Extensions of the results on powers of -hyponormal and -hyponormal operators

Firstly, we will show the following extension of the results on powers of -hyponormal and -hyponormal operators: let and be positive integers, if is -hyponormal for , then: (i) in case , and hold, (ii) in case , and hold. Secondly, we will show an estimation on powers of -hyponormal operators for which implies the best possibility of our results. Lastly, we will show a parallel estimation on powers of -hyponormal operators as follows: let , then the following hold for each positive integer and : (i) there exists a log-hyponormal operator such that , (ii) there exists a -hyponormal operator such that .