Abstract. The authors [5] called a bounded linear operator $T$ $\infty $-hyponormal if $T$ is $p$-hyponormal for every $p>0$. They investigated the spectral properties of a pure $\infty $-hyponormal operator $T$ under the condition that $T$ has dense range and has no nontrivial reducing subspace. In this paper it is shown that these properties of a pure $\infty $-hyponormal operator $T$ still hold without this condition.

AMS Subject Classification (1991): 47B20, 47A10

Received April 24, 2002, and in revised form January 21, 2003. (Registered under 2936/2009.)