ON SOME SPECTRAL PROPERTIES OF OPERATORS GENERATED BY QUASI-DIFFERENTIAL MULTI-INTERVAL SYSTEMS

Abstract

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.