Cyclic $m$-isometries, and Dirichlet type spaces

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.