We study dichotomous behavior of solutions to a non-autonomous linear difference equation in a Hilbert space. The evolution operator of this equation is not continuously invertible and the corresponding unstable subspace is of infinite dimension in general. We formulate a condition ensuring the dichotomy in terms of a sequence of indefinite metrics in the Hilbert space. We also construct an example of a difference equation in which dichotomous behavior of solutions is not compatible with the signature of the indefinite metric.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm177-3-1,
author = {V. Khatskevich and L. Zelenko},
title = {Plus-operators in Krein spaces and dichotomous behavior of irreversible dynamical systems with discrete time},
journal = {Studia Mathematica},
volume = {173},
year = {2006},
pages = {195-210},
zbl = {1113.47024},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm177-3-1}
}
V. Khatskevich; L. Zelenko. Plus-operators in Krein spaces and dichotomous behavior of irreversible dynamical systems with discrete time. Studia Mathematica, Tome 173 (2006) pp. 195-210. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm177-3-1/