Journal of Inequalities and Applications
Volume 5 (2000), Issue 4, Pages 367-380

On powers of p-hyponormal and log-hyponormal operators

Takayuki Furuta and Masahiro Yanagida

Department of Applied Mathematics, Faculty of Science, Science University of Tokyo, 1-3 Kagurazaka, Shinjuku, Tokyo 162-8601, Japan

Received 17 May 1999; Revised 14 July 1999

Copyright © 2000 Takayuki Furuta and Masahiro Yanagida. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A bounded linear operator Ton a Hilbert space H is said to be p-hyponormal for p>0 if (TT)p(TT)p, and T is said to be log-hyponormal if T is invertible and logTTlogTT. Firstly, we shall show the following extension of our previous result: If T is p-hyponormal for p(0,1], then (TnTn)(p+1)/n(T2T2)(p+1)/2(TT)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 (TnTn)1/n(T2T2)1/2TT and TT(T2T2)1/2(TnTn)1/n hold for all positive integer n.