Remarks on an operator Wielandt inequality

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 $Φ$, define $Γ=\left(Φ(X^{*}AY)Φ(Y^{*}AY)^{-1}Φ(Y^{*}AX)\right)^{p}Φ(X^{*}AX)^{-p}, \, \, \, p>0.$ We consider several upper bounds for $\frac{1}{2}|Γ+Γ^{*}|$. These bounds complement a recent result on operator Wielandt inequality.

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