Abstract. It is known that any positive integer power of an $\omega $-hyponormal operator is $\omega $-hyponormal. In this note we show that, for any $0< p\leq1$, there exists an invertible operator whose integer powers are all $p-\omega $-hyponormal. We also show that there exists a ${1\over2}-\omega $-hyponormal operator $T$ such that $T^3$ is not ${1\over2}-\omega $-hyponormal.

AMS Subject Classification (1991): 47B20, 47A63

Keyword(s): p-$\omega$, $\omega$-hyponormal, Furuta inequality

Received June 26, 2003. (Registered under 5874/2009.)