In search of the invisible spectrum

Nikolski, Nikolai

Annales de l'Institut Fourier, Tome 49 (1999) no. 6, pp. 1925-1998.

Le sujet principal de ce travail est l’aspect quantitatif du phénomène du “spectre invisible” dans les algèbres de Banach commutatives. De ce point de vue, nous étudions certaines algèbres de fonctions, les algèbres de séries entières formelles, ainsi que certaines algèbres d’opérateurs. Cela nous permet de donner une interprétation quantitative du célèbre effet de Wiener, Pitt et Sreider pour les algèbres convolutives des mesures sur les groupes abéliens localement compacts. L’approche développée permet de trouver des majorants explicites (et parfois exacts), dépendant uniquement des bornes inférieures spectrales, pour les résolvantes et des solutions des équations de Bezout d’ordres supérieurs. Nous utilisons ces résultats pour définir et calculer la “moindre enveloppe spectrale” d’un ensemble donné, ainsi que le calcul fonctionnel uniformément continu. Dans ce travail, ce programme est réalisé pour les algèbres suivantes : les algèbres des mesures sur des groupes et semi-groupes abéliens localement compacts; leurs sous-algèbres, comme l’algèbre $ℱL^1\left(G\right)$ des fonctions presque périodiques, l’algèbre des séries de Dirichlet absolument convergentes, etc. Pour toutes ces algèbres, nous trouvons les meilleurs majorants pour les inverses, ainsi que les “constantes critiques” correspondantes.

In this paper, we begin the study of the phenomenon of the “invisible spectrum” for commutative Banach algebras. Function algebras, formal power series and operator algebras will be considered. A quantitative treatment of the famous Wiener-Pitt-Sreider phenomenon for measure algebras on locally compact abelian (LCA) groups is given. Also, our approach includes efficient sharp estimates for resolvents and solutions of higher Bezout equations in terms of their spectral bounds. The smallest “spectral hull” of a given closed set is introduced and studied; it permits the definition of a uniformly bounded functional calculus. In this paper, the program traced above is realized for the following algebras: the measure algebras of LCA groups; the measure algebras of a large class of topological abelian semigroups; their subalgebras - the (semi)group algebra of LCA (semi)groups, the algebra of almost periodic functions, the algebra of absolutely convergent Dirichlet series. Upper and lower estimates for the best majorants and critical constants are obtained.

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