Operator inequalities related to the Corach--Porta--Recht inequality

TL;DR

This paper refines and extends operator inequalities related to the Corach--Porta--Recht inequality in complex Hilbert spaces, introducing new variants and characterizations of operators satisfying specific bounds.

Contribution

It provides new refinements of existing inequalities, presents several related inequalities, and characterizes operators meeting certain norm conditions.

Findings

Refined inequalities based on Heinz inequality

New variants of the Corach--Porta--Recht inequality

Characterization of operators satisfying specific norm bounds

Abstract

We prove some refinements of an inequality due to X. Zhan in an arbitrary complex Hilbert space by using some results on the Heinz inequality. We present several related inequalities as well as new variants of the Corach--Porta--Recht inequality. We also characterize the class of operators satisfying $\left\Vert SXS^{-1}+S^{-1}XS+kX\right\Vert \geq (k+2)\left\Vert X\right\Vert$ under certain conditions.