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Journal of Operator Theory

Volume 50, Issue 1, Summer 2003  pp. 77-106.

Shifts as models for spectral decomposability on Hilbert space

Authors:  Earl Berkson (1) and T.A. Gillespie (2)
Author institution: (1) Department of Mathematics, University of Illinois, 1409 W. Green St., Urbana, IL 61801, USA
(2) Department of Mathematics and Statistics, University of Edinburgh, James Clerk Maxwell Building, Edinburgh EH9 3JZ, Scotland, UK

Summary:  Let $U$ be a bounded invertible linear mapping of the Hilbert space ${\frak K} $ onto itself. Let ${\cal W} = \{(U^{j})^{*}U^{j}\}_{j=-\infty }^{\infty }$, and denote by $\ell^{2}({\cal W})$ the corresponding weighted Hilbert space. Our main result shows that the right bilateral shift ${\cal R}$ on $\ell^{2}({\cal W})$ serves as a model for spectral decomposability of $U$. Further aspects of this for multiplier transference are treated, and lead to an example wherein the discrete Hilbert kernel defines a bounded convolution operator on $\ell^{2}({\cal W}^{(0)})$, but the analogues of the classical Marcinkiewicz Multiplier Theorem and the classical Littlewood-Paley Theorem fail to hold for $\ell ^{2}({\cal W}^{(0)}).$

Keywords:  Operator, Hilbert space, spectral family, operator-valued weight sequence, shift

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