On weakly unitarily invariant norm and the Aluthge transformation

In this paper we show that where , • is a semi-norm on which satisfies some conditions, T=UP (polar decomposition), 0 λ 1 and f is a polynomial. As a consequence of this fact, we will show that some semi-norms • including the ρ-radii (0<ρ 2) satisfy the inequality f(PλUP1−λ) f(T) . We also give some related results.