Normal extensions of a first order differential operator

Abstract

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-, l1,... ln, l+), l-+ = d/dt + A-+, lk = d/dt + Ak where the coefficients A-+, Ak are selfadjoint operator in separable Hilbert spaces H-+, Hk, k= 1,..., n, n ? N respectively, are researched in the direct sum of Hilbert spaces of vector-functions L2(H_, (-? a))? L2(H1, (a1, b1)) ?...? L2(Hn, (an, bn)) ? L2(H+, (b,+?)) -? < a < a1 < b1 < . .. < an < bn < b < +?. 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?lder space in regular case.