Numerical Radius and Operator Norm Inequalities

Abstract

A general inequality involving powers of the numerical radius for sums and products of Hilbert space operators is given. This inequality generalizes several recent inequalities for the numerical radius, and includes that if and are operators on a complex Hilbert space , then for . It is also shown that if is normal , then . Related numerical radius and usual operator norm inequalities for sums and products of operators are also presented.

1. Introduction

Let be a complex Hilbert space with inner product , and let denote the - algebra of all bounded linear operators on . For , the usual operator norm of an operator is defined by

(1.1)

where .

The numerical range of , known also as the field of values of , is defined as the set of complex numbers given by

(1.2)

The most important properties of the numerical range are that it is convex and its closure contains the spectrum of the operator.

A unitarily invariant norm on is a norm on the ideal of , satisfying for all and all unitary operators and in . It is called weakly unitarily invariant norm (or invariant under similarities) if for all and all unitary operators .

The most familiar example of weakly unitarily invariant norm is the numerical radius defined by

(1.3)

It is well known that defines a norm on and for every , we have

(1.4)

Thus, the usual operator norm and the numerical radius norm are equivalent. The inequalities in (1.4) are sharp: if , then the first inequality becomes an equality, while the second inequality becomes an equality if is normal. In fact, for a nilpotant operator with , Haagerup and Harpe [1] show that . In particular, when , we get the reverse inequality of the first inequality in (1.4). For a comprehensive account on the theory of the numerical range and numerical radius, the reader is referred to [2, 3]. A detailed study for the field of values of a matrix is given in [4].

The inequalities in (1.4) have been improved considerably by Kittaneh in [5, 6]. It has been shown that if , then

(1.5)

(1.6)

where is the absolute value of . The second inequality in (1.5) refines the second inequality in (1.4). For diverse applications of these inequalities we refer to [5, 7].

Considerable generalizations of the first inequality in (1.5) and the second inequality in (1.6) have been established in [8] for the numerical radius of one operator and for the sum of two operators. It has been shown that if ,, then

(1.7)

(1.8)

for and . Other recent inequalities have been obtained in [9, 10], which are related to the Euclidean radius of two Hilbert space operators and -normal operators in Hilbert spaces, respectively.

A general numerical radius inequality has been proved by Kittaneh, it has been shown in [6] that if , then

(1.9)

for all . In particular,

(1.10)

Usual operator norm inequalities for sums of operators have attracted the attention of several mathematicians. Some of these inequalities have been introduced in [3, 11]. It has been shown in [6] that if and are normal and , then

(1.11)

Another important norm inequalities for unitarily invariant norms, which are related to (1.11) assert that if are positive and , then

(1.12)

(1.13)

(see, e.g., [12]).

In Section 2 of this paper, we establish a general numerical radius inequality that generalizes (1.6), (1.7), (1.8), and (1.9), from which numerical radius inequalities for sums, products, and commutators of operators are obtained. Usual operator norm inequalities that generalize (1.11) and related to (1.13) are presented in Section 3.

2. A General Numerical Radius Inequality

In this section, we establish a general numerical radius inequality for Hilbert space operators which yields well known and new numerical radius inequalities as special cases. To prove our generalized inequality, we need the following basic lemmas. The first lemma is a generalized form of the mixed Schwarz inequality, which has been proved by Kittaneh [13].

Lemma 2.1.

Let be an operator in , and let and be nonnegative functions on which are continuous and satisfy the relation for all Then

(2.1)

for all and in .

The second lemma, which is called Hölder-McCarthy inequality, is a well-known result that follows from the spectral theorem for positive operators and Jensen's inequality (see [13]).

Lemma 2.2.

Let be a positive operator in and let be any unit vector. Then

(2.2)

The third lemma concerned with positive real numbers, and it is a consequence of the convexity of the function , .

Lemma 2.3.

Let be a positive real number Then

(2.3)

The fourth lemma is a norm inequality for the sum of two operators, which can be found in [14].

Lemma 2.4.

If and are positive operators in , then

(2.4)

Another important usual operator norm inequality which will be used in this section says that for any positive operators , we have (see [11])

(2.5)

Our main result of this paper, which leads to a generalization of (1.6), (1.7), (1.8), and (1.9), can be stated as follows.

Theorem 2.5.

Let ,,, and let and be nonnegative functions on which are continuous and satisfy the relation for all . Then

(2.6)

for all .

Proof.

For every unit vector , we have

(2.7)

Now the result follows by taking the supremum over all unit vectors in .

