Remarks on an operator Wielandt inequality

Abstract

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

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