H.S.V. de Snoo

Boundary relations and generalized resolvents of symmetric operators

The Kreĭn-Naĭmark formula provides a parametrization of all selfadjoint exit space extensions of a (not necessarily densely defined) symmetric operator in terms of maximal dissipative (in ℂ+) holomorphic linear relations on the parameter space (the so-called Nevanlinna families). The new notion of boundary relation makes it possible to interpret these parameter families as Weyl families of boundary relations and to establish a simple coupling method to construct generalized resolvents from given parameter families. A general version of the coupling method is introduced and the role of the boundary relations and their Weyl families in the Kreĭn-Naĭmark formula is investigated and explained. These notions lead to several new results and new types of solutions to problems involving generalized resolvents and their applications, e.g., in boundary-value problems for (ordinary and partial) differential operators. For instance, an old problem going back to M. A. Naĭmark and concerning the analytic characterization of the so-called Naĭmark extensions is solved.

On the class of extremal extensions of a nonnegative operator

A nonnegative selfadjoint extension Aof a nonnegative operator A is called extremal if inf {(A)(ϕ) - f),ϕ - f) : ∈ dom A} = 0 for all ϕ ∈ dom A.A new construction of all extremal extensions of a nonnegative densely defined operator will be presented.It employs a fixed auxiliary Hilbert space to factorize each extremal extension.Various functional-analytic interpretations of extremal extensions are studied and some new types of characterizations are obtained.In particular,a purely analytic description of extremal extensions is established,based on a class of functions introduced by M.G.Krein and I.E.Ovearenko.

Smooth rank one perturbations of selfadjoint operators

Let A be a selfadjoint operator in a Hilbert space aleph with inner product [.,.]. The rank one perturbations of A have the form A+tau [.,omega]omega, tau epsilon R, for some element omega epsilon aleph. In this paper we consider smooth perturbations, i.e. we consider omega epsilon dom \A\(k/2) for some k epsilon N boolean OR {0}. Function-theoretic properties of their so-called Q-functions and operator-theoretic consequences will be studied.

Subordinate solutions and spectral measures of canonical systems

The theory of 2×2 trace-normed canonical systems of differential equations on ℝ+ can be put in the framework of abstract extension theory, cf. [9]. This includes the theory of strings as developed by I.S. Kac and M.G. Kreįn. In the present paper the spectral properties of such canonical systems are characterized by means of subordinate solutions. The theory of subordinacy for Schrödinger operators on the halfline ℝ+, was originally developed D.J. Gilbert and D.B. Pearson. Its extension to the framework of canonical systems makes it possible to describe the spectral measure of any Nevanlinna function in terms of subordinate solutions of the corresponding trace-normed canonical system, which is uniquely determined by a result of L. de Branges.

The sum of matrix Nevanlinna functions and selfadjoint extensions in exit spaces

For a closed symmetric operator or relation, Kre 137-1 n’s formula describes all its self-adjoint extensions in terms of certain holomorphic parameters. Our interest is in self-adjoint extensions of a symmetric relation which extends itself an orthogonal sum of two symmetric relations. The corresponding class of parameters in Kre 137-2 n’s formula is idcntificd. This leads to a description of (minimal) self-adjoint extensions in a fixed exit space.

Eigenfunction Expansions for Nondensely Defined Differential Operators

The operator L, is closed, densely defined and symmetric with domain ID(&). Let 5, be a p-dimensional (p is finite) subspace of !+j and let S,, be the restriction of L, to a(&,) = a(L,) n !&l. Then S, is a closed operator but not densely defined on 53. By identifying graph and operator we may speak of the adjoint subspace S = S,,* of S, in a2 = fi x

Holomorphic Operators Between Krein Spaces and the Number of Squares of Associated Kernels

j and then we have S, C S. We consider self-adjoint subspace extensions H of S,,: S,, C H = H* C S. Such a self-adjoint subspace H gives rise to a self-adjoint operator H, in the smaller Hilbert space @ 0 H(O), where H(0) is the multivalued part of H. The above theory is given by Coddington [3]. The same author characterizes the self-adjoint subspace extensions of nondensely defined operators in terms of abstract boundary conditions, cf. [4], [Sj. Now, in the boundary conditions determining H appear integral terms and in the expression for H, appear both boundary and integral terms. For such operators Coddington [6], [7] has shown the existence of a matrix valued spectral distribution function p in terms of which he gave eigenfunction expansions converging in the Hilbertspace norm. The principal aim of this paper is to derive an expansion theorem in which point-wise convergence is allowed. Coddington’s norm convergence theorem is a straightforward consequence of our theorem. We will prove our results by adaptation of a method used by Niessen [lo], cf. Rellich [ll]. This adaptation is required to take care of the boundary and integral terms

A relation for the spectral shift function of two self-adjoint extensions

Suppose that Θ(z) is a bounded linear mapping from the Kreĭ space \(\Im \) to the Kreĭn space G, which is defined and holomorphic in a small neighborhood of z = 0. Then often Θ admits realizations as the characteristic function of an isometric, a coisometric and of a unitary colligation in which for each case the state space is a Kreĭn space. If the colligations satisfy minimality conditions (i.e., are controllable, observable or closely connected, respectively) then the positive and negative indices of the state space can be expressed in terms of the number of positive and negative squares of certain kernels associated with Θ, depending on the kind of colligation. In this note we study the relations between the numbers of positive and negative squares of these kernels. Using the Potapov-Ginzburg transform we give a reduction to the case where the spaces \(\Im \) and G are Hilbert spaces. For this case these relations has been considered in detail in [DLS1].

Selfadjoint extensions of the orthogonal sum of symmetric relations II

The spectral shift function for two self-adjoint extensions of a symmetric operator with defect index (1, 1) is expressed by means of the normalized spectral functions of all ‘intermediate’ self-adjoint extensions.

Realization and Factorization in Reproducing Kernel Pontryagin Spaces

Let S 1 and S 2 be closed symmetric linear relations in Hilbert spaces H 1 and H 2 with finite and equal defect numbers. The selfadjoint extensions of the closed symmetric linear relation S = S 1 ⊕ S 2 are studied and a description for these extensions in the Hilbert space H 1 is given. The results are applied to a class of differential operators.