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- W2024561452 abstract "For a generalized Toeplitz operator acting on a Hilbert space with unilateral shift, necessary and sufficient conditions are obtained for the existence of a nontrivial unitary part, and an explicit description of this unitary part is given. Let T be a contractive linear operator on a separable complex Hilbert space X. Such an operator is said to be completely nonunitary if it has no nontrivial reducing subspace N such that the restriction TIN of T to N is unitary. It is known [2, Theorem 1.3.2] that for any contraction T on X we can find a unique orthogonal decomposition X = M D M1 such that M and M1 reduce T, TIM is unitary and TIM1 is completely nonunitary. It is not excluded that M or M1 is possibly the subspace {O}. Furthermore, M is given by M = {x E X: 11Tnxll = llxll = 11T*nxll, n = 1, 2, . . . } and is called the unitary subspace of T. TI M is called the unitary part of T. In this paper, we describe explicitly the above decomposition for the class of generalized Toeplitz operators, which are obtained as follows. We say that an isometry S: X -* X is a unilateral shift if there exists a subspace C in X for which (SJC) I(SkC) for nonnegative integers j # k, and" @default.
- W2024561452 created "2016-06-24" @default.
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- W2024561452 date "1977-01-01" @default.
- W2024561452 modified "2023-09-26" @default.
- W2024561452 title "Unitary parts of generalized Toeplitz operators" @default.
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- W2024561452 doi "https://doi.org/10.1090/s0002-9939-1977-0454695-4" @default.
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