In search of the invisible spectrum
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.
algebra of absolutely convergent Dirichlet series
scientific article; zbMATH DE number 1368754
scientific article