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.

Singular solitons and spectral meromorphy

Along with regular solutions of soliton equations one usually can also construct interesting classes of singular solutions. The conditions for the compatibility of their singularities with the dynamics prescribed by the equation impose stringent restrictions on the form of the singular points. For instance, the known meromorphic solutions of the Korteweg-de Vries equation have second-order poles with respect to the space variable, and the leading coefficient is always a triangular number. Singular finite-gap solutions are an important example of this type of solution. In the case of one space dimension the eigenfunctions of the auxiliary linear operators with pole singularities that are compatible with the dynamics turn out to be also locally meromorphic for all values of the spectral parameter. This property, which will be called spectral meromorphy, makes it possible to define a natural indefinite metric on the space spanned by the eigenfunctions, and the number of negative squares of this metric is a new integral of motion. Also discussed are analogues of these results for two-dimensional problems. Bibliography: 50 titles.

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.

An extension of Herglotz's theorem to the quaternions

A classical theorem of Herglotz states that a function n↦r(n) from Z into Cs×s is positive definite if and only if there exists a Cs×s-valued positive measure μ on [0,2π] such that r(n)=∫02πeintdμ(t) for n∈Z. We prove a quaternionic analogue of this result when the function is allowed to have a number of negative squares. A key tool in the argument is the theory of slice hyperholomorphic functions, and the representation of such functions which have a positive real part in the unit ball of the quaternions. We study in great detail the case of positive definite functions.

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.

On the negative squares of a class of self‐adjoint extensions in Krein spaces

AbstractA description of all exit space extensions with finitely many negative squares of a symmetric operator of defect one is given via Krein's formula. As one of the main results an exact characterization of the number of negative squares in terms of a fixed canonical extension and the behaviour of a function τ (that determines the exit space extension in Krein's formula) at zero and at infinity is obtained. To this end the class of matrix valued \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {D}_\kappa ^{n\times n}$\end{document}‐functions is introduced and, in particular, the properties of the inverse of a certain \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {D}_\kappa ^{2\times 2}$\end{document}‐function which is closely connected with the spectral properties of the exit space extensions with finitely many negative squares is investigated in detail. Among the main tools here are the analytic characterization of the degree of non‐positivity of generalized poles of matrix valued generalized Nevanlinna functions and some extensions of recent factorization results.

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.

On the Negative Spectrum of One-Dimensional Schrödinger Operators with Point Interactions

We investigate negative spectra of one-dimensional (1D) Schrodinger operators with δ- and δ′-interactions on a discrete set in the framework of a new approach. Namely, using the technique of boundary triplets and the corresponding Weyl functions, we complete and generalize the results of Albeverio and Nizhnik (Lett Math Phys 65:27–35, 2003; Methods Funct Anal Topol 9(4):273–286, 2003). For instance, we propose an algorithm for determining the number of negative squares of the operator with δ-interactions. We also show that the number of negative squares of the operator with δ′-interactions equals the number of negative strengths.

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.

Classification of all noncommutative polynomials whose Hessian has negative signature one and a noncommutative second fundamental form

Every symmetric polynomial p = p(x) = p(x1,..., xg) (with real coefficients) in g noncommuting variables x1,..., xg can be written as a sum and difference of squares of noncommutative polynomials: $$ (SDS) p(x) = \sum\limits_{j = 1}^{\sigma _ + } {f_j^ + (x)^T f_j^ + (x)} - \sum\limits_{\ell = 1}^{\sigma _ - } {f_\ell ^ - (x)^T f_\ell ^ - (x)} , $$ where fj+, fl− are noncommutative polynomials. Let σ−min(p), the negative signature of p, denote the minimum number of negative squares used in this representation; and let the Hessian of p be defined by the formula $$ p''(x)[h]: = \frac{{d^2 p(x + th)}} {{dt^2 }}|_{t = 0} . $$ In this paper, we classify all symmetric noncommutative polynomials p(x) such that $$ \sigma _ - ^{min} (p'') \leqslant 1. $$ We also introduce the relaxed Hessian of a symmetric polynomial p of degree d via the formula $$ p''_{\lambda ,\delta } (x)[h]: = p''(x)[h] + \delta \sum {m(x)^T h_j^2 m(x) + } \lambda p'(x)[h]^T p'(x)[h] $$ for λ, δ, ∈, ℝ and show that if this relaxed Hessian is positive semidefinite in a suitable and relatively innocuous way, then p has degree at most 2. Here the sum is over monomials m(x) in x of degree at most d − 1 and 1 ≤ j ≤ g. This analysis is motivated by an attempt to develop properties of noncommutative real algebraic varieties pertaining to curvature, since, as will be shown elsewhere, \( - \left\langle {p''_{\lambda ,\delta } (x)[h]v,v} \right\rangle \) (appropriately restricted) plays the role of a noncommutative second fundamental form.

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.

On the negative squares of indefinite Sturm–Liouville operators

The number of negative squares of all self-adjoint extensions of a simple symmetric operator of defect one with finitely many negative squares in a Krein space is characterized in terms of the behaviour of an abstract Titchmarsh–Weyl function near 0 and ∞. These results are applied to a large class of symmetric and self-adjoint indefinite Sturm–Liouville operators with indefinite weight functions.

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.

Completely J -positive linear systems of finite order

Completely J — positive linear systems of finite order are introduced as a generalization of completely symmetric linear systems. To any completely J — positive linear system of finite order there is associated a defining measure with respect to which the transfer function has a certain integral representation. It is proved that these systems are asymptotically stable. The observability and reachability operators obey a certain duality rule and the number of negative squares of the Hankel operator is estimated. The Hankel operator is bounded if and only if a certain measure associated with the defining measure is of Carleson type. We prove that a real symmetric operator valued function which is analytic outside the unit disk has a realization with a completely J — symmetric linear space which is reachable, observable and parbalanced. Uniqueness and spectral minimality of the completely J — symmetric realizations are discussed.

On Representations of Matrix Valued Nevanlinna Functions by u - Resolvents

We show that every matrix valued generalized Nevanlinna function can be represented as a u-resolvent of a certain selfadjoint relation acting in a Pontryagin space. The negative index of this Pontryagin space may be larger than the number of negative squares of the given function. The minimal index of negative squares which is needed to obtain such a representation is determined. In the case of scalar functions, the results presented give rise to some new classes of generalized Nevanlinna functions.