We construct the common and the ordered spectral representation
for operators,
generated as direct sums of self-adjoint extensions of
quasi-differential minimal operators on a multiinterval
set (self-adjoint vector-operators), acting in a Hilbert space.
The structure of the ordered
representation is investigated for the case of differential
coordinate operators. Results, connected
with other spectral properties of such vector-operators, such
as the introduction of the identity
resolution and the spectral multiplicity have also been
obtained.
Vector-operators have been mainly studied by W.N. Everitt,
L. Markus and A. Zettl. Being a
natural continuation of Everitt-Markus-Zettl theory, the
presented results reveal the internal structure
of self-adjoint differential vector-operators and are essential
for the further study of their spectral
properties.