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:
[1] | Bart, H.; Gohberg, I.; Kaashoek, M. A., Stable factorization of monic matrix polynomials and stable invariant subspaces, Integral Equations Operator Theory, 1, 496-517 (1978) · Zbl 0398.47011 |
[2] | Bart, H.; Gohberg, I.; Kaashoek, M. A., Minimal Factorization of Matrix and Operator Functions (1979), Birkhauser: Birkhauser Basel · Zbl 0424.47001 |
[3] | Bognar, J., Indefinite Inner Product Spaces (1974), Springer: Springer New York · Zbl 0277.47024 |
[4] | Campbell, S.; Daughtry, J., The stable solutions of quadratic matrix equations, Proc. Amer. Math. Soc., 74, 19-23 (1979) · Zbl 0403.15012 |
[5] | Glazman, I. M.; Ljubič, Ju. I., Finite Dimensional Linear Analysis: A Systematic Presentation in Problem Form (1974), M.I.T. Press: M.I.T. Press Cambridge, Mass · Zbl 1115.46001 |
[6] | Gohberg, I.; Lancaster, P.; Rodman, L., Spectral analysis of selfadjoint matrix polynomials, Ann. of Math., 112, 34-71 (1980) · Zbl 0446.15007 |
[7] | Gohberg, I.; Lancaster, P.; Rodman, L., Matrix Polynomials (1982), Academic: Academic New York · Zbl 0482.15001 |
[8] | Gohberg, I.; Lancaster, P.; Rodman, L., Perturbation of \(H\)-selfadjoint matrices, with applications to differential equations, Integral Equations Operator Theory, 5, 718-757 (1982) · Zbl 0511.15010 |
[9] | Gohberg, I.; Lancaster, P.; Rodman, L., Spectral analysis of selfadjoint matrix polynomials, (Res. Report 419 (1979), Dept. of Math. and Statist., Univ. of Calgary: Dept. of Math. and Statist., Univ. of Calgary Canada) · Zbl 0446.15007 |
[10] | Gohberg, I. C.; Sigal, E. I., An operator generalization of the logarithmic residue theorem and the theorem of Rouché, Math. USSR—Sb., 13, 603-625 (1971) · Zbl 0254.47046 |
[11] | Hua, Loo-Keng, On the theory of automorphic functions of a matrix variable, II. The classification of hypercircles under the symplectic group, Amer. J. Math., 66, 531-563 (1944) · Zbl 0063.02920 |
[12] | Iohvidov, I. S.; Krein, M. G., Spectral theory of operators in spaces with an indefinite metric, Amer. Math. Soc. Transl., 34, 283-373 (1963) · Zbl 0184.16401 |
[13] | Kaashoek, M. A.; van der Mee, C. V.M.; Rodman, L., Analytic operator functions with compact spectrum, II. Spectral pairs and factorization, Integral Equations Operator Theory, 5, 791-827 (1982) · Zbl 0493.47015 |
[14] | Kato, T., Perturbation Theory for Linear Operators (1966), Springer: Springer New York · Zbl 0148.12601 |
[15] | Lancaster, P., Theory of Matrices (1969), Academic: Academic New York · Zbl 0186.05301 |
[16] | Lancaster, P.; Rodman, L., Existence and uniqueness theorems for the algebraic Riccati equation, Internat. J. Control, 32, 285-309 (1980) · Zbl 0439.49011 |
[17] | Mal’cev, A. I., Foundations of Linear Algebra (1963), Freeman: Freeman San Francisco |
[18] | Ran, A. C.M., Minimal factorization of selfadjoint rational matrix functions, Integral Equations Operator Theory, 5, 850-869 (1982) · Zbl 0492.47013 |
[19] | A.C.M. Ran and L. Rodman, Stability of neutral invariant subspaces and stable symmetric factorizations, Integral Equations Operator Theory; A.C.M. Ran and L. Rodman, Stability of neutral invariant subspaces and stable symmetric factorizations, Integral Equations Operator Theory · Zbl 0516.47022 |
[20] | Rellich, F., Perturbation Theory of Eigenvalue Problems (1953), New York Univ, Lecture Notes · Zbl 0053.08801 |
[21] | L. Rodman, Maximal invariant neutral subspaces and an application to the algebraic Riccati equation, to appear.; L. Rodman, Maximal invariant neutral subspaces and an application to the algebraic Riccati equation, to appear. · Zbl 0521.15017 |
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