Stability of invariant maximal semidefinite subspaces. I. (English) Zbl 0561.15001
Let the \(n\times n\) invertible Hermitian matrix H define an indefinite inner product by \([x,y]=<Hx,y>=y*Hx\). Suppose A is H-selfadjoint. This paper is interested in A-invariant subspaces N (AN\(\subset N)\) which are maximal H-nonnegative subspaces ([x,x]\(\geq 0\) for all \(x\in N)\) or hypermaximal H-neutral subspaces \((HN=N^{\perp})\). The stability of such subspaces is examined. For example, a hypermaximal H-neutral subspace N is stable if for every \(\epsilon >0\) there is a \(\delta >0\) so that \(0<\| A-A^ 1\| +\| H-H^ 1\| <\delta\) implies there is a subspace \(N^ 1\) which is \(A^ 1\)-invariant and hypermaximal \(H^ 1\)-neutral such that the orthogonal projections onto N, \(N^ 1\) are within \(\epsilon\) of each other.
Reviewer: S.L.Campbell
MSC:
| 15A03 | Vector spaces, linear dependence, rank, lineability |
| 15B57 | Hermitian, skew-Hermitian, and related matrices |
| 15A63 | Quadratic and bilinear forms, inner products |
Keywords:
stable; Hermitian matrix; indefinite inner product; H-selfadjoint; A- invariant subspaces; maximal H-nonnegative subspaces; hypermaximal H- neutral subspacesReferences:
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