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Shifts as models for spectral decomposability on Hilbert spaces. (English) Zbl 1041.47011

Summary: Let \(U\) be a bounded invertible linear mapping of the Hilbert space \({\mathfrak K}\) into itself. Let \({\mathcal W}=\{(U^j)^*U^j\}_{j=-\infty}^\infty\), and denote by \(\ell^2({\mathcal W})\) the corresponding weighted Hilbert space. Our main result shows that the right bilateral shift \({\mathcal R}\) on \(\ell^2({\mathcal W})\) 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 \(\ell^2({\mathcal W}^{(0)})\), but analogues of the classical Marcinkiewicz multiplier theorem and the classical Littlewood-Paley theorem fail to hold on \(\ell^2({\mathcal W}^{(0)})\).

MSC:

47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
42A45 Multipliers in one variable harmonic analysis
46C99 Inner product spaces and their generalizations, Hilbert spaces
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
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