Journal of Inequalities and Applications 
Volume 2006 (2006), Article ID 36919, 14 pages
doi:10.1155/JIA/2006/36919

Extensions of the results on powers of p-hyponormal and log⁡-hyponormal operators

Changsen Yang and Jiangtao Yuan

 College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China
Received 22 November 2004; Revised 27 April 2005; Accepted 10 May 2005

Abstract

Firstly, we will show the following extension of the results on powers of p-hyponormal and log⁡-hyponormal operators: let n and m be positive integers, if T is p-hyponormal for p∈(0,2], then: (i) in case m≥p,(Tn+m∗Tn+m)(n+p)/(n+m)≥(Tn∗Tn)(n+p)/n and (TnTn∗)(n+p)/n≥(Tn+mTn+m∗)(n+p)/(n+m) hold, (ii) in case m<p,Tn+m∗Tn+m≥(Tn∗Tn)(n+m)/n and (TnTn∗)(n+m)/n≥Tn+mTn+m∗ hold. Secondly, we will show an estimation on powers of p-hyponormal operators for p>0 which implies the best possibility of our results. Lastly, we will show a parallel estimation on powers of log⁡-hyponormal operators as follows: let α>1, then the following hold for each positive integer n and m: (i) there exists a log-hyponormal operator T such that (Tn+m∗Tn+m)nα/(n+m)≱(Tn∗Tn)α , (ii) there exists a log⁡-hyponormal operator T such that (TnTn∗)α≱(Tn+mTn+m∗)nα/(n+m).