Illinois Journal of Mathematics

Intertwining relations and extended eigenvalues for analytic Toeplitz operators

Paul S. Bourdon and Joel H. Shapiro

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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.

Article information

Illinois J. Math., Volume 52, Number 3 (2008), 1007-1030.

First available in Project Euclid: 1 October 2009

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Zentralblatt MATH identifier

Primary: 37B35: Gradient-like and recurrent behavior; isolated (locally maximal) invariant sets
Secondary: 47B33: Composition operators


Bourdon, Paul S.; Shapiro, Joel H. Intertwining relations and extended eigenvalues for analytic Toeplitz operators. Illinois J. Math. 52 (2008), no. 3, 1007--1030. doi:10.1215/ijm/1254403728.

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