Relative Inversion in der Störungstheorie von Operatoren und \(\Psi\)- Algebren. (Relative inversions in perturbation theory of operators and \(\Psi\)-algebras)
type: Item
instance of: scholarly article
title: Relative Inversion in der Störungstheorie von Operatoren und \(\Psi\)- Algebren. (Relative inversions in perturbation theory of operators and \(\Psi\)-algebras)
zbMATH Open document ID: 0661.47037
DOI: 10.1007/BF01455995
author: Bernhard Gramsch
published in: Mathematische Annalen
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/163920
review text: In this substantial paper the author considers a separated topological algebra \({\mathcal B}\) under the assumption that its group of invertible elements is open. A function defined on subsets of \({\mathcal B}^ n\) is called \({\mathcal B}\)-rational if it is composed of projections onto coordinate spaces of \({\mathcal B}^ n\) and of constant functions by a finite number of algebraic operations (addition, multiplication and inversion). A topological space is called a locally \({\mathcal B}\)-rational (\(\ell {\mathcal B}r\)-) manifold with an atlas \(\{(U_{\alpha},\phi_{\alpha},V_{\alpha})\); \(\alpha\in A\}\) of charts, iff: 1) \(\{U_{\alpha}\), \(\alpha\in A\}\) is an open covering of M, for \(\alpha\in A\), \(V_{\alpha}\) is an open subset of the topological subspace \((T_{\alpha},\tau (T_{\alpha}))\) and \(\phi_{\alpha}: U_{\alpha}\to V_{\alpha}\) is a homeomorphism, 2) for \(\alpha\in A\), \(T_{\alpha}\) is a linear subspace of \({\mathcal B}^{n_{\alpha}}\) with a topology \(\tau (T_{\alpha})\) which is finer than the subspace topology of the product topology of \(\tau\) (\({\mathcal B})\) on \({\mathcal B}^{n_{\alpha}}\) and 3) all mappings \(\phi_{\alpha}\phi_{\beta}^{-1}: \phi_{\beta}(U_{\alpha}\cap U_{\beta})\to \phi_{\alpha}(U_{\alpha}\cap U_{\beta})\) (\(\alpha\),\(\beta\in A)\) are \({\mathcal B}\)-rational functions. A homogeneous space \((G<\Pi,M)\) is called an \(\ell {\mathcal B}r\) homogeneous space and M an \(\ell {\mathcal B}r\) homogeneous manifold, iff: 1) G and M are \(\ell {\mathcal B}_ r\)-manifolds, 2) the group operations \(G\times G\to G\) are \(\ell {\mathcal B}r\)-mappings, 3) \(\Pi\) : \(G\times M\to M\) is an \(\ell {\mathcal B}r\)-mapping and 4) for all \(p\in M\) there is an open neighbourhood of p and an \(\ell {\mathcal B}r\)-submanifold \({\mathcal T}_ p\) of G such that \(\Pi |_{{\mathcal T}_ p}: {\mathcal T}_ p\to U\) is an \(\ell {\mathcal B}r\)-homeomorphism. If J is a two-sided ideal in \({\mathcal B}\), then \(G_ e(J)\) i…
Mathematics Subject Classification ID: 47L10
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 47Gxx
Mathematics Subject Classification ID: 58B25
Mathematics Subject Classification ID: 58J40
Mathematics Subject Classification ID: 47A53
zbMATH DE Number: 4080235
zbMATH Keywords: perturbation theory
zbMATH Keywords: Fredholm operators
zbMATH Keywords: ideals
zbMATH Keywords: topological algebra
zbMATH Keywords: group of invertible elements
zbMATH Keywords: locally \({\mathcal B}\)-rational (\(\ell {\mathcal B}r\)-) manifold
zbMATH Keywords: \(\ell {\mathcal B}r\) homogeneous space
zbMATH Keywords: \(\Psi\)-algebras
zbMATH Keywords: pseudodifferential operators
zbMATH Keywords: Cartan lemma on holomorphic factorization
zbMATH Keywords: Oka principle
zbMATH Keywords: Fréchet algebra
reviewed by: Reinhard Mennicken
MaRDI profile type: Publication
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Relative Inversion in der Störungstheorie von Operatoren und \(\Psi\)- Algebren. (Relative inversions in perturbation theory of operators and \(\Psi\)-algebras) (cites work)
cites work: BANACH BUNDLES AND LINEAR OPERATORS

Relative Inversion in der Störungstheorie von Operatoren und \(\Psi\)- Algebren. (Relative inversions in perturbation theory of operators and \(\Psi\)-algebras) (OpenAlex ID)
OpenAlex ID: W2911870966