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
College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China
Received 22 November 2004; Revised 27 April 2005; Accepted 10 May 2005
Copyright © 2006 Changsen Yang and Jiangtao Yuan. 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.
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).