Selfadjoint Singular Differential Operators for First Order

Abstract

Abstract. The parametrization of all selfadjoint extensions of the minimal operator generated by …rst order linear symmetric singular di¤erential-operator expression in the Hilbert space of vector-functions de…ned at the right semi-axis has been given. To this end we use the Calkin-Gorbachuk method. Finally, the structure of spectrum set of such extensions is researched. IntroductionIt is known that fundamental question on the parametrization of selfadjoint extensions of the linear closed densely de…ned with equal de…ciency indices symmetric operators in a Hilbert space has been investigated by J. von Neumann [11] and M. H. Stone [10] …rstly. Applications of these results to any scaler linear even order symmetric di¤erential operators and representation of all selfadjoint extensions in terms of boundary conditions have been investigated by I. M. Glazman-M. G. Krein-M. A. Naimark (see [5,8]). In mathematical literature there is co-called Calkin-Gorbachuk method (see [6,9]).The motivation of this paper originates from the interesting researches of W. N. Everitt, L. Markus, A. Zettl, J. Sun, D. O'Regan, R. Agarwal [2,3,4,12] in scaler cases. Throughout this paper A. Zettl's and J. Suns's view about these topics is to be taken into consideration [12]. A selfadjoint ordinary di¤erential operator in a Hilbert space is generated by two things: (1) a symmetric ( formally selfadjoint) di¤erential expression; (2) a boundary condition which consists selfadjoint di¤erential operators. And also the geometrical place in plane of the spectrum of given selfadjoint di¤er-ential operator is one of the important questions of this theory.In this work in Section 3 the representation of all selfadjoint extensions of the symmetric singular di¤erential operator, generated by …rst order symmetric c 2 0 1 8 A n ka ra U n ive rsity. C o m m u n ic a tio n s Fa c u lty o f S c ie n c e s U n ive rsity o f A n ka ra -S e rie s A 1 M a th e m a tic s a n d S ta tistic s. C o m m u n ic a tio n s d e la Fa c u lté d e s S c ie n c e s d e l'U n ive rsité d 'A n ka ra -S é rie s A 1 M a th e m a tic s a n d S ta tistic s.