Inequality (2.6) includes several numerical radius inequalities as special cases. Samples of inequalities are demonstrated in what follows.

For and , , in inequality (2.6), we get the following inequality that generalizes (1.9).

Corollary 2.6.

Let ,,, , and . Then

(2.8)

In particular,

(2.9)

For in inequality (2.6), we get the following numerical radius inequalities for sums of operators that generalizes (1.8).

Corollary 2.7.

Let , and let and be as in Lemma 2.1. Then

(2.10)

In particular,

(2.11)

It should be mentioned here that the inequality in (2.11) generalizes (1.7) in the case .

Remark 2.8.

The case in (2.11) gives

(2.12)

which generalizes the second inequality in (1.6), while the choice will give a generalization of the first inequality in (1.5) and can be stated as

(2.13)

Note that using (2.4) and (2.5), a related inequality can be derived from (2.13). Indeed,

(2.14)

The above inequality generalizes the second inequality in (1.5). In fact, for , we have

(2.15)

The last equality can be proved using the polar decomposition. In fact, if and are the polar decompositions of and , respectively, then .

It is known that . However, the numerical radius is not submultiplicative, even for commuting operators. On the other hand, we have the power inequality, which asserts that if , then

(2.16)

It is evident from the first inequality in (1.4) that if , then

(2.17)

Moreover, if , then

(2.18)

These inequalities, among other related ones, can be found in [2].

For in inequality (2.6), we get the following numerical radius inequalities for products of operators that are related to the above inequalities.

Corollary 2.9.

Let , and . Then

(2.19)

In particular,

(2.20)

Remark 2.10.

The case in (2.19), provides the following inequality

(2.21)

which is a numerical radius inequality for the product of operators and is related to the arithmetic-geometric mean inequality for operators. Note that a more general inequality can be obtained by letting and in (2.8). In fact, we have

(2.22)

For in (2.22), we obtain the inequality

(2.23)

as well as

(2.24)

which follows from the the arithmetic-geometric mean inequality for operators (see [15]). Inequalities (2.23) and (2.24) are not equivalent. This can be seen from the example , , .

The inequality in (2.22) can be used to give an upper bound for the numerical radius of and . In fact, we have

(2.25)

(2.26)

The commutator of and is the operator . Commutators play an important role in operator theory. It follows by the triangle inequality that if , then .

For in inequality (2.6), we get the following numerical radius inequalities that generalize (1.9), and give an estimate for the numerical radius of commutators.

Corollary 2.11.

Let , and let and be as in Theorem 2.5. Then

(2.27)

for . In particular,

(2.28)

We end this section by the following remark.

Remark 2.12.

Inequality (2.28) gives a numerical radius inequality for commutators of operators that generalizes (1.10). If , , and , then

(2.29)

In particular,

(2.30)

In fact, by letting in (2.29) and (2.30), respectively, we get the following inequalities for the generalized commutator and the self commutator

(2.31)

(2.32)

3. A General Norm Inequality

In this section, we introduce a general norm inequality for Hilbert space operators, from which new inequalities for operators and generalizations of earlier results can be derived. The proof of this general inequality is similar to that of Theorem 2.5 under slight modification.

Theorem 3.1.

Let ,,, and let and be as in Theorem 2.5. Then

(3.1)

for

Inequality (3.1) yields several norm inequalities as special cases. Samples of these inequalities are demonstrated below.

Corollary 3.2.

Let ,,, , and . Then

(3.2)

In particular,

(3.3)

For in inequality (3.2), we get the following operator inequalities for sums of operators.

Corollary 3.3.

Let , , and . Then

(3.4)

In particular, if is normal , then

(3.5)

Remark 3.4.

The inequality (3.5) is a generalized form of (1.11). The normality of is necessary, this inequality is not true for arbitrary operators , as may be seen for and .

For in inequality (3.2), we get norm inequalities for products of operators.

Corollary 3.5.

Let and . Then

(3.6)

In particular,

(3.7)

For in inequality (3.2), we get the following norm inequalities that give an estimate for the usual norm of commutators.

Corollary 3.6.

Let , and let . Then

(3.8)

Finally, we end this paper by the following remark.

Remark 3.7.

Inequality (3.8) gives a norm inequality for commutators of operators. If , , and , then we get

(3.9)

In particular,

(3.10)

In fact, by letting in (3.10), we get the following inequality for self commutator

(3.11)

Moreover a related inequality to (3.11) can be derived from (1.12) and (1.13). Indeed,

(3.12)