Normal Extensions of a First Order Differential Operator

Z. I. Ismailov, M. Sertbas, B. Ö. Güler

In the paper of W.N. Everitt and A. Zettl [26] in scalar case, all selfadjoint extensions of the minimal operator generated by Lagrange-symmetric any order quasi-differential expression with equal deficiency indexes in terms of boundary conditions are described by Glazman--Krein--Naimark method for regular and singular cases in the direct sum of corresponding Hilbert spaces of functions. In this work, by using the method of Calkin--Gorbachuk theory all normal extensions of the minimal operator generated by fixed order linear singular multipoint differential expression $l=(l_\_,l_1,ldots,l_n,l_+),l_\mp=\frac d{dt}+A_\mp$, $l_k=\frac d{dt}+A_k$ where the coefficients $A_\mp$, $A_k$ are selfadjoint operator in separable Hilbert spaces $H_\mp$, $H_k$, $k=1,\ldots,n$, $n\in\Bbb N$ respectively, are researched in the direct sum of Hilbert spaces of vector-functions \[ L_2(H_\_,(-\infty,a))\oplus L_2(H_1,(a_1,b_1))\oplus\ldots\oplus L_2(H_n(a_n,b_n))\oplus L_2(H_+,(b,+\infty)) \] $-\infty<a<a_1<b_1<\ldots<a_n<b_n<b<+\infty$. Moreover, the structure of the spectrum of normal extensions is investigated. Note that in the works of A. Ashyralyev and O. Gercek [2, 3] the mixed order multipoint nonlocal boundary value problem for parabolic-elliptic equation is studied in weighed H\"older space in regular case.