Some generalized numerical radius inequalities for Hilbert space operators

TL;DR

This paper introduces new generalized inequalities for the numerical radius of products of Hilbert space operators, including positive operators and Heinz means, expanding the theoretical understanding of operator bounds.

Contribution

It presents novel inequalities involving powers of the numerical radius for products of positive operators and Heinz means, generalizing existing bounds.

Findings

Established inequalities for $A^ ext{alpha} X B^ ext{alpha}$ and $A^ ext{alpha} X B^{1- ext{alpha}}$

Extended numerical radius bounds to Heinz means

Provided conditions under which these inequalities hold

Abstract

We generalize several inequalities involving powers of the numerical radius for product of two operators acting on a Hilbert space. For any $A, B, X\in \mathbb{B}(\mathscr{H})$ such that $A,B$ are positive, we establish some numerical radius inequalities for $A^\alpha XB^\alpha$ and $A^\alpha X B^{1-\alpha}\,\,(0 \leq \alpha \leq 1)$ and Heinz means under mild conditions.