HYPERCYCLICITY OF OPERATORS THAT $$\lambda$$-COMMUTE WITH THE HARDY BACKWARD SHIFT

Abstract

AbstractAn operator T acting on a separable complex Banach space $$\mathcal {B}$$ B is said to be hypercyclic if there exists $$f\in \mathcal {B}$$ f ∈ B such that the orbit $$\{T^n f:\ n\in \mathbb {N}\}$$ { T n f : n ∈ N } is dense in $$\mathcal {B}$$ 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 $$\lambda$$ λ -commute with the Hardy backward shift B, that is, $$BX=\lambda XB$$ B X = λ X B .