A nonlinear characterization of stable invariant subspaces
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.
scientific article; zbMATH DE number 3895688