# 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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Vadim Olshevsky
olshevsk@isl.stanford.edu
Last modified: Fri Sep 15 1995