Communications of the
Korean Mathematical Society CKMS

A Hilbert space operator $T\in\bh$ is said to be $n$-$\ast$-paranormal, $T\in\sP(n)$ for short, if $\norm{T^*x}^n\leq \norm{T^nx}\norm{x}^{n-1}$ for all $x\in\h$. We proved some properties of class $\sP(n)$ and we proved an asymmetric Putnam-Fuglede theorem for $n$-$\ast$-paranormal. Also, we study some invariants of Weyl type theorems. Moreover, we will prove that a class $n$-$\ast$ paranormal operator is finite and it remains invariant under compact perturbation and some orthogonality results will be given.

Keywords: $\ast$-paranormal operators, $n$-$\ast$-paranormal operators, finite operators, Fuglede-Putnam theorem, Weyl theorem

MSC numbers: Primary 47A10, 47A12, 47B20