Algebraic multiplicity of eigenvalues of linear operators
type: Item
instance of: scholarly article
title: Algebraic multiplicity of eigenvalues of linear operators
zbMATH Open document ID: 1144.47001
author: Julián López-Gómez
author: Carlos Mora-Corral
published in: Operator Theory: Advances and Applications
publication date: 2007-06-08T00:00:00Z
review text: This book has three parts. Part~1 consists of three chapters and concerns the basics of finite-dimensional spectral theory. The exposition is clear and suitable even for an undergraduate audience. The main body of the book is Part~2 which consists of eight chapters and concerns a generalized concept of algebraic multiplicity. Part~3 is an illustration of some basic results on the topological degree and is aimed at illustrating some application to nonlinear problems of the results of Part~2. We now briefly illustrate the content of the Chapters 4--10 of Part~2. Chapter~4 introduces a concept of algebraic multiplicity \(\chi[{\mathfrak L},\lambda_{0}]\) of a family \({\mathfrak L}\in C^{r}(\Omega, {\mathcal{L}}(U,V))\) at any algebraic eigenvalue \(\lambda_{0}\in \Omega\). Here, \(\Omega\) denotes a connected subset of the real or complex numbers, \(U\) and \(V\) are Banach spaces, and \({\mathcal{L}}(U,V)\) denotes the space of bounded linear operators of \(U\) to \(V\). Chapter~5 describes an equivalent approach to the concept of multiplicity of Chapter~4 and analyzes the relations between the two types of multiplicity. In Chapter~6, the authors select some fundamental properties of the algebraic multiplicity which essentially determine it. In Chapter~7, the authors introduce the concept of algebraic multiplicity by exploiting the theory of Jordan chains, which extend the classical concept of chains of generalized eigenvectors. Such a concept of algebraic multiplicity is then shown to coincide with those introduced in Chapters~4 and 5. Here, the authors also introduce the concept of local Smith forms for a family \({\mathfrak L}\) at an eigenvalue \(\lambda_{0}\), which is a canonical form obtained by multiplying \({\mathfrak L}\) on the left and on the right by isomorphisms, and show that the existence of such a form at \(\lambda_{0}\) is equivalent for \(\lambda_{0}\) to be an algebraic eigenvalue. The concept of local Smith form has then been employed to analyze …
reviewed by: Massimo Lanza de Cristoforis
Mathematics Subject Classification ID: 47-02
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A22
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A11
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 47H11
Mathematics Subject Classification ID: 47J15
zbMATH DE Number: 5162723
zbMATH Keywords: eigenvalues and eigenvectors
zbMATH Keywords: algebraic multiplicity
zbMATH Keywords: real analytic and holomorphic families of operators
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5162723