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Ewing Hall, University of Delaware, Newark, DE 19716, USA
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Title: A Survey of Aleksandrov-Clark Theory and Generalizations

Abstract: We will begin by explaining Aleksandrov-Clark Theory: First note that Beurling’s Theorem says that any non-trivial shift-invariant subspace of the Hardy space $H^2(\mathbb{D})$ is of the form $\theta H^2(\mathbb{D})$ for an inner function $\theta$. Now, for a fixed inner $\theta$, we form the model space, that is, the orthogonal complement of the corresponding shift-invariant subspace in the Hardy space. Consider the compressed shift, which is the application of the shift to functions from the model space followed by the projection to the model space. Clark observed that all rank one perturbations of the compressed shift that are also unitary have a particular, simple form. Following this disc​overy, a rich theory was developed connecting the spectral properties of those unitary rank one perturbations with properties of functions from the model space, more precisely, with their non-tangential boundary values. Some intriguing perturbation results were obtained via complex function theory.

Generalizations we will consider in the mini course include the following. Mo​del spaces can be defined but turn out considerably more complicated when $\theta$ is not inner. Finite rank perturbations were investigated. Parts of the theory have been studied for some several variables settings.​

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