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Formal and Analytic Diagonalization of Operator functions

  • We give conditions for local diagonalization of analytic operator families acting between real or complex Banach spaces.The transformations are constructed from an operator Töplitz matrix obtained from Jordan chains of increasing length. The basic assumption is given by stabilization of the Jordan chains at length k in the sense that no root elements with finite rank above k are allowed to exist. Jordan chains with infinite rank may appear. These assumptions ensure finite pole order equal to k of the generalized inverse. The Smith form arises immediately. Smooth continuation of kernels and ranges towards appropriate limit spaces is considered using associated families of analytic projection functions. No Fredholm properties or other finiteness assumptions, besides the pole order, are assumed. Real and complex Banach spaces are treated without difference by elementary analysis of the system of undetermined coefficients. Formal power series solutions of the system of undetermined coefficients are constructed, which are turningWe give conditions for local diagonalization of analytic operator families acting between real or complex Banach spaces.The transformations are constructed from an operator Töplitz matrix obtained from Jordan chains of increasing length. The basic assumption is given by stabilization of the Jordan chains at length k in the sense that no root elements with finite rank above k are allowed to exist. Jordan chains with infinite rank may appear. These assumptions ensure finite pole order equal to k of the generalized inverse. The Smith form arises immediately. Smooth continuation of kernels and ranges towards appropriate limit spaces is considered using associated families of analytic projection functions. No Fredholm properties or other finiteness assumptions, besides the pole order, are assumed. Real and complex Banach spaces are treated without difference by elementary analysis of the system of undetermined coefficients. Formal power series solutions of the system of undetermined coefficients are constructed, which are turning into convergent solutions, as soon as analyticity of the operator family and continuity of the projections is assumed. Along these lines, results concerning linear Artin approximation follow immediately, which are well known in finite dimensions. The main technical tool is given by a defining equation of Nakayama Lemma type.show moreshow less

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Metadaten
Author:Matthias StiefenhoferORCiD
URL / DOI:https://arxiv.org/abs/2305.14228
DOI:https://doi.org/10.48550/arXiv.2305.14228
Publisher:arXiv
Document Type:Preprint
Language:English
Date of Publication (online):2023/05/23
Year of first Publication:2023
Tag:Artin approximation; Diagonal Operator Function; Diagonalization; Generalized Inverse; Jordan chain; Smith Form; Toeplitz Matrix; Transversalization
Number of pages:54 Seiten
First Page:1
Last Page:54
Institutes:Fakultät Maschinenbau
Dewey Decimal Classification:6 Technik, Medizin, angewandte Wissenschaften
Open Access:open_access
Research focus:Eigenständige Forschungsfelder
Publication Lists:Stiefenhofer, Matthias
Publication reviewed:nicht begutachtet
Release Date:2023/07/21
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