In this paper we connect the function-theoretic properties of the matrix Titchmarsh-Weyl M(λ) function with the spectrum of a linear Hamiltonian system. Characterizations are given for the resolvent set, the point spectrum, the continuous spectrum and the point-continuous spectrum. The Hamiltonian system is considered under a limit-point hypothesis. In section 5 we establish invariance of certain parts of the spectrum with respect to boundary conditions.