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 $X^{\ast }Y=0$. For every 2-positive linear map $\mathrm{\Phi }$, define $$\mathrm{\Gamma }={(\mathrm{\Phi }(X^{\ast }AY)\mathrm{\Phi }(Y^{\ast }AY{)}^{-1}\mathrm{\Phi }(Y^{\ast }AX))}^p\mathrm{\Phi }(X^{\ast }AX{)}^{-p},p>0.$$ We consider several upper bounds for $\frac{1}{2}\mathrm{\mid }\mathrm{\Gamma }+{\mathrm{\Gamma }}^{\ast }\mathrm{\mid }$. These bounds complement a recent result on operator Wielandt inequality.