For aw-hyponormal operatorTacting on a separable complex Hilbert
spaceH, we prove that: 1) the quasi-nilpotent partHo(T - λI ) is equal toKer(T-λI); 2)T has Bishop’s property<i>β</i>; 3) ifσw(T)={0}, then it is a compact normal operator; 4) IfTis
an algebraicallyw-hyponormal operator, then it is polaroid and reguloid. Among
other things, we prove that ifTn andTn* arew-hyponormal, thenT is normal.
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