Weyl's theorem for algebraically absolute-(p,r)-paranormal operators

Abstract

An operator $T \in B(H)$ is said to be absolute-$(p, r)$-paranormal if $\| |T|^{p} |T^{*}|^{r} x \|^{r} \|x\| \geq \| |T^{*}|^{r} x\|^{p + r}$ for all $x \in H$ and for positive real number $p > 0$ and $r > 0$, where $T=U |T|$ is the polar decomposition of $T$. In this paper, we discuss some properties of absolute-$(p, r)$-paranormal operators and show that Weyl's theorem holds for algebraically absolute-$(p, r)$-paranormal operators.