For a symmetric differential
expression associated with a first order system
where A0 and A are n × n matrices and x is an n × 1 vector, a spectral
decomposition will be developed. That is, if S is a closed symmetric differential
operator determined by the differential system, the explicit nature of the generalized
resolutions of the identity for all the self-adjoint extensions of S in any Hilbert space
will be determined in terms of a fundamental matrix and spectral matrices associated
with these extensions. An important aspect is that these self-adjoint extensions may
be defined in Hilbert spaces larger than the natural one ℋ in which the operator S is
defined.
