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Norm Inequalities for Commutators of Self-adjoint Operators

Authors:

Fuad Kittaneh

Institution:

(1) Department of Mathematics, University of Jordan, Amman, Jordan

Abstract:

Let A, B, and X be bounded linear operators on a complex separable Hilbert space. It is shown that if A and B are self-adjoint with $a_{1} \leq A \leq a_{2}$ and $b_{1} \leq B \leq b_{2}$ for some real numbers a ₁, a ₂, b ₁, and b ₂, then for every unitarily invariant norm|||·|||,

$|||AX - XB||| \leq {\rm max}(a_2 - b_1, b_2 - a_1) |||X|||$

. If, in addition, X is positive, then

$|||AX - XA||| \leq \frac{1}{2} (a_2 - a_1) |||X \oplus X|||$

. These norm inequalities generalize recent related inequalities due to Kittaneh, Bhatia-Kittaneh, and Wang-Du.

Keywords:

" target="_blank"> Commutator normal operator self-adjoint operator positive operator unitarily invariant norm norm inequality

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