Remarks on an operator Wielandt inequality

TL;DR

This paper investigates new upper bounds for a specific operator expression related to the Wielandt inequality, extending recent results in operator theory with implications for positive operators and linear maps.

Contribution

It introduces several novel upper bounds for an operator expression involving 2-positive linear maps, complementing recent advances in the Wielandt inequality.

Findings

Established new upper bounds for the operator expression rac12|\u03b3+\u03b3^*|.

Extended the understanding of inequalities involving positive operators and isometries.

Complemented recent results on the operator Wielandt inequality.

Abstract

Let $A$ be a positive operator on a Hilbert space $\mathcal{H}$ with $0<m\leq A\leq M$ and $X$ and $Y$ are two isometries on $\mathcal{H}$ such that $X^{*}Y=0$. For every 2-positive linear map $\Phi$, define $$\Gamma=\left(\Phi(X^{*}AY)\Phi(Y^{*}AY)^{-1}\Phi(Y^{*}AX)\right)^{p}\Phi(X^{*}AX)^{-p}, \, \, \, p>0.$$ We consider several upper bounds for $\frac{1}{2}|\Gamma+\Gamma^{*}|$. These bounds complement a recent result on operator Wielandt inequality.