Abstract
A bounded linear operator Ton a Hilbert space H is said to be p-hyponormal for p>0 if (T∗T)p≥(TT∗)p, and T is said to be log-hyponormal if T is invertible and logT∗T≥logTT∗. Firstly, we shall show the following extension of our previous result: If T is p-hyponormal for p∈(0,1], then (Tn∗Tn)(p+1)/n≥⋯≥(T2∗T2)(p+1)/2≥(T∗T)p+1 and (TT∗)p+1≥(T2T2∗)(p+1)/2≥⋯≥(TnTn∗)(p+1)/n hold for all positive integer n. Secondly, we shall discuss the best possibilities of the following parallel result for log-hypponormal operators by Yamazaki: If T is log-hyponormal, then (Tn∗Tn)1/n≥⋯≥(T2∗T2)1/2≥T∗T and TT∗≥(T2T2∗)1/2≥⋯≥(TnTn∗)1/n hold for all positive integer n.