Different exponential spectra in Banach algebras
type: Item
instance of: scholarly article
title: Different exponential spectra in Banach algebras
zbMATH Open document ID: 0969.46038
author: L. Lindeboom
author: Heinrich Raubenheimer
published in: Rocky Mountain Journal of Mathematics
publication date: 2001-09-23T00:00:00Z
full work available at URL: CONT29-3.html
review text: 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.
Mathematics Subject Classification ID: 46H05
Mathematics Subject Classification ID: 46H10
zbMATH DE Number: 1439643
zbMATH Keywords: exponential spectrum
zbMATH Keywords: connected components
zbMATH Keywords: groups of invertible elements
zbMATH Keywords: perturbation by radical elements
zbMATH Keywords: rank one elements
zbMATH Keywords: Riesz elements
reviewed by: Srinivasa Swaminathan
MaRDI profile type: Publication
OpenAlex ID: W2094153510
cites work: On the 0-spektrum of regular operators and the spectrum of measures
cites work: A primer on spectral theory
cites work: Q3317595
cites work: Q3215824
cites work: Comparisons between different spectra of an element in a Banach algebra
cites work: Extension of Stokes series for flow in a circular boundary
cites work: Spectral properties of elements in different Banach algebras
cites work: A NOTE ON THE DIFFERENT EXPONENTIAL SPECTRA OF AN ELEMENT IN A BANACH ALGEBRA
cites work: PERTURBATION BY INESSENTIAL AND RIESZ ELEMENTS
cites work: A NOTE ON THE SINGULAR AND EXPONENTIAL SPECTRUM IN BANACH ALGEBRAS
cites work: The Exponential Spectrum in Banach Algebras
cites work: On rank one elements
cites work: Q4881257
cites work: Semigroups of operators and measures of noncompactness
cites work: ON INESSENTIAL IDEALS IN BANACH ALGEBRAS
cites work: FREDHOLM THEORY RELATIVE TO TWO BANACH ALGEBRA HOMOMORPHISMS
cites work: Q4282384
cites work: Q3363575
cites work: Q4176556
cites work: Q5667036
DOI: 10.1216/RMJM/1181071617
description: scientific article; zbMATH DE number 1439643