A condition for the closeness of the sets of invariant subspaces of the close matrices in terms of their Jordan structures

Vadim Olshevsky

In this paper we consider the problem of the behavior of the distance     dist ( Inv A, Inv B)     between the sets     Inv A     and     Inv B     of invariant subspaces of the operators A and B, acting in a finite dimensional Hilbert space, depending upon on the Jordan structures of these operators and on distance between A and B. By distance between subspaces we mean their gap, and dist ( Inv A, Inv B)     is undersood in a Hausdorff sense.

The main result of this paper is that     lim dist ( Inv A, Inv B) = 0    , while     B --> A    , if and only if A and B have the same Gohberg-Kaashoek numbers. It is also proved that     inf dist ( Inv A, Inv B) > 0    , where infimum is taken over all possible pairs of operators A and B, for which the Gohberg-Kaashoek numbers do not coincide.

Finally we give an example that refutes two Gohberg-Rodman conjectures.

Related papers: In 1980 Markus and Parilis, and den Boer and Thijsse, confirming an earlier conjecture of Gohberg and Kaashoek, obtained a description of the possible domain of variation of Gohberg-Kaashoek numbers, under small perturbation of a given matrix. In [MO94] we obtained two new proofs for the Markus-Parilis-den Boer-Thijsse theorem, which are based on the analysis of cyclic dimensions and of kernel multiplicities. The change of Gohberg-Kaashoek numbers of G-selfadjoint operators and selfadjoint analytic operator functions, under small perturbations, was studied in [O90], where additinal restrictions were obtained, revealing the role, played by the sign characteristic.

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Last modified: Fri Sep 15 1995