Unbounded Wiener–Hopf Operators and Isomorphic Singular Integral Operators

Some basics of a theory of unbounded Wiener–Hopf operators (WH) are developed. The alternative is shown that the domain of a WH is either zero or dense. The symbols for non-trivial WH are determined explicitly by an integrability property. WH are characterized by shift invariance. We study in detail WH with rational symbols showing that they are densely defined, closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains, ranges, spectral and Fredholm points are explicitly determined. Another topic concerns semibounded WH. There is a canonical representation of a semibounded WH using a product of a closable operator and its adjoint. The Friedrichs extension is obtained replacing the operator by its closure. The polar decomposition gives rise to a Hilbert space isomorphism relating a semibounded WH to a singular integral operator of Hilbert transformation type. This remarkable relationship, which allows to transfer results and methods reciprocally, is new also in the thoroughly studied case of bounded WH.

The Krein-von Neumann extension of a regular even order quasi-differential operator

We characterize by boundary conditions the Krein-von Neumann extension of a strictly positive minimal operator corresponding to a regular even order quasi-differential expression of Shin-Zettl type. The characterization is stated in terms of a specially chosen basis for the kernel of the maximal operator and employs a description of the Friedrichs extension due to Möller and Zettl.

Self-Adjoint Extensions with Friedrichs Lower Bound

We produce a simple criterion and a constructive recipe to identify those self-adjoint extensions of a lower semi-bounded symmetric operator on Hilbert space which have the same lower bound as the Friedrichs extension. Applications of this abstract result to a few instructive examples are then discussed.

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.

Estimates of Solutions of Elliptic Differential-Difference Equations with Degeneration

We consider a second-order differential-difference equation in a bounded domain Q ⊂ Rn. We assume that the differential-difference operator contains some difference operators with degeneration corresponding to differentiation operators. Moreover, the differential-difference operator under consideration cannot be expressed as a composition of a difference operator and a strongly elliptic differential operator. Degenerated difference operators do not allow us to obtain the G˚arding inequality. We prove a priori estimates from which it follows that the differential-difference operator under consideration is sectorial and its Friedrichs extension exists. These estimates can be applied to study the spectrum of the Friedrichs extension as well. It is well known that elliptic differential-difference equations may have solutions that do not belong even to the Sobolev space W 1(Q). However, using the obtained estimates, we can prove some smoothness of solutions, though not in the whole domain Q, but inside some subdomains Qr generated by the shifts of the boundary, where U Qr = Q.

Sesquilinear Forms in Hilbert Spaces

The centre-pieces of this chapter are the Lax–Milgram Theorem and the existence of weak or variational solutions to problems involving sesquilinear forms. An important application is to Kato’s First Representation Theorem, which associates a unique m-sectorial operator with a closed, densely defined sesquilinear form, thus extending the Friedrichs extension for a lower bounded symmetric operator. Stampacchia’s generalization of the Lax–Milgram Theorem to variational inequalities is also discussed.