On weakly unitarily invariant norm and the λ-Aluthge transformation for invertible operator

Let T∈B(H) be an invertible operator with polar decomposition T = UP and B∈B(H) commute with T. In this paper we prove that ∣∣∣PλBUP1−λ∣∣∣ ⩽ ∣∣∣BT∣∣∣, where ∣∣∣ · ∣∣∣ is a weakly unitarily invariant norm on B(H) and 0 ⩽ λ ⩽ 1. As the consequence of this result, we have ∣∣∣f(PλUP1−λ)∣∣∣ ⩽ ∣∣∣f(T)∣∣∣ for any polynomial f.