Rank one perturbations at infinite coupling in Pontryagin spaces

In this paper we relate the operators in the operator representations of a generalized Nevanlinna function N(z) and of the function −N(z)⁻¹ under the assumption that z=∞ is the only (generalized) pole of nonpositive type. The results are applied to the Q-function for S and H and the Q-function for S and H^∞, where H is a self-adjoint operator in a Pontryagin space with a cyclic element w, H^∞ is the self-adjoint relation obtained from H and w via a rank one perturbation at infinite coupling, and S is the symmetric operator given by S=H∩H^∞.     

Un théorème de type Beurling–Lax dans la boule unitéA Beurling–Lax type theorem in the unit ball

We prove a Beurling–Lax theorem for a family of reproducing kernel Hilbert spaces of functions analytic in an open subset of the unit ball containing the origin. The spaces under consideration are characterized by functions called Schur multipliers. Using the theory of linear relations in Pontryagin spaces we also give coisometric realizations of Schur multipliers. To cite this article: D. Alpay et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 349–354.     

On the Loewner problem in the class

Loewner's theorem on boundary interpolation of functions is proved under rather general conditions. In particular, the hypothesis of Alpay and Rovnyak (1999) that the function , which is to be extended to an 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 is defined contains an accumulation point at which satisfies some kind of differentiability condition. The proof of the theorem in this note uses the representation of functions in terms of selfadjoint relations in Pontryagin spaces and the extension theory of symmetric relations in Pontryagin spaces.

     

On the Nevanlinna-Pick interpolation problem for generalized stieltjes functions

The solutions of the Nevanlinna-Pick interpolation problem for generalized Stieltjes matrix functions are parametrized via a fractional linear transformation over a subset of the class of classical Stieltjes functions. The fractional linear transformation of some of these functions may have a pole in one or more of the interpolation points, hence not all Stieltjes functions can serve as a parameter. The set of excluded parameters is characterized in terms of the two related Pick matrices.Dedicated to the memory of M. G. Kreîn     

High Order Singular Rank One Perturbations of a Positive Operator

In this paper self-adjoint realizations in Hilbert and Pontryagin spaces of the formal expression are discussed and compared. Here L is a positive self-adjoint operator in a Hilbert space with inner product 〈·,·〉, α is a real parameter, and φ in the rank one perturbation is a singular element belonging to with n ≥ 3, where is the scale of Hilbert spaces associated with L in      

Closedness and Adjoints of Products of Operators, and Compressions

We reprove and slightly improve theorems of Nudelman and Stenger about compressions of maximal dissipative and self-adjoint operators to subspaces of finite codimension and discuss related results concerning the closedness and the adjoint of a product of two operators on a Hilbert space.     

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

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. Present address:Department of Mathematics Ben-Gurion University of the Negev Beersheva Israel     

Index formulae for subspaces of Kreîn spaces

For a subspaceS of a Kreîn spaceK and an arbitrary fundamental decompositionK=K ^?[+]K ⁺ ofK, we prove the index formula $$\kappa ^ - \left( \mathcal{S} \right) + \dim \left( {\mathcal{S}^ \bot \cap \mathcal{K}^ + } \right) = \kappa ^ + \left( {\mathcal{S}^ \bot } \right) + \dim \left( {\mathcal{S} \cap \mathcal{K}^ - } \right)$$ where κ^±(S) stands for the positive/negative signature ofS. The difference dim(S∩K ^?)?dim(S ^⊥∩K ⁺), provided it is well defined, is called the index ofS. The formula turns out to unify other known index formulac for operators or subspaces in a Kreîn space.