Hypercyclicity of operators that λ-commute with the hardy backward shift

Abstract

An operator T acting on a separable complex Banach space B is said to be hypercyclic if there exists f ∈ B such that the orbit {Tnf ∶ n ∈ ℕ} is dense in B. Godefroy and Shapiro (J. Funct. Anal., 98(2):229–269, 1991) characterized those elements, which are hypercyclic, in the commutant of the Hardy backward shift. In this paper, we study some dynamical properties of operators X that 휆-commute with the Hardy backward shift B, that is, BX = 휆XB.