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

Abstract

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

1. Introduction

Let be a complex Hilbert space. The numerical range of an operator is the subset of the complex numbers given by [1, page 1]:

(1.1)

The numerical radius of an operator on is given by [1, page 8]:

(1.2)

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 any one has

(1.3)

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.

If are two bounded linear operators on the Hilbert space then

(1.4)

In the case that then

(1.5)

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

Theorem 1.3.

If is a unitary operator that commutes with another operator then

(1.6)

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).

If the operators and double commute, then

(1.7)

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

Corollary 1.5.

Let be a normal operator commuting with Then

(1.8)

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.

Let be two bounded linear operators on the Hilbert space then

(1.9)

respectively.

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

Theorem 1.7.

Let be two bounded linear operators on , and is invertible such that, for a given

(1.10)

Then

(1.11)

respectively.

Motivated by the natural questions that arise, in order to compare the quantity with other expressions comprising the norm or the numerical radius of the involved operators and (or certain expressions constructed with these operators), we establish in this paper some natural inequalities of the form

(1.12)

or

(1.13)

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

Applications in providing upper bounds for the non-negative quantities

(1.14)

and the superunitary quantities

(1.15)

are also given.

2. Numerical Radius Inequalities of Grüss Type

For the complex numbers and the bounded linear operator , we define the following transform:

(2.1)

where by we denote the adjoint of .

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

(i)For any and we have

(2.2)

• (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

Utilizing the following identity

(2.3)

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 transformis accretive.

(ii)The transformis accretive.

(iii)One has the norm inequality

(2.4)

or, equivalently,

(2.5)

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.

Let and be such that the transforms and are accretive, then

(2.6)

Proof.

Since and are accretive, then, on making use of Lemma 2.1, we have that

(2.7)

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]).

Let be an inner product space over the real or complex number field , and such that

(2.8)

or equivalently,

(2.9)

then

(2.10)

Applying (2.10) for , and we deduce

(2.11)

for any which is an inequality of interest in itself.

Observing that

(2.12)

then by (2.10), we deduce the inequality

(2.13)

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.

Let and be such that the transform is accretive, then

(2.14)

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.

Let and be such that the transform is accretive. Then

(2.15)

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.

Let and be such that and the transforms are accretive, then

(2.16)

respectively.

Proof.

With the assumptions (2.8) (or, equivalently, (2.9) in the proof of Theorem 2.3) and if then

(2.17)

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.

Applying (2.10) for , and we deduce

(2.18)

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.

Let and be such that and the transform is accretive, then

(2.19)

respectively.

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

Remark 2.8.

Let and be such that the transform is accretive. Then, on making use of Corollary 2.7, we may state the following simpler results:

(2.20)

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

The following result is well known in the literature (see, e.g., [10]):

(3.1)

for each positive integer and any operator

The following reverse inequalities for can be stated.

Proposition 3.1.

Let and be such that the transform is accretive, then

(3.2)

Proof.

On applying inequality (2.11) from Theorem 2.3 for the choice we get the following inequality of interest in itself:

(3.3)

for any Since obviously,

(3.4)

then by (3.3), we get

(3.5)

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

Remark 3.2.

Let and be such that the transform is accretive. Then

(3.6)

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

Finally, we also have the following proposition.

Proposition 3.3.

Let and be such that and the transform is accretive, then

(3.7)

respectively.

Proof.

On applying inequality (2.18) from Theorem 2.6 for the choice we get the following inequality of interest in itself:

(3.8)

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.

Let and be such that the transform is accretive. Then, on making use of Proposition 3.3, we may state the following simpler results:

(3.9)

respectively.