Some spectral problems of multipoint normal differential operators for the first-order

In this study, a connection between normal extensions of formally normal minimal operator generated by differential expression for first-order regular type with normal operator coefficient in the Hilbert space of Hilbert space-valued vector-functions in the countable many separated bounded subintervals of real line and its normal extensions of coordinate minimal operators has been investigated. Moreover, it is described all normal extensions in direct sum of Hilbert space in terms of boundary values at the end points of the subintervals.Furthermore, some spectral properties like discreteness of spectrum, structure of discrete spectrum, analytic expression of eigenvalues, asymptotical behaviour of eigenvalues at the infinity and completeness of eigenvectors are researched.Finally, a compactness connections between of inverses of this type extensions and its coordinate extensions are established.The obtained results have been supported by applications.