In search of the invisible spectrum
type: Item
instance of: scholarly article
title: In search of the invisible spectrum
zbMATH Open document ID: 0947.46035
author: N. K. Nikol'skij
publication date: 1999-11-24T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75407
review text: This long paper has 6 chapters. 0. Introduction. 1. Preliminaries on \((\delta-n)\)-visibility, best majorants, and critical constants, 2. Upper estimates for measure algebras on groups and semigroups, 3. A lower estimate for measure algebras on groups and semigroups, 4. Remarks on finite groups and semigroups, 5. Spectral hulls and norm-controlled functional calculi. The author's abstract: 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 Bézout 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)group, 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.
Mathematics Subject Classification ID: 46J20
Mathematics Subject Classification ID: 43A10
Mathematics Subject Classification ID: 46H30
Mathematics Subject Classification ID: 47A25
zbMATH DE Number: 1368754
zbMATH Keywords: invisible spectrum
zbMATH Keywords: LCA groups
zbMATH Keywords: measure algebra
zbMATH Keywords: Wiener-Pitt-Sreider phenomenon
zbMATH Keywords: norm controlled inversion
zbMATH Keywords: cyclic groups
zbMATH Keywords: spectral hulls
zbMATH Keywords: function algebras
zbMATH Keywords: norm-controlled functional calculi
zbMATH Keywords: commutative Banach algebras
zbMATH Keywords: formal power series
zbMATH Keywords: operator algebras
zbMATH Keywords: Bézout equations
zbMATH Keywords: algebra of almost periodic functions
zbMATH Keywords: algebra of absolutely convergent Dirichlet series
zbMATH Keywords: best majorants
zbMATH Keywords: critical constants
reviewed by: Takahiko Nakazi
MaRDI profile type: Publication
OpenAlex ID: W2330241241
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DOI: 10.5802/AIF.1743
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published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1368754