In search of the invisible spectrum

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 hounds. 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, thf algebra of almost periodic functions, the algebra of absolutely convergent Dirichlet series. Upper and lower estimates for the best majorants anti critical constants are obtained.