A. Dijksma

A Basic Interpolation Problem for Generalized Schur Functions and Coisometric Realizations

The basic interpolation problem for Schur functions is: Find all Schur functions s(z)for which s (0) has a given value. In this paper we consider the same basic interpolation problem but now for the class of generalized Schur functions with finitely many negative squares which are holomorphic at z = 0. In Section3 its solutions are given by three fractional linear transformations in which the main parameter runs through a subset of the class of generalized Schur functions.

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

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].

The Schur Algorithm for Generalized Schur Functions IV: Unitary Realizations

The generalized Schur transform as defined in [13] (see also [2][6]) is applied to the class A° of all complex-valued functions, which are holomorphic at z = O. Each such function has a coisometric and a unitary realization in some Krein space. We study the effect of this generalized Schur transform to the unitary realization; in [2], [3] we studied similar questions for the coisometric realizations. The main difference with [2], [3] is that a certain one-sidedness is replaced by a two-sidedness, comparable to the difference between the unilateral shift on one-sided sequences and the shift on two-sided sequences. We follow a direct approach in line with [2, 3, 6].

The Schur Transformation for Nevanlinna Functions: Operator Representations, Resolvent Matrices, and Orthogonal Polynomials

A Nevanlinna function is a function which is analytic in the open upper half-plane and has a non-negative imaginary paxt there. In this paper we study a fractional linear transformation for a Nevanlinna function n with a suitable asymptotic expansion at ∞, that is an analogue of the Schur transformation for contractive analytic functions in the unit disk. Applying the transformation p times we find a Nevanlinna function n p which is a fractional linear transformation of the given function n. The main results concern the effect of this transformation to the realizations of n and n p by which we mean their representations through resolvents of self-adjoint operators in Hilbert space. Our tools are block operator matrix representations, u-resolvent matrices, and reproducing kernel Hilbert spaces.

Quadratic (Weakly) Hyperbolic Matrix Polynomials: Direct and Inverse Spectral Problems

Let L be a monic quadratic weakly hyperbolic or hyperbolic n × n matrix polynomial. We solve some direct spectral problems: We prove that the eigenvalues of a compression of L to an (n − 1)-dimensional subspace of ℂ n block-interlace and that the eigenvalues of a one-dimensional perturbation of L (−,+)-interlace the eigenvalues of L. We also solve an inverse spectral problem: We identify two given block-interlacing sets of real numbers as the sets of eigenvalues of L and its compression.

On the Loewner problem in the class _

Loewners theorem on boundary interpolation of N κ functions is proved under rather general conditions. In particular, the hypothesis of Alpay and Rovnyak (1999) that the function f, which is to be extended to an N κ function, is defined and continuously differentiable on a nonempty open subset of the real line, is replaced by the hypothesis that the set on which f is defined contains an accumulation point at which f satisfies some kind of differentiability condition. The proof of the theorem in this note uses the representation of N κ functions in terms of selfadjoint relations in Pontryagin spaces and the extension theory of symmetric relations in Pontryagin spaces.

Colligations in Pontryagin Spaces with a Symmetric Characteristic Function

A symmetry in the characteristic function of a colligation is invest-tigated for its effect on the main operator of the colligation.

Realization and Factorization in Reproducing Kernel Pontryagin Spaces

New proofs are constructed of some realization theorems for Schur functions on arbitrary domains. A generalization of Leech’s factorization theorem is obtained in which the nonnegativity condition on kernels is replaced by a hypothesis of finitely many negative squares.

Augmented Schur Parameters for Generalized Nevanlinna Functions and Approximation

Schur parameters of a Schur function are a well-known concept in Schur analysis. Here we define the Schur transformation and the sequence of Schur parameters for a Nevanlinna function and a generalized Nevanlinna function. They are applied to approximate the Nevanlinna function or the generalized Nevanlinna. function by rational ones.

Nonstationary Analogs of the Herglotz Representation Theorem: Realizations Centered at an Arbitrary Point

In this paper we prove generalized Herglotz representation theorems for bounded upper triangular operators with nonnegative real part when the base “point” (in fact a diagonal operator) is different from 0.