Property $(UW_{\Pi })$ for Functions of Operators and Compact Perturbations

Abstract

Let ${\mathcal {H}}$ be a complex infinite dimensional Hilbert space and $\mathcal {B(H)}$ be the algebra of all bounded linear operators on ${\mathcal {H}}$. $T\in \mathcal {B(H)}$ is said to satisfy property $(UW_{\Pi })$ if $\sigma _{a}(T)\setminus \sigma _{ea}(T)=\sigma (T)\setminus \sigma _{D}(T)$, where $\sigma _{a}(T)$, $\sigma _{ea}(T)$, $\sigma (T)$ and $\sigma _{D}(T)$ denote the approximate point spectrum, the Weyl essential approximate point spectrum, the spectrum and the Drazin spectrum of T, respectively. In this paper, we talk about the property $(UW_{\Pi })$ for functions of operators. Also, the compact perturbations of property $(UW_{\Pi })$ are also discussed.