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An operator $T$ acting on a separable complex Hilbert space $H$ is said to behypercyclic 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} characterizedthose elements in the commutant of the Hardy backward shift which arehypercyclic. In this paper we study some dynamics properties of operators $X$that $\lambda$-commute with the Hardy backward shift $B$, that is, $BX=\lambdaXB$.

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