A note on property $(UW_{\Pi})$

Abstract

Let $\mathcal{H}$ be a complex infinite dimensional Hilbert space.
For $T\in \mathcal{B(H)}$, $T$ is said to satisfy property $(UW_{\Pi})$ if the complement in the approximate point spectrum of the Weyl essential approximate point spectrum coincides with the poles of $T$. In this paper, we deeply talk about the property $(UW_{\Pi})$ under some perturbations and property $(UW_{\Pi})$ for functions of operators. In addition, if $T$ is Drazin invertible, then property $(UW_{\Pi})$ for functions of $T$ can be transmitted to the functions of its Drazin inverse.