

The following particular case provides an upper bound for the nonnegative quantity when some information about the operator is available.

Corollary 2.4.

Proof.

Follows on applying Theorem 2.3 above for the choice taking into account that is accretive implies that is the same and

Remark 2.5.

A sufficient simple condition for to be accretive is that is a self-adjoint operator on and such that in the partial operator order of

The following result may be stated as well.

Theorem 2.6.

respectively.

Proof.

The first inequality has been established in [7] (see [6, page 62]) while the second one can be obtained in a canonical manner from the reverse of the Schwarz inequality given in [8]. The details are omitted.

for any which are of interest in themselves.

A similar argument to that in the proof of Theorem 2.3 yields the desired inequalities (2.16). The details are omitted.

Corollary 2.7.

respectively.

The proof is obvious from Theorem 2.6 on choosing and the details are omitted.

Remark 2.8.

respectively. These two inequalities were obtained earlier by the author using a different approach (see [9]).

Problem 1.

Find general examples of bounded linear operators realizing the equality case in each of inequalities (2.6), (2.16), respectively.

## 3. Some Particular Cases of Interest

for each positive integer and any operator

The following reverse inequalities for can be stated.

Proposition 3.1.

Proof.

for any Taking the supremum over in (3.5), we deduce the desired result (3.2).

Remark 3.2.

If in the partial operator order of then (3.6) is valid.

Finally, we also have the following proposition.

Proposition 3.3.

respectively.

Proof.

for any

Now, on making use of a similar argument to the one in the proof of Proposition 3.1, we deduce the desired results (3.7). The details are omitted.

Remark 3.4.

respectively.

## Authors’ Affiliations

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## Copyright

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