Abstract. This article deals with operators that are $p$-hyponormal for every $p>0$ under the name of $\infty $-hyponormal operators. A characterization of such operators is given, and their properties are investigated. Special attention is paid to the spectral property of $\infty $-hyponormal operators having dense ranges and no non-trivial reducing subspaces.

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

Keyword(s): p, -hyponormal operator, spectral resolution, polar decomposition, spectrum, \infty, -hyponormal operator

Received March 1, 2000, and in revised form September 12, 2000. (Registered under 2788/2009.)