On powers of class A (k) operators including p-hyponormal and log-hyponormal operators

In [11], we introduced class A (k) as a class of operators including p-hyponormal and log-hyponormal operators. In this paper, we shall show that "if T is an invertible class A(k) operator for k is an element of (0, 1], then T-n is a class A (k/n) operator for all positive integer n." Moreover, we shall show a similar result on powers of class AI(s, t) operators which were introduced in [7] as extensions of class A (k) operators, that is, "if T is a class AI(s, t) operator for s, t is an element of (0, 1], then T-n is a class AI(s/n, t/n) operator for all positive integer n.".