Remarks on an operator Wielandt Inequality

Let A be a positive operator on a Hilbert space H with 0 < m ≤ A ≤ M, and let X and Y be isometries on H such that X*Y = 0, p > 0, and Φ be a 2-positive unital linear map. Define Γ = (Φ(X*AY )Φ(Y*AY )^(−1)Φ(Y*AX)^p Φ(X*AX)^(−p). Several upper bounds for (1/2) |Γ + Γ*| are established. These bounds complement a recent result on the operator Wielandt inequality.