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.
Highlights
Let H be a complex Hilbert space with inner product ·, ·, and let B H denote the C∗algebra of all bounded linear operators on H
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 H is a norm on the ideal C|||·||| of B H, satisfying |||UAV ||| |||A||| for all A ∈ B H and all unitary operators U and V in B H
Summary
Let H be a complex Hilbert space with inner product ·, · , and let B H denote the C∗algebra of all bounded linear operators on H. The most familiar example of weakly unitarily invariant norm is the numerical radius w A , defined by w A sup |λ| : λ ∈ W A. 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. A general numerical radius inequality has been proved by Kittaneh, it has been shown in 6 that if A, B, C, D, S, T ∈ B H , w AT B. 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.
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