Similarity of Cowen-Douglas operators to the backward Dirichlet shift

TL;DR

This paper extends the similarity characterization of Cowen-Douglas operators from the backward shift on reproducing kernel Hilbert spaces to the Dirichlet space, using a model theorem for eigenvector bundle structure.

Contribution

It generalizes existing similarity criteria to the Dirichlet space setting, providing a unified approach for Cowen-Douglas operators.

Findings

Similarity characterization applies to Dirichlet space operators

Model theorem crucial for eigenvector bundle analysis

Unified framework for Cowen-Douglas operators

Abstract

We show that the same similarity characterization obtained for Cowen-Douglas operators to the backward shift operators on reproducing kernel Hilbert spaces with analytic kernels can be used to describe similarity in the Dirichlet space setting. As in previous proofs, a model theorem that allows one to get the eigenvector bundle structure of the operator plays a crucial role.