Intertwining relations and extended eigenvalues for analytic Toeplitz operators

Abstract

We study the intertwining relation $XT_\varphi=T_\psi X$ where $T_\varphi $ and $T_\psi$ are the Toeplitz operators induced on the Hardy space $H^2$ by analytic functions $\varphi$ and $\psi$, bounded on the open unit disc~$\mathbb{U}$, and $X$ is a nonzero bounded linear operator on $H^2$. Our work centers on the connection between intertwining and the image containment $\psi(\mathbb{U})\subset\varphi (\mathbb{U})$, as well as on the nature of the intertwining operator $X$. We use our results to study the ``extended eigenvalues'' of analytic Toeplitz operators $T_\varphi$, i.e., the special case $XT_{\lambda\varphi}=T_\varphi X$, where $\lambda$ is a complex number.