ANALYTIC m-ISOMETRIES AND WEIGHTED DIRICHLET-TYPE SPACES

Corresponding to any ( m-1)-tuple of semispectral measures on the unit circle, a weighted Dirichlet-type space is introduced and studied. We prove that every analytic m-isometry which satisfies a certain set of operator inequalities can be represented as the operator of multiplication by the coordinate function on such a weighted Dirichlet-type space. This extends a result of Richter as well as of Olofsson on analytic 2-isometries. We also prove that all left invertible m-concave operators satisfying the aforementioned operator inequalities admit a Wold-type decomposition. This result serves as a key ingredient in our model theorem and it also generalizes a result of Shimorin on a class of 3=concave operators.