EXTENSIONS OF THE RESULTS ON POWERS OFp-HYPONORMAL AND log-HYPONORMAL OPERATORS

Firstly, we will show the following extension of the results on powers of p-hyponormal and log-hyponormal operators: let n andm be positive integers, if T is p-hyponormal for p ∈ (0,2], then: (i) in case m ≥ p, (Tn+mTn+m)(n+p)/(n+m) ≥ (TnTn)(n+p)/n and (TnTn ∗ )(n+p)/n ≥ (Tn+mTn+m)(n+p)/(n+m) hold, (ii) in case m < p, Tn+mTn+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) ≥ (TnTn)α, (ii) there exists a log-hyponormal operator T such that (TnTn ∗ )α ≥ (Tn+mTn+m)nα/(n+m).