Unitarily invariant norm inequalities involving $G_1$ operators

TL;DR

This paper derives new upper bounds for unitarily invariant norms, especially the Hilbert-Schmidt norm, involving $G_1$ operators and functions analytic on the unit disk, extending inequalities for Hermitian matrices.

Contribution

It introduces novel upper bounds for unitarily invariant norms involving $G_1$ operators and analytic functions, generalizing existing inequalities for Hermitian matrices.

Findings

Established upper bounds for Hilbert-Schmidt norm involving $A, B, X$

Extended inequalities to functions with positive real part on the unit disk

Provided bounds applicable to Hermitian matrices with spectra in the unit disk

Abstract

In this paper, we present some upper bounds for unitarily invariant norms inequalities. Among other inequalities, we show some upper bounds for the Hilbert-Schmidt norm. In particular, we prove \begin{align*} \|f(A)Xg(B)\pm g(B)Xf(A)\|_2\leq \left\|\frac{(I+|A|)X(I+|B|)+(I+|B|)X(I+|A|)}{d_Ad_B}\right\|_2, \end{align*} where $A, B, X\in\mathbb{M}_n$ such that $A$, $B$ are Hermitian with $\sigma (A)\cup\sigma(B)\subset\mathbb{D}$ and $f, g$ are analytic on the complex unit disk $\mathbb{{D}}$, $g(0)=f(0)=1$, $\textrm{Re}(f)>0$ and $\textrm{Re}(g)>0$.