Cyclic $m$-isometries, and Dirichlet type spaces

TL;DR

This paper characterizes cyclic m-isometries on Hilbert spaces using shifts on weighted Dirichlet type spaces, providing new models and examples, including unbounded cases, with spaces contained in Hardy space subspaces.

Contribution

It introduces a novel model for cyclic m-isometries via Dirichlet integral-based spaces, extending previous work and enabling the construction of unbounded operators.

Findings

Model spaces are contained in specific Hardy space subspaces.

Constructed a variety of examples of m-isometries.

Demonstrated the framework's ability to produce unbounded m-isometries.

Abstract

We consider cyclic $m$-isometries on a complex separable Hilbert space. Such operators are characterized in terms of shifts on abstract spaces of weighted Dirichlet type. Our results resemble those of Agler and Stankus, but our model spaces are described in terms of Dirichlet integrals rather than analytic Dirichlet operators. The chosen point of view allows us to construct a variety of examples. An interesting feature among all of these is that the corresponding model spaces are contained in a certain subspace of the Hardy space $H^2$, depending only on the order of the corresponding operator. We also demonstrate how our framework allows for the construction of unbounded $m$-isometries.