Dissipative canonical type differential operators for first order

Rukiye Öztürk Mert ,Zameddin I Ismailov ,Pembe Ipek Al

doi:10.3906/mat-1912-52

Abstract

In this paper, using the Calkin-Gorbachuk method, the general form of all maximally dissipative extensions of the minimal operator generated by the first order linear symmetric canonical type quasi-differential expression in the weighted Hilbert space of vector functions has been found. Also, the spectrum set of these extensions has been investigated.

Highlights

In the development of the studies on the spectral properties of an operator related with a boundary value problem acting on a Hilbert space, operator theory has a big importance
The minimal operator is generated by formally symmetric differential-operator expression in the Hilbert space of vector-functions defined in one finite or infinite interval case
We obtain the representation of all maximally dissipative extensions of the minimal operator. This minimal operator is generated by the first order linear symmetric canonical type quasi-differential expression with operator coefficient in the weighted Hilbert spaces of vector-functions defined on right semi-axis

Summary

Introduction

In the development of the studies on the spectral properties of an operator related with a boundary value problem acting on a Hilbert space, operator theory has a big importance. A dissipative operator which does not have any proper dissipative extension is called maximally dissipative [6] It is well-known that their spectrum lies in the closed upper half-plane. The minimal operator is generated by formally symmetric differential-operator expression in the Hilbert space of vector-functions defined in one finite or infinite interval case. We obtain the representation of all maximally dissipative extensions of the minimal operator This minimal operator is generated by the first order linear symmetric canonical type quasi-differential expression with operator coefficient in the weighted Hilbert spaces of vector-functions defined on right semi-axis. We investigate the structure of the spectrum of such extensions

Statement of the problem

Description of maximally dissipative extensions

The spectrum of the maximally dissipative extensions