Lectures on the shift operator. III
Copyright policy )Lectures on the shift operator. III
This is the continuation of the papers I-III published in Vols. 39, 47, and 56 of the same “Zapiski.” One considers the shift operator f → zf in the spaces of (generalized) functions on the circumference T and their spectral subspaces. If X is such a space and e⊂T, then Open image in new window . What is the class of all X-negligible sets e, i.e., such that Xe={O} ? This is the wellknown question in harmonic analysis regarding sets of uniqueness. We prove known theorems about such sets (due to O. Frostman, D. Newman, and Y. Katznelson) as well as new ones. Among these: if $$X = \left\{ {f.\sum\limits_{n \in \mathbb{Z}} {\left| {\hat f(n)} \right|^p } \log ^{ - \beta p} (\left| n \right| + 1)< \infty } \right\}, 0 = \beta< 1 \leqslant p< \frac{2}{{1 + \beta }}$$ , then there exists a set e with Xe={O} for which the Lebesgue measure of Te is arbitrarily small.
Shift Operator, shift operator, Research exposition (monographs, survey articles) pertaining to operator theory, Invariant subspaces of linear operators, truncated shift operator, Canonical models for contractions and nonselfadjoint linear operators, unicellularity, spectral subspaces, Spectral synthesis on groups, semigroups, etc., Banach algebras of differentiable or analytic functions, \(H^p\)-spaces, Linear operators on function spaces (general), Toeplitz operators, Hankel operators, Wiener-Hopf operators, Hermitian and normal operators (spectral measures, functional calculus, etc.), Spectrum, resolvent, semigroups of shifts, (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces, Subnormal operators, hyponormal operators, etc., sets of uniqueness
Shift Operator, shift operator, Research exposition (monographs, survey articles) pertaining to operator theory, Invariant subspaces of linear operators, truncated shift operator, Canonical models for contractions and nonselfadjoint linear operators, unicellularity, spectral subspaces, Spectral synthesis on groups, semigroups, etc., Banach algebras of differentiable or analytic functions, \(H^p\)-spaces, Linear operators on function spaces (general), Toeplitz operators, Hankel operators, Wiener-Hopf operators, Hermitian and normal operators (spectral measures, functional calculus, etc.), Spectrum, resolvent, semigroups of shifts, (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces, Subnormal operators, hyponormal operators, etc., sets of uniqueness
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