Numerical Radius and Operator Norm Inequalities
© K. Shebrawi and H. Albadawi. 2009
Received: 4 November 2008
Accepted: 2 March 2009
Published: 8 March 2009
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.
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 .
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  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 .
(see, e.g., ).
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 .
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 ).
The third lemma concerned with positive real numbers, and it is a consequence of the convexity of the function , .
The fourth lemma is a norm inequality for the sum of two operators, which can be found in .
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.
for all .
For every unit vector , we have
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).
For in inequality (2.6), we get the following numerical radius inequalities for sums of operators that generalizes (1.8).
It should be mentioned here that the inequality in (2.11) generalizes (1.7) in the case .
The last equality can be proved using the polar decomposition. In fact, if and are the polar decompositions of and , respectively, then .
These inequalities, among other related ones, can be found in .
For in inequality (2.6), we get the following numerical radius inequalities for products of operators that are related to the above inequalities.
which follows from the the arithmetic-geometric mean inequality for operators (see ). Inequalities (2.23) and (2.24) are not equivalent. This can be seen from the example , , .
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.
We end this section by the following remark.
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.
Inequality (3.1) yields several norm inequalities as special cases. Samples of these inequalities are demonstrated below.
For in inequality (3.2), we get the following operator inequalities for sums of operators.
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.
For in inequality (3.2), we get the following norm inequalities that give an estimate for the usual norm of commutators.
Finally, we end this paper by the following remark.
The authors thank the anonymous referee for his valuable comments and suggestions for improving this paper.
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