  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, $BX=\lambda
XB$.

Functional Analysis