Trace conditions for regular spectral behavior of vector-valued analytic functions
type: Item
instance of: scholarly article
title: Trace conditions for regular spectral behavior of vector-valued analytic functions
zbMATH Open document ID: 1188.47014
author: Harm Bart
author: B. Silbermann
author: Torsten Ehrhardt
published in: Linear Algebra and its Applications
publication date: 2009-04-29T00:00:00Z
review text: The authors consider the so-called logarithmic residue \[ \frac{1}{2\pi i} \int_\Delta F'(\lambda) F'(\lambda)^{-1} \] of a function \(F\) with values in a complex Banach algebra and \(\Delta\) a bounded Cauchy domain in \(\mathbb C\) such that \(F\) is invertible on \(\partial \Delta\). If \(F\) is invertible in every \(\lambda\in\Delta\), then the logarithmic residue is zero by Cauchy's theorem. A function is called spectrally regular if also the reverse holds, that is, if the vanishing of its logarithmic residue on every Cauchy domain where it makes sense, implies that \(F(\lambda)\) is invertible for every \(\lambda\) in the domain. The authors are interested in giving conditions when \(F\) is spectrally regular. To this end, they consider plain functions, that is, functions that are analytically equivalent to elementary polynomials, or more generally, to \(J\)-plain functions, where \(J\) is an ideal in the Banach algebra. As main results, the authors show that a \(J\)-plain function is spectrally regular if there exists a \(J\)-resolving family of traces.
reviewed by: Monika Winklmeier
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 46H99
Mathematics Subject Classification ID: 30G30
zbMATH DE Number: 5549253
zbMATH Keywords: logarithmic residue
zbMATH Keywords: analytic vector-valued function
zbMATH Keywords: elementary factor
zbMATH Keywords: plain function
zbMATH Keywords: idempotent
zbMATH Keywords: annihilating family of idempotents
zbMATH Keywords: resolving family of traces
zbMATH Keywords: integer combination of idempotents
MaRDI profile type: Publication
full work available at URL: j.laa.2008.11.004
OpenAlex ID: W2001214681
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DOI: 10.1016/J.LAA.2008.11.004
description: scientific article; zbMATH DE number 5549253