Local research of manifolds generated by families of self-adjoint operators
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.
Arnold's manifold of selfadjoint operators with fixed eigenvalue multiplicity
scientific article; zbMATH DE number 5706264