Krein Space Research Articles

The Virozub–Matsaev Condition and Spectrum of Definite Type for Self-adjoint Operator Functions

We establish sufficient conditions for the so-called Virozub–Matsaev condition for twice continuously differentiable self-adjoint operator functions. This condition is closely related to the existence of a local spectral function and to the notion of positive type spectrum. Applications to self-adjoint operators in Krein spaces and to quadratic operator polynomials are given.

Boundary Relations and Generalized Resolvents of Symmetric Operators in Krein Spaces

The classical Krein-Naimark formula establishes a one-to-one correspondence between the generalized resolvents of a closed symmetric operator in a Hilbert space and the class of Nevanