On k-quasi- $$*$$ ∗ -paranormal operators

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 \(\mathscr {R}(E)=\ker (T^k)\).