  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.
