Algebraically paranormal operators on Banach spaces

Abstract

In this paper we show that a bounded linear operator on a Banach space $X$ is polaroid if and only if $p(T)$ is polaroid for some polynomial $p$. Consequently, algebraically paranormal operators defined on Banach spaces are hereditarily polaroid. Weyl type theorems are also established for perturbations $f(T+K)$, where $T$ is algebraically paranormal, $K$ is algebraic and commutes with $T$, and $f$ is an analytic function, defined on an open neighborhood of the spectrum of $T+K$, such that $f$ is nonconstant on each of the components of its domain. These results subsume recent results in this area.