Different exponential spectra in Banach algebras
The exponential spectrum of a Banach algebra element was introduced by \textit{R. Harte} [Proc. Am. Math. Soc. 58, 114-118 (1976; Zbl 0338.46043)]. Let \(A\) and \(B\) be complex Banach algebras with unit 1. The connected component of 1 in the group of invertible elements is denoted by Exp \(A\). First the relationship between homomorphisms between \(A\) and \(B\) and Exp is examined. Then it is investigated how the exponential spectrum of \(b \in B \subset A\) depends on the algebra if \(B\) is merely a subalgebra of \(A\) with the same unit and also in the case where \(B\) has a unit different from that in \(A\). After a study of the exponential spectrum in quotient algebras, the behaviour of exponential spectrum under perturbation by radical elements, rank one elements and Riesz elements is examined. Finally it is indicated briefly how the exponential spectrum shares many analytic properties with the ordinary spectrum.
scientific article; zbMATH DE number 1439643
scientific article