Analytic perturbation theory for matrices and operators. Licensed ed
type: Item
instance of: scholarly article
title: Analytic perturbation theory for matrices and operators. Licensed ed
zbMATH Open document ID: 0591.47013
published in: Operator Theory: Advances and Applications
publication date: 1985-01-01T00:00:00Z
review text: This valuable monograph is an extended and partly modified English version of the author's book ``Endlichdimensionale analytische Störungstheorie.'' Berlin: Akademie-Verlag (1972; Zbl 0303.47012). Both the original version and the present book present essential supplements to \textit{T. Kato}'s fundamental monograph ``Perturbation theory for linear operators.'' Berlin etc.: Springer(1966; Zbl 0148.12601). In his book the author gves a systematic presentation of analytic perturbation theory for matrices and the implications of this theory for analytic perturbations of isolated eigenvalues of finite algebraic multiplicity of closed linear operators in Banach spaces. The central topic of the book is a complete description of how the structure of a matrix is changed by analytic perturbation. The main results can be described as follows: Let \(A(\cdot)\) be a matrix-valued function, defined in a subset \(G\) of the complex plane. Let \(\xi\in G\). Denote the eigenvalues of \(A(\xi)\) by \(\lambda_i(\xi)\) and the corresponding eigenprojections by \(P_i(\xi)\), \[ P_i(\xi) := \frac{1}{2\pi i}\int_{\Gamma_i}(\lambda - A(\xi))^{-1}\,d\lambda \quad (i=1,2,\ldots,r(\xi)); \] each path \(\Gamma_i\) of integration is a positively oriented circle with center \(\lambda_i(\xi)\) which separates \(\lambda_i(\xi)\) from the remaining spectrum of \(A(\xi)\). The operator \[ D(\xi) := A(\xi)- \sum^{r(\xi)}_{i=1}\lambda_i(\xi)P_i(\xi) \] is nilpotent. The function \(A(\cdot)\) is said to have an invariant Jordan structure in \(G_0\subset G\) if \(r(\xi)\) is constant in \(G_0\), the dimensions of the ranges of the projections \(P_i(\xi)\) are constant in \(G_0\) (for a suitable numbering) and there is a constant nilpotent matrix \(N\) such that \(D(\xi)\) is similar to \(N\) for each \(\xi \in G_0\). Theorem: Let \(G\) be a domain and \(A(\cdot)\) be holomorphic (meromorphic) in \(G\). Then there is an at most countable set \(A\subset G\), having no accumulation points in \(G\), such t…
reviewed by: Reinhard Mennicken
Mathematics Subject Classification ID: 47-02
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 47A70
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 81Q15
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 3949073
zbMATH Keywords: analytic perturbation theory for matrices
zbMATH Keywords: analytic perturbations of isolated eigenvalues of finite algebraic
zbMATH Keywords: multiplicity of closed linear operators in Banach spaces
zbMATH Keywords: invariant Jordan structure
zbMATH Keywords: Puiseux series
zbMATH Keywords: expansions of eigenprojections and irreducible projections
zbMATH Keywords: Laurent series
zbMATH Keywords: meromorphic operator-valued functions
zbMATH Keywords: reduction theory
zbMATH Keywords: analytic perturbations of isolated eigenvalues
zbMATH Keywords: finite algebraic multiplicity of closed linear operators
zbMATH Keywords: Banach spaces
author: Hellmut Baumgärtel
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3949073