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

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