Hypercyclicity of operators that $λ$-commute with the Hardy backward shift

Abstract: An operator $T$ acting on a separable complex Hilbert space $H$ is said to be hypercyclic if there exists $f\in H$ such that the orbit ${T^n f:\ n\in \mathbb{N}}$ is dense in $H$. Godefroy and Shapiro \cite{GoSha} characterized those elements in the commutant of the Hardy backward shift which are hypercyclic. In this paper we study some dynamics properties of operators $X$ that $\lambda$-commute with the Hardy backward shift $B$, that is, $BX=\lambda XB$.