Logarithmic residues, Rouché's theorem, and spectral regularity: The \(C^{\ast}\)-algebra case
type: Item
instance of: scholarly article
title: Logarithmic residues, Rouché's theorem, and spectral regularity: The \(C^{\ast}\)-algebra case
zbMATH Open document ID: 1279.47028
author: B. Silbermann
author: Harm Bart
author: Torsten Ehrhardt
published in: Indagationes Mathematicae. New Series
publication date: 2012-12-06T00:00:00Z
review text: The paper extends the theory of analytic Fredholm operator valued functions to \(C^*\)-algebras. An element \(a\) of a unital \(C^*\)-algebra \(\mathcal B\) is called Fredholm if it is invertible modulo elements in \(\mathcal C(\mathcal B)\), where \(\mathcal C(\mathcal B)\) denotes the closure of the ideal consisting of all finite rank elements of \(\mathcal B\). The authors review and extend the results on Fredholm elements in [\textit{R. Hagen}, \textit{S. Roch} and \textit{B. Silbermann}, \(C^*\)-algebras and numerical analysis. New York, NY: Marcel Dekker (2001; Zbl 0964.65055)]. In the case where \(\mathcal B\) is \(L(H)\) for some Hilbert space \(H\), the aforementioned theory reduces to the well-known theory of Fredholm operators. The authors investigate Fredholm valued analytic functions \(f:D\to \mathcal B\), where \(D\) is a non-empty connected open subset of \(\mathbb C\). If \(f(\lambda)\) is invertible for at least one \(\lambda\in D\), then the set of all \(\lambda\in D\) where \(f\) is not invertible has no accumulation point in \(D\). At such points, the principal part of the Laurent expansion of \(f\) is finite and its coefficients are finite rank elements of \(\mathcal B\). For a bounded Cauchy domain \(\Delta\subset \overline \Delta \subset D\), it is shown that the logarithmic residue \(\frac{1}{2\pi \mathrm{i}} \int_{\partial \Delta} f'(\lambda) f(\lambda)^{-1}\mathrm{d}\lambda\) is a finite sum of finite rank idempotents in \(\mathcal B\), given that \(f\) is invertible on \(\partial \Delta\). The authors proceed to prove the following analogue of Rouché's theorem: If \(g\) is a \(\mathcal B\)-valued function on \(D\) and \(\max_{\lambda\in\partial \Delta} \left\| \left( g(\lambda) - f(\lambda) \right) f(\lambda)^{-1} \right\| < 1\), then \(g\) is a Fredholm function on \(\Delta\), it is invertible on \(\partial \Delta\) and the trace of the logarithmic residues of \(f\) and \(g\) are equal. The paper concludes with some examples; the example …
reviewed by: Monika Winklmeier
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 30E20
Mathematics Subject Classification ID: 30G30
zbMATH DE Number: 6112934
zbMATH Keywords: Fredholm elements
zbMATH Keywords: Rouché's theorem
zbMATH Keywords: spectral regularity
zbMATH Keywords: Calkin algebra
zbMATH Keywords: \(C^{\ast }\)-Fredholm theory
zbMATH Keywords: abstract analytic Fredholm valued function
zbMATH Keywords: logarithmic residue
MaRDI profile type: Publication
full work available at URL: j.indag.2012.08.001
OpenAlex ID: W1969192496
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DOI: 10.1016/J.INDAG.2012.08.001
description: scientific article; zbMATH DE number 6112934