On an approach to the construction of the Friedrichs and Neumann-Krein extensions of nonnegative linear relations

Let $L_{0}$ be a closed linear nonnegative (probably, positively defined) relation ("multivalued operator") in a complex Hilbert space $H$. In terms of the so called boundary value spaces (boundary triples) and corresponding Weyl functions and Kochubei-Strauss characteristic ones, the Friedrichs (hard) and Neumann-Krein (soft) extensions of $L_{0}$ are constructed. It should be noted that every nonnegative linear relation $L_{0}$ in a Hilbert space $H$ has two extremal nonnegative selfadjoint extensions: the Friedrichs extension $L_{F}$ and the Neumann-Krein extension $L_{K},$ satisfying the following property: $$(\forall \varepsilon > 0) (L_{F} + \varepsilon 1)^{-1} \leq (\widetilde{L} + \varepsilon 1)^{-1} \leq (L_{K} + \varepsilon 1)^{-1}$$ in the set of all nonnegative selfadjoint subspace extensions $\widetilde{L}$ of $L_{0}.$ The boundary triple approach to the extension theory was initiated by F.S. Rofe-Beketov, M.L. and V.I. Gorbachuk, A.N. Kochubei, V.A. Mikhailets, V.O. Dercach, M.N. Malamud, Yu. M. Arlinskii and other mathematicians. In addition, it is showed that the construction of the mentioned extensions may be realized in a more simple way under the assumption that initial relation is a positively defined one.

Normal extensions of a first order differential operator

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.