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\(m\)-isometric transformations of Hilbert space. I. (English) Zbl 0836.47008

Summary: A model for operators satisfying the equation \(\sum_{k=0}^m (-1)^m {m\choose k} (T^* )^{m-k} T^{m-k}=0\) is given as multiplication by \(e^{i\varphi}\) on a Hilbert space whose inner product is defined in terms of periodic distributions and we relate this model theory for the case when \(m=2\) to a disconjugacy theory for a subclass of Toeplitz operators of the type studied by Boutet de Monvel and Guilliman, classical function theoretic ideas on the Dirichlet space, and the theory of nonstationary stochastic processes. This is presented in a series of three papers. In this first paper, we concentrate on a model for these \(T\).

MSC:

47A45 Canonical models for contractions and nonselfadjoint linear operators
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
46E20 Hilbert spaces of continuous, differentiable or analytic functions
46F05 Topological linear spaces of test functions, distributions and ultradistributions
46F10 Operations with distributions and generalized functions
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