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

TL;DR

This paper investigates spectral properties and normal extensions of first-order differential operators in weighted Hilbert spaces, establishing conditions for normality, spectrum structure, and Schatten class membership.

Contribution

It derives conditions for the minimal operator to be formally normal, finds all normal extensions, and analyzes their spectral and asymptotic properties in weighted Hilbert spaces.

Findings

Characterized when the minimal operator is formally normal.

Derived the general form of all normal extensions.

Analyzed the spectrum and Schatten class properties of extensions.

Abstract

In this article, in order to 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 extension 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 behaviour of the singular numbers of the inverse of such normal extensions. Lastly, an approach is developed on all normal extension expressed in the weighted Hilbert spaces.