Finite Number Of Negative Squares Research Articles

Commuting operators over Pontryagin spaces with applications to system theory

In this paper we extend vessel theory, or equivalently, the theory of overdetermined 2D systems to the Pontryagin space setting. We focus on realization theorems of the various characteristic functions associated to such vessels. In particular, we develop an indefinite version of de Branges-Rovnyak theory over real compact Riemann surfaces. To do so, we use the theory of contractions in Pontryagin spaces and the theory of analytic kernels with a finite number of negative squares. Finally, we utilize the indefinite de Branges-Rovnyak theory on compact Riemann surfaces in order to prove a Beurling type theorem on indefinite Hardy spaces on finite bordered Riemann surfaces.

Krein-von Neumann extension of an even order differential operator on a finite interval

We describe the Krein-von Neumann extension of minimal operator associated with the expression \(\mathcal{A}:=(-1)^n\frac{d^{2n}}{dx^{2n}}\) on a finite interval \((a,b)\) in terms of boundary conditions. All non-negative extensions of the operator \(A\) as well as extensions with a finite number of negative squares are described.

Representations of Nilpotent Groups on Spaces with Indefinite Metric

The paper studies the structure of J-unitary representations of connected nilpotent groups on \(\Pi _{k}\)-spaces, that is, the representations on a Hilbert space preserving a quadratic form “with a finite number of negative squares”. Apart from some comparatively simple cases, such representations can be realized as double extensions of finite-dimensional representations by unitary ones. So their study is based on some special cohomological technique. We concentrate mostly on the problems of the decomposition of these representations and the classification of “non-decomposable” ones.

Singular soliton operators and indefinite metrics

We consider singular real second order 1D Schrodinger operators such that all local solutions to the eigenvalue problems are x-meromorphic for all λ. All algebrogeometrical potentials (i.e. “singular finite-gap” and “singular solitons”) satisfy to this condition. A Spectral Theory is constructed for the periodic and rapidly decreasing potentials in the classes of functionswith singularities: The corresponding operators are symmetric with respect to some natural indefinite inner product as it was discovered by the present authors. It has a finite number of negative squares in the both (periodic and rapidly decreasing) cases. The time dynamics provided by the KdV hierarchy preserves this number. The right analog of Fourier Transform on Riemann Surfaces with good multiplicative properties (the R-Fourier Transform) is a partial case of this theory. The potential has a pole in this case at x = 0 with asymptotics u ∼ g(g + 1)/x 2. Here g is the genus of spectral curve.

A Schur transformation for functions in a general class of domains

In this paper we present a framework in which the Schur transformation and the basic interpolation problem for generalized Schur functions, generalized Nevanlinna functions and the like can be studied in a unified way. The basic object is a general class of functions for which a certain kernel has a finite number of negative squares. The results are based on and generalize those in previous papers of the first three authors on the Schur transformation in an indefinite setting.

Representation of generalized Toeplitz kernels with a finite number of negative squares

Representation of generalized Toeplitz kernels with a finite number of negative squares

Singular solitons and indefinite metrics

The singular real second order 1D Schrodinger operators are considered here with such potentials that all local solutions near singularities to the eigenvalue problem are meromorphic for all values of the spectral parameter. All algebro-geometrical or "singular finite-gap" potentials satisfy to this condition. A Spectral Theory is constructed here for the periodic and rapidly decreasing cases in the special classes of functions with singularities and indefinite inner product. It has a finite number of negative squares if the unimodular Bloch multipliers are fixed in the periodic case, and in the rapidly decreasing case. The time dynamics provided by the KdV hierarchy preserves this number. The right analog of Fourier Transform for the Riemann Surfaces preserving remarkable multiplicative properties of the ordinary (i.e. genus zero) Fourier Transform based on the standard exponential basis, leads to such operators as it was shown in our previous works.

Generalized Schur Functions and Related de Branges-Rovnyak Spaces in the Banach Space Setting

We study the analogues of de Branges-Rovnyak spaces in the Banach space case. An important role is played by self-adjoint operators from the dual of a Banach space into the Banach space itself. A factorization theorem for such operators is proved in the case when they have a finite number of negative squares.

Integral Representations for Generalized Difference Kernels Having a Finite Number of Negative Squares

An integral representation is derived for matrix-valued generalized difference kernels which have a finite number of negative squares. The representation is used to extend such kernels to the real line with a bound on the number of negative squares. The main results are obtained by means of an operator interpolation theorem. The nondegenerate case is assumed.

Reproducing Kernel Quaternionic Pontryagin Spaces

This paper studies various aspects of reproducing kernel spaces with a possibly indefinite metric when the field of scalar is replaced by the skew–field of quaternions. We first discuss in some details the positive case. A key fact which allows to consider the non–positive case is that Hermitian matrices with quaternionic entries have only real eigenvalues. This permits to extend the notion of functions with a finite number of negative squares to the present setting and we prove in particular that there is a one–to–one correspondence between such functions and reproducing kernel Pontryagin quaternionic spaces.

Krein-Langer Factorizations via Pole Triples

A theorem by Krein and Langer asserts existence of factorizations of special type for operator functions in a generalized Schur class, i. e., meromorphic operator functions defined on the unit disk and such that their Nevanlinna-Pick kernel has a fixed finite number of negative squares. A different view and proof of this theorem are presented, based on description of pole data of meromorphic operator functions in terms of pole pairs and pole triples. A criterion for existence, and a parametrization, of operator functions in a generalized Schur class with given pole triple is obtained.

Loewner’s theorem for kernels having a finite number of negative squares

By a theorem of Loewner, a continuously differentiable real-valued function on a real interval whose difference quotient is a nonnegative kernel is the restriction of a holomorphic function which has nonnegative imaginary part in the upper half-plane and is holomorphic across the interval. An analogous result is obtained when the difference-quotient kernel has a finite number of negative squares.

Functions with a finite number of negative squares on compact groups

Functions with a finite number of negative squares on compact groups

Functions with a finite number of negative squares on semigroups

We consider indefinite functions on semigroups and their relation to representations in spaces with an indefinite metric. Special attention is given to functions of finite rank, where the space of representation is of finite dimension, and to functions for which the corresponding representation consists of bounded operators in Pontrjagin spaces.

On bounded functions with a finite number of negative squares

In the case of a locally compact abelian groupG we give a characterization of bounded continuous functions onG which have a finite number of negative squares. We show that these functions can be regarded in a certain sense as a special case of positive definite functions.

Functions with a Finite Number of Negative Squares

Let f be a complex-valued function defined on the real line R with the property that for every x∊R. If k is a nonnegative integer,f is said to have k negative squares, or to be indefinite of order k, if the Hermitian form