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. |