On k-quasi-$*$-paranormal operators

Abstract

For a positive integer k,  an operator $T\in \mathscr {B}(\mathscr {H})$ is called k-quasi-$*$-paranormal if $\Vert T^*T^kx\Vert ^2\le \Vert T^kx\Vert \Vert T^{k+2}x\Vert$ for all $x\in \mathscr {H}$, which is a common generalization of $*$-paranormal and quasi-$*$-paranormal. In this paper, firstly we prove some inequalities of this class of operators; secondly we give a necessary and sufficient condition for T to be k-quasi-$*$-paranormal. Using these results, we prove that: (1) if $\Vert T^*T^n\Vert =\Vert T^n\Vert \Vert T\Vert$ for some positive integer $n\ge k,$ then a k-quasi-$*$-paranormal operator T is normaloid; (2) if E is the Riesz idempotent for an isolated point $\mu _0$ of the spectrum of a k-quasi-$*$-paranormal operator T, then (i) if $\mu _0\ne 0$, then $\mathscr {R}(E)=\ker (T-\mu _0)$ (ii) if $\mu _0=0$, then $\mathscr {R}(E)=\ker (T^k)$.