Daniel Alpay

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.

On Applications of Reproducing Kernel Spaces to the Schur Algorithm and Rational J Unitary Factorization

The main theme of the first half of this paper rests upon the fact that there is a reproducing kernel Hilbert space of vector valued functions B (X) associated with each suitably restricted matrix valued analytic function X. The deep structural properties of certain classes of these spaces, and the theory of isometric and contractive inclusion of pairs of such spaces, which originates with de Branges, partially in collaboration with Rovnyak, is utilized to develop an algorithm for constructing a nested sequence B(X) ⊃ B(X 1) ⊃... of such spaces, each of which is included isometrically in its predecessor. This leads to a new and pleasing viewpoint of the Schur algorithm and various matrix generalizations thereof. The same methods are used to reinterpret the factorization of rational J inner matrices and a number of related issues, from the point of view of isometric inclusion of certain associated sequences of reproducing kernel Hilbert spaces.

Hilbert spaces of analytic functions, inverse scattering and operator models.II

This is the second and final part of a paper which appeared in a preceding issue of this journal. Herein the methods developed in the earlier sections of this paper are used first, in conjunction with some ideas of Krein, to develop models for simple, closed symmetric [resp. isometric]operators with finite and equal deficiency indices. A number of other related issues and applications are then discussed briefly. These include entropy inequalities, interpolation, parametrization ofJ inner matrices, the Schur algorithm and canonical equations. Finally, a list of misprints for Part I is incorporated at the end.

Unitary Rational Matrix Functions

This paper contains the theory of realization and minimal factorization of rational matrix valued functions unitary on the unit circle or on the imaginary line (in the framework of a, generally speaking, indefinite scalar product). The Blaschke-Potapov decomposition for unitary or J-inner rational matrix-valued functions is obtained as a corollary, and the inertia theorems are explained from the point of view of the theory of rational matrix-valued functions. The main results are also used to obtain decomposition theorems for selfadjoint matrix-valued functions.

The Schur Algorithm, Reproducing Kernel Spaces and System Theory

Introduction Reproducing kernel spaces Theory of linear systems Schur algorithm and inverse scattering problem Operator models Interpolation The indefinite case The non-stationary case Riemann surfaces Conclusion Bibliography Index.

Reproducing kernel spaces and applications

The tools developed in the previous chapters allow us to define and study in the operator-valued case the various families of functions appearing in classical Schur analysis. In this section we obtain realization formulas for these functions. These formulas in turn have important consequences, such as the existence of slice hyperholomorphic extensions and results in function theory such as Bohr’s inequality. Recall that all two-sided quaternionic vector spaces are assumed to satisfy condition (5.4). An important tool in this chapter is Shmulyan’s theorem on densely defined contractive relations between Pontryagin spaces with the same index, see Theorem 5.7.8, and this forces us to take two-sided quaternionic Pontryagin spaces with the same index for coefficient spaces, and not Krein spaces. The rational case, studied in the following chapter, corresponds to the setting where both the coefficient spaces and the reproducing kernel Pontryagin spaces associated to the various functions are finite-dimensional.

Lossless Inverse Scattering and Reproducing Kernels for Upper Triangular Operators

In this paper the topics mentioned in the title are studied in the algebra of upper triangular bounded linear operators acting on the space l N 2 of “square summable” sequences f = (..., f −1, f 0, f l,...) with components f i in a complex separable Hilbert space N. Enroute analogues of point evaluation (where in this case the points are operators and the functions are operator valued functions of operators) and simple Blaschke products are developed in this more general context. These tools are then used to establish a theory of structured reproducing kernel Hilbert spaces in the class of upper triangular Hilbert Schmidt operators on l N 2. (An application of these tools and spaces to solve a Nevanlinna Pick interpolation problem wherein both the interpolation points and the values assigned at those points are block diagonal operators, will be considered in a future publication.)

Interpolation problems, extensions of symmetric operators and reproducing kernel spaces II

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.

Some finite-dimensional backward-shift-invariant subspaces in the ball and a related interpolation problem

AbstractWe solve Gleasons problem in the reproducing kernel Hilbert space with repoducing kernel