An application of the smooth variational principle to the existence of nontrivial invariant subspaces
type: Item
instance of: scholarly article
title: An application of the smooth variational principle to the existence of nontrivial invariant subspaces
zbMATH Open document ID: 1008.47006
DOI: 10.1016/S0764-4442(00)01803-6
author: Aharon Atzmon
author: Gilles Godefroy
publication date: 2003-04-01T00:00:00Z
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46B40
Mathematics Subject Classification ID: 47D03
zbMATH DE Number: 1587992
zbMATH Keywords: invariant subspace
zbMATH Keywords: invariant subset
zbMATH Keywords: moment sequence
zbMATH Keywords: Gâteaux smooth norm
MaRDI profile type: Publication
full work available at URL: s0764-4442(00)01803-6
OpenAlex ID: W2067885337
Wikidata QID: Q126781575
published in: Comptes Rendus de l'Académie des Sciences. Série I. Mathématique
review text: The main result of the paper under review is the following theorem: Let \(X\) be a real Banach space which has an equivalent Gâteaux smooth norm and let \(F\) be a nontrivial closed subset of \(X\). Let \(\mathcal S\) be a semigroup of (bounded linear) operators on \(X\). If for all \(A\in\mathcal S\) there is a differentiable function \(f_A\:\mathcal J_A=(-\delta_A,\delta_A)\to L(X)\) such that (i)~\(f_A(0)=I\), (ii)~\((f_A)'(0)=A\), (iii)~\(F\) is an invariant subset of each element of \(f_A(\mathcal I_A)\), then there is a nontrivial invariant subspace for \(\mathcal S\). NEWLINENEWLINENEWLINEUsing this theorem, some invariant subspace results are proved in the paper, e.g., an operator \(X\) which admits a moment sequence has a nontrivial invariant subspace.
reviewed by: Michal Zajac
description: scientific article