Aad Dijksma

Schur functions, operator colligations, and reproducing kernel Pontryagin spaces

Using the theory of linear relations in Pontryagin spaces we extend to the nonpositive case the theory of reproducing kernel spaces associated with contractions in Hilbert spaces.

A factorization result for generalized Nevanlinna functions of the class N-k

AbstractLetQ∈Nk. It is shown that if α is a nonreal pole or a real generalized pole of nonpositive type and β is a nonreal zero or a real generalized zero of nonpositive type of the functionQ then the function

Self-adjoint operators with inner singularities and pontryagin spaces

Q_1 (z): = \frac{{(z - \alpha )(z - \bar \alpha )}}{{(z - \beta )(z - \bar \beta )}}Q(z)

Minimal Realizations of Scalar Generalized Nevanlinna Functions Related to Their Basic Factorization

belongs to the classNk−1.

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

The aim of part I and this paper is to study interpolation problems for pairs of matrix functions of the extended Nevanlinna class using two different approaches and to make explicit the various links between them. In part I we considered the approach via the Kreîn-Langer theory of extensions of symmetric operators. In this paper we adapt Dyms method to solve interpolation problems by means of the de Branges theory of Hilbert spaces of analytic functions. We also show here how the two solution methods are connected.

The linearization of boundary eigenvalue problems and reproducing kernel Hilbert spaces

Let A 0 be an unbounded self-adjoint operator in a Hilbert space H 0 and let χ be a generalized element of order — m — 1 in the rigging associated with A 0 and the inner product 〈·, ·〉0 of H 0. In [S1, S2, S3] operators H t , t · R ∪ ∞, are defined which serve as an interpretation for the family of operators A 0 + t -1 〈·, χ〉0 χ. The second summand here contains the inner singularity mentioned in the title. The operators H t act in Pontryagin spaces of the form π m = H 0⊕C m ⊕C m where the direct summand space C m ⊕ C m is provided with an indefinite inner product. They can be interpreted both as a canonical extension of some symmetric operator S in π m and also as extensions of a one-dimensional restriction S 0 of A 0 in H 0 and hence they can be characterized by a class of Straus extensions of S 0 as well as via M.G. Krein’s formulas for (generalized) resolvents. In this paper we describe both these realizations explicitly and study their spectral properties. A main role is played by a special class of Q-functions. Factorizations of these functions correspond to the separation of the nonpositive type spectrum from the positive spectrum of H t . As a consequence, in Subsection 7.3 a family of self-adjoint Hilbert space operators is obtained which can serve as a nontrivial quantum model associated with the operators A 0 + t -1 〈·, χ〉0 χ.

The Schur algorithm for generalized Schur functions I: coisometric realizations

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).

Notes on a Nevanlinna-Pick interpolation problem for generalized Nevanlinna functions

In this paper we present a minimal realization of a scalar generalized Nevanlinna function q which corresponds to the basic factorization of q as a product of a Nevanlinna function qo and of a rational function r # r, which collects the generalized poles and generalized zeros of q that are not of positive type. The key tool are reproducing kernel Pontryagin spaces.