Local research of manifolds generated by families of self-adjoint operators
type: Item
instance of: scholarly article
title: Local research of manifolds generated by families of self-adjoint operators
zbMATH Open document ID: 1197.47034
author: B. E. Eshmatov
published in: Topology
publication date: 2010-05-10T00:00:00Z
review text: The article is devoted to the study of manifolds associated with eigenvalues of selfadjoint operators. These manifolds first introduced by Arnold are used for the study of a spectrum perturbation problem. A~local diffeomorphism is constructed with the help of a decomposition of a selfadjoint operator into antisymmetric and diagonal parts and their exponential mappings. Some properties of these mappings are given and the analyticity of the manifolds and their tangent spaces is investigated. Moreover, Fredholm indices and dimensions of projections and their tangent mappings of bundles over manifolds of eigenvalues for their given multiplicity and order number are studied. The results of this paper develop preceding results of \textit{V.\,I.\thinspace Arnol'd} [Sel.\ Math., New Ser.\ 1, 1--19 (1995; Zbl 0841.58008)], \textit{K.\,Uhlenbeck} [Am.\ J.\ Math.\ 98, 1059--1078 (1976; Zbl 0355.58017)], \textit{D.\,Fujiwara, M.\,Tanikawa} and \textit{Sh.\,Yukita} [Proc.\ Japan Acad., Ser.\,A 54, 87--91 (1978; Zbl 0416.35054)]. Applications to a finite-multiplicity eigenvalue splitting problem are given.
Mathematics Subject Classification ID: 47A65
Mathematics Subject Classification ID: 47B15
Mathematics Subject Classification ID: 58C40
Mathematics Subject Classification ID: 58B15
Mathematics Subject Classification ID: 58D15
zbMATH DE Number: 5706264
zbMATH Keywords: Arnold's manifold of selfadjoint operators with fixed eigenvalue multiplicity
zbMATH Keywords: Uhlenbeck's manifold of eigenvectors
zbMATH Keywords: eigenvalue splitting problem
reviewed by: Sergey V. Ludkovsky
MaRDI profile type: Publication
full work available at URL: j.top.2009.11.021
OpenAlex ID: W2013370729
DOI: 10.1016/J.TOP.2009.11.021
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description: scientific article; zbMATH DE number 5706264