Strongly Invariant Subspaces of Nonautonomous Linear Periodic Systems and Solutions Whose Period Is Incommensurable with the Period of the System Itself

Abstract

We introduce the notions of quasi-invariant and strongly invariant subspaces of a one-parameter family of linear operators acting on a finite-dimensional vector space. The geometric meaning of these notions is that the restrictions of all operators of the family to a quasiinvariant subspace coincide and that the restrictions to a strongly invariant subspace are, in addition, an endomorphism of that subspace. These notions are used to reduce the well-known problem on Ω-periodic solutions of an ω-periodic linear differential system with incommensurable Ω and ω to the algebraic problem on the eigenvalues and eigenvectors of some matrix constructed from the right-hand side of the system.