Conni Liaw: Aleksandrov-Clark Theory and its generalizations

Speaker
Conni Liaw, University of Delaware

Abstract
We will begin by recalling the origination of Aleksandrov-Clark Theory: First note that Beurling’s Theorem says that any 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 discovery, 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, some of which we will consider, include the following. Model spaces can be defined but turn out considerably more complicated when \(\theta\) is not inner. Finite rank perturbations were investigated. A generalization to the non-commutative setting has been formulated.