Almost isometric operators: Functional model, invariant subspaces, commutant
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A new function model for an arbitrary bounded operator on a Hilbert space is constructed. This model generalizes the model of Sz.-Nagy and Foiaş for contractions and seems to be useful for operators close to an isometry (in a sense). All the model spaces are Hilbert spaces, but instead of dilation a generalization of it is used. The model admits a symmetry with respect to the map z→1/z of the complex plane. In terms of the model the question of lifting the commutant is investigated, a relationship between invariant subspaces of a unitary operator is established, and the characteristic function of the model operator is calculated. Some other problems are solved as well. Bibliography: 8 titles.
KeywordsHilbert Space Unitary Operator Invariant Subspace Functional Model Invertible Operator
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