Property $(R)$ for bounded linear operators and its perturbation

Abstract

Let $\mathcal{H}$ be a complex infinite dimensional Hilbert space, and $\mathcal{B(H)}$ denotes the algebra of bounded linear operators acting on $\mathcal{H}$. For $T\in\mathcal{B(H)}$, we say $T$ satisfy property $(R)$ if $\sigma_{a}(T)\backslash\sigma_{ab}(T)=\pi_{00}(T)$, where $\sigma_{a}(T)$ and $\sigma_{ab}(T)$ denote the approximate point spectrum and the Browder essential approximate point spectrum of $T$, respectively, and $\pi_{00}(T)$ denotes the set of all finite dimensional isolated eigenvalues. In this paper, we introduce a new judgment method for bounded linear operators and their function calculus satisfying property $(R)$. Moreover, the perturbation of property $(R)$ is explored.