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.
Vadim Olshevsky's Home page
Vadim Olshevsky
olshevsk@isl.stanford.edu
Last modified: Fri Sep 15 1995