On the definition of eigenvalues for nonlinear operators
type: Item
instance of: scholarly article
title: On the definition of eigenvalues for nonlinear operators
zbMATH Open document ID: 0956.47038
DOI: 10.1016/S0362-546X(00)85034-8
author: Paola Santucci
author: Martin Väth
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2000-07-09T00:00:00Z
review text: The ``naive'' definition of eigenvalue for a nonlinear operator \(F\) in a Banach space \(X\) as a scalar \(\lambda\) with the property that the nullset \(N(\lambda I-F)= \{x\in X:\lambda x-F(x)= 0\}\) contains some nonzero element is not suitable for many purposes. In the present paper, the authors propose some other notions of eigenvalue, the most appropriate being the following: a scalar \(\lambda\) is called ``connected eigenvalue'' if \(N(\lambda I-F)\) contains an unbounded connected subset. With this definition, several well-known results carry over from the linear case; for example, the union of \(0\) and the set of connected eigenvalues of a compact nonlinear operator is always a compact subset of the complex plane.
reviewed by: Jürgen Appell
Mathematics Subject Classification ID: 47J10
zbMATH DE Number: 1472237
zbMATH Keywords: connected eigenvalue
zbMATH Keywords: compact nonlinear operator
Wikidata QID: Q57363492
MaRDI profile type: Publication
cites work: A new spectral theory for nonlinear operators and its applications
cites work: Contributions to the spectral theory for nonlinear operators in Banach spaces
cites work: Q3913956
cites work: Q5508167
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full work available at URL: s0362-546x(00)85034-8
OpenAlex ID: W35157156
description: scientific article; zbMATH DE number 1472237