Change of Jordan structure of G-selfadjoint operators and selfadjoint operator functions under small perturbations

Vadim Olshevsky

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 the lengths of Jordan chains ( partial multiplicities ) of linear operators and analytic operator functions, and of Gohberg-Kaashoek numbers, under small perturbations. In this paper similar problems are extended to the classes of G-selfadjoint operators, and selfadjoint operator functions. For this case, we obtained new inequalities on the variation of partial multiplicities, revealing the role played by the sign characteristic.

Related papers: Two new proofs for the Markus-Parilis-den Boer-Thijsse theorem, which are based on the analysis of cyclic dimensions and of kernel multiplicities, appeared in [MO94]. The role played by the Gohberg-Kaashoek numbers in the description of the behavior of the lattice of all invariant subspaces, under a small perturbation of a matrix, is revealed in [O89a].

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Last modified: Fri Sep 15 1995