  We study the spectral theory of operators, generated as direct sums of
self-adjoint extensions of quasi-differential minimal operators on a
multi-interval set (self-adjoint vector-operators), acting in a Hilbert space.
  Spectral theorems for such operators are discussed, the structure of the
ordered spectral representation is investigated for the case of differential
coordinate operators. One of the main results is the construction of spectral
resolutions. Finally, we study the matters connected with analytical
decompositions of generalized eigenfunctions of such vector-operators and build
a matrix spectral measure leading to the matrix Hilbert space theory. Results,
connected with other spectral properties of self-adjoint 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
vector-operators and are essential for the further study of their spectral
properties.
