Henk de Snoo

Boundary relations and their Weyl families

The concepts of boundary relations and the corresponding Weyl families are introduced. Let S be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space h, let H be an auxiliary Hilbert space, let [GRAPHICS] and let JH be defined analogously. A unitary relation G from the Krein space (h(2), J(h)) to the Kre. in space (H-2, J(H)) is called a boundary relation for the adjoint S* if ker Gamma = S. The corresponding Weyl family M(lambda) is de. ned as the family of images of the defect subspaces (n) over cap (lambda), lambda is an element of C \ R under Gamma. Here Gamma need not be surjective and is even allowed to be multi-valued. While this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex Hilbert space H and the class of unitary relations Gamma : ( H-2, J(H)) -> (H-2, J(H)), it also generalizes the notion of so-called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems. Moreover, these new notions yield, for instance, the following realization theorem: every H-valued maximal dissipative (for lambda is an element of C+) holomorphic family of linear relations is the Weyl family of a boundary relation, which is unique up to unitary equivalence if certain minimality conditions are satisfied. Further connections between analytic and spectral theoretical properties of Weyl families and geometric properties of boundary relations are investigated, and some applications are given.

Hamiltonian Systems with Eigenvalue Depending Boundary Conditions

In earlier papers [DLS1–6] we have described the selfadjoint extensions, in indefinite inner product spaces and with nonempty resolvent sets of a symmetric closed relation S in a Hilbert space ℌ by means of generalized resolvents, characteristic functions and Straus extensions. In this paper we show how these results can be applied when S comes from a 2n×2n Hamiltonian system of ordinary differential equations on an interval [a, b), (1.1) Jy′(t) = (lΔ(t) + H(t)) y(t) + Δ(t)f(t), t ∈[a, b), l∈ℂ, which is regular in a and in the limit point case in b; for further specifications see Section 5. We pay special attention to selfadjoint extensions beyond the given space ℌ, as they give rise to eigenvalue and boundary value problems with boundary conditions of the form (1.2) A(l)y 1 (a) + B(l)y 2 (a) = 0, in which the matrix coefficients A(l) and В(l) depend holomorphically on the eigenvalue parameter l, see Theorem 7.1 below. The eigenvalue problem for such a selfadjoint extension in a larger space can be considered as a linearization of the corresponding boundary value problem (1.1) and (1.2).

Interpolation problems, extensions of symmetric operators and reproducing kernel spaces II: Missing section 3

By an oversight on the part of the authors this section was not included in the paper previously published in Integral Equations Operator Theory, volume 14/4 (1991), 466–500.

Extremal extensions for the sum of nonnegative selfadjoint relations

The sum A + B of two nonnegative selfadjoint relations (multivalued operators) A and B is a nonnegative relation. The class of all extremal extensions of the sum A + B is characterized as products of relations via an auxiliary Hilbert space associated with A and B. The so-called form sum extension of A+B is a nonnegative selfadjoint extension, which is constructed via a closed quadratic form associated with A and B. Its connection to the class of extremal extensions is investigated and a criterion for its extremality is established, involving a nontrivial dependence on A and B.

Sesquilinear forms corresponding to a non-semibounded Sturm-Liouville operator

Let - DpD be a differential operator on the compact interval [-b, b] whose leading coefficient is positive on (0, b] and negative on [b,0), with fixed, separated, self-adjoint boundary conditions at h and b and an additional interface condition at 0. The self-adjoint extensions of the corresponding minimal differential operator are non-semibounded and are related to non-semibounded sesquilinear forms by a generalization of Katos representation theorems. The theory of non-semibounded sesquilinear forms is applied to this concrete situation. In particular, the generalized Friedrichs extension is obtained as the operator associated with the unique regular closure of the minimal sesquilinear form. Moreover, among all closed forms associated with the self-ad joint extensions, the regular closed forms are identified. As a consequence, eigenfunction expansion theorems are obtained for the differential operators as well as for certain indefinite Krein-Feller operators with a single concentrated mass.

Monotone convergence theorems for semi-bounded operators and forms with applications

Let Hn be a monotone sequence of non-negative self-adjoint operators or relations in a Hilbert space. Then there exists a self-adjoint relation H∞ such that Hn converges to H∞ in the strong resolvent sense. This result and related limit results are explored in detail and new simple proofs are presented. The corresponding statements for monotone sequences of semi-bounded closed forms are established as immediate consequences. Applications and examples, illustrating the general results, include sequences of multiplication operators, Sturm–Liouville operators with increasing � ∞ � ∞ �

Domain and Range Descriptions for Adjoint Relations, and Parallel Sums and Differences of Forms

The adjoint of a linear operator or relation from a Hilbert space \( \mathfrak{H} \) to a Hilbert space \( \mathfrak{K} \) is a closed linear relation. The domain and the range of the adjoint are characterized in terms of certain mappings defined on K and \( \mathfrak{H} \), respectively. These characterizations are applied to contractions between Hilbert spaces and to the form domains and ranges of the Friedrichs and Kreĭn-von Neumann extensions of a nonnegative operator or relation. Furthermore these characterizations are used to introduce and derive properties of the parallel sum and the parallel difference of a pair of forms on a linear space.

Generalized Friedrichs Extensions Associated with Interface Conditions for Sturm-Liouville Operators

For a class of Sturm-Liouville operators with an interface condition at an interior point all selfadjoint realizations are determined. This result is obtained via a description of the selfadjoint extensions of the coupling of two symmetric operators. The (generalized) Friedrichs extension, when it exists, is determined. Sufficient conditions for the (generalized) Friedrichs extension to exist are given.

Generalized Nevanlinna Functions with Polynomial Asymptotic Behaviour at Infinity and Regular Perturbations

Let N K be the class of meromorphic functions Q(z) defined on ℂ\ℝ with \( Q\left( {\overline z } \right) = \overline {Q\left( z \right)}, \) and such that on its domain of holomorphy the kernel