In this paper we consider two linear differential systems on a time scale. Both systems depend linearly on a complex spectral parameter lambda. We prove that if all solutions of these two systems are square integrable with respect to a given weight matrix for one value lambda, then this property is preserved for all complex values lambda. This result extends and improves the corresponding continuous time statement, which was derived by Walker (1975) for two non-hermitian linear Hamiltonian systems, to appropriate differential systems on arbitrary time scales. The result is new even in the purely discrete case, or in the scalar time scale case, as well as when both time scale systems coincide. The latter case also generalizes a limit circle invariance criterion for symplectic systems on time scales, which was recently derived by the authors.

Last change: June 4, 2014. (c) Roman Simon Hilscher