Some Spectral Problems for First-Order Normal Differential Operators in the Weighted Hilbert Spaces of Vector Functions

In this article, in order for the minimal operator generated by the first-order differential-operator expression in the weighted Hilbert space of vector functions in the finite interval to be formal normal, the relationship between the variable operator coefficient of this differential-operator expression and the weight function is established. Afterwards, the general form of all normal extensions of the minimal operator is found using the Glazman–Krein–Naimark Method. Then, the structure of spectrum of such extensions is investigated. Later on, the issue of belonging to Schatten–von Neumann classes is explored, as well as the asymptotic behavior of the singular numbers of the inverse of such normal extensions. Lastly, an approach is developed on all normal extensions expressed in the weighted Hilbert spaces.