On polynomially *-paranormal operators

Abstract

Let T be a bounded linear operator on a complex Hilbert space H. T is called a -paranormal
operator T if kTxk2 kT2xkkxk for all x 2 H. ”-paranormal” is a generalization of hyponormal (TT TT),
and it is known that a -paranormal operator has several interesting properties. In this paper, we prove
that if T is polynomially -paranormal, i.e., there exists a nonconstant polynomial q(z) such that q(T) is
-paranormal, then T is isoloid and the spectral mapping theorem holds for the essential approximate point
spectrum of T. Also, we prove related results.