Completely hyperexpansive operators with finite rank defect operator and de Branges-Rovnyak spaces

Abstract

The process of identifying a Dirichlet-type space $D(\hat{I}¼)$ for a positive, Borel measure $\hat{I}¼$, supported on the unit circle $\mathbb{T},$ with a de Branges-Rovnyak space was initiated by Sarason. A characterization of the symbol for a de Branges-Rovnyak spaces for which the shift operator is a $2$-isometry, was provided in an article by Kellay and Zarrabi. In this paper, capitalizing on the Aleman's model for the cyclic, analytic, completely hyperexpansive operators, we provide a characterization of cyclic, analytic, completely hyperexpansive operator with finite rank defect operator in terms of the symbol for a de Branges-Rovnyak space.