Introduction to the spectral theory of self-adjoint differential vector-operators

Abstract

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.