Unitarily invariant norm inequalities for elementary operators involving $G_{1}$ operators

TL;DR

This paper establishes new unitarily invariant norm inequalities for elementary operators involving G1 operators, providing bounds related to perturbation theory and analytic functions, with applications to Schatten norms.

Contribution

It introduces novel upper bounds for operator expressions involving G1 operators and analytic functions, extending perturbation theory results.

Findings

Derived bounds for $|||f(A)Xg(B)+ X|||$ and $|||f(A)Xg(B)- X|||$

Established new upper bounds for Schatten 2-norm of $f(A)X \pm Xg(B)$

Discussed special cases illustrating the inequalities

Abstract

In this paper, motivated by perturbation theory of operators, we present some upper bounds for $|||f(A)Xg(B)+ X|||$ in terms of $|||\,|AXB|+|X|\,|||$ and $|||f(A)Xg(B)- X|||$ in terms of $|||\,|AX|+|XB|\,|||$, where $A, B$ are $G_{1}$ operators, $|||\cdot|||$ is a unitarily invariant norm and $f, g$ are certain analytic functions. Further, we find some new upper bounds for the the Schatten $2$-norm of $f(A)X\pm Xg(B)$. Several special cases are discussed as well.