A nonlinear characterization of stable invariant subspaces
type: Item
instance of: scholarly article
title: A nonlinear characterization of stable invariant subspaces
zbMATH Open document ID: 0562.47004
DOI: 10.1007/BF01691911
author: Gregory T. Adams
published in: Integral Equations and Operator Theory
publication date: 1983-01-01T00:00:00Z
review text: An invariant subspace \({\mathcal M}\) for a Hilbert space operator \(T\in {\mathcal L}({\mathcal H})\) is called stable invariant if for each \(\epsilon >0\) there is a \(\delta >0\) such that if \(\| T-T'\| <\delta\), then T' has an invariant subspace \({\mathcal M}'\) such that \(\| P_{{\mathcal M}'}-P_{{\mathcal M}}\| <\epsilon.\) The present author shows that this is equivalent to the fact that the function \(F:{\mathcal L}({\mathcal M},{\mathcal M}^{\perp})\to {\mathcal L}({\mathcal M},{\mathcal M}^{\perp}),\) defined by \(F(X)=XA-DA+XBX\) is surjective, where \(T=\left(\begin{matrix} A &B\\ 0&D\end{matrix}\right)\) with respect to the decomposition \({\mathcal H}={\mathcal M}\oplus {\mathcal M}^{\perp}\). He also proves a sort of implicit function theorem, and uses it for studying the surjectivity of F. Various results on stable invariant subspaces and examples are then infered.
Mathematics Subject Classification ID: 47A15
zbMATH DE Number: 3895688
zbMATH Keywords: implicit function theorem
zbMATH Keywords: surjectivity
zbMATH Keywords: stable invariant subspaces
MaRDI profile type: Publication
cites work: Stable factorizations of monic matrix polynomials and stable invariant subspaces
cites work: The Stable Solutions of Quadratic Matrix Equations
cites work: Solving Linear Operator Equations
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description: scientific article; zbMATH DE number 3895688