Spectrum of k-quasi-class A_n operators

Abstract

In this paper, we introduce a new class of operators, called $k$-quasi-class $A_n$ operators, which is a superclass of class $A$ and a subclass of $(n,k)$-quasiparanormal operators. We will show basic structural properties and some spectral properties of this class of operators. We show that, if $T$ is of $k$-quasi-class $A_n$ then $T-\lambda$ has finite ascent for all $\lambda\in\c$. Also, we will prove $T$ is polaroid and Weyl's theorem holds for $T$ and $f(T)$, where $f$ is an analytic function in a neighborhood of the spectrum of $T$. Moreover, we show that if $\lambda$ is an isolated point of $\s(T)$ and $E$ is the Riesz idempotent of the spectrum of a $k$-quasi-class $A_n$ operator $T$, then $E\h=\n(T-\lambda)$ if $\lambda\neq 0$ and $E\h=\n(T^{n+1})$ if $\lambda=0$.