Property (R) and hypercyclicity for bounded linear operators

Abstract

Let $\mathcal{H}$ be a complex infinite dimensional separable Hilbert space and let $\mathcal{B(H)}$ denote the algebra of bounded linear operators acting on $\mathcal{H}$. In this paper, we mainly characterize these bounded linear operators $T$ on $\mathcal{H}$ and their function calculus that satisfy property $(R)$ by the new spectrum originated from the single-valued extension property. Meanwhile, the relationship between property $(R)$ and hypercyclic property is also explored.