zbMATH — the first resource for mathematics

Contractions with countable spectra and unicity properties of closed countable sets of the circle. (Contractions à spectre dénombrable et propriétés d’unicité des fermés dénombrables du cercle.) (French) Zbl 0766.47002
On montre que si \(T\) est une contraction à spectre dénombrable et telle que, pour tout \(\varepsilon>0\) \(\| T^{- 1}\|=O(e^{\varepsilon n^{1/2}})\) \((n\to+\infty)\), alors \(T\) est une isometrie. On montre aussi que ce résultat est lié à une propriété d’unicité forte des fermés dénombrable du cercle unité.
Reviewer: M.Zarrabi

47A30 Norms (inequalities, more than one norm, etc.) of linear operators
42A63 Uniqueness of trigonometric expansions, uniqueness of Fourier expansions, Riemann theory, localization
43A45 Spectral synthesis on groups, semigroups, etc.
Full Text: DOI Numdam EuDML
[1] G.R. ALLAN, T.J. RANSFORD, Power dominated elements in Banach algebras, Studia Math., 94 (1989), 63-79. · Zbl 0705.46021
[2] H.H. SCHAEFER, M. WOLFF and W. ARENDT, On lattice isomorphisms with positive real spectrum and groups of positive operators, Math. Z., 164 (1978), 115-123. · Zbl 0377.47026
[3] A. ATZMON, Operators which are annihilated by analytic functions and invariant subspaces, Acta Math., 144 (1980), 27-63. · Zbl 0449.47007
[4] J. ESTERLE, E. STROUSE, F. ZOUAKIA, Theorems of katznelson tzafriri type for contractions, à paraître au Journal of Functional Analysis. · Zbl 0723.47013
[5] J.P. KAHANE, Y. KATZNELSON, Sur LES algèbres de restrictions de séries de Taylor absolument convergentes à un fermé du cercle, J. Analyse Math., 23 (1970), 185-197. · Zbl 0228.43002
[6] J.P. KAHANE, R. SALEM, Ensembles parfaits et séries trigonométriques, Paris, Hermann, 1963. · Zbl 0112.29304
[7] Y. KATZNELSON, An introduction to harmonic analysis, Wiley, New York, 1968. · Zbl 0169.17902
[8] M. ZARRABI, Synthèse spectrale dans certaines algèbres de Beurling sur le cercle unité, Bull. Soc. Math. France, 118 (1990), 241-249. · Zbl 0751.43006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.