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

Zameddin I. Ismailov, Pembe Ipek Al, Mohammad Sababheh

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.

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

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.
Paper Structure (7 sections, 16 theorems, 100 equations)

This paper contains 7 sections, 16 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On normality of first order linear differential operator in weighted Hilbert space
  4. Description of the normal extensions
  5. Spectrum of the normal extensions
  6. Asymptotical behaviour of $s-$ numbers of inverse for normal extensions
  7. One generalization

Key Result

Theorem 1

If the minimal operator $L_{0}$ is formally normal in $L_{\alpha}^{2}$, then where $A_{r}(t)=ReA(t)=\frac{1}{2} \overline{\left( A(t)+A^{*}(t)\right)}, \ t\in (0,1)$ and $I$ is identity operator in $H.$

Theorems & Definitions (19)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Example 1
  • Definition 1
  • Theorem 7
  • ...and 9 more