Journal of Operator Theory

Volume 88, Issue 2, Fall 2022 pp. 445-477.

Analytic $m$-isometries and weighted Dirichlet-type spaces

Authors: Soumitra Ghara (1), Rajeev Gupta (2), Md. Ramiz Reza (3)
Author institution:(1) Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, India
(2) School of Mathematics and Computer Science, Indian Institute of Technology Goa, India
(3) Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, India

Summary: 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.

DOI: http://dx.doi.org/10.7900/jot.2021mar26.2335
Keywords: $m$-Isometry, $m$-concave operators, wandering subspace property, Wold-type decomposition, Dirichlet-type spaces

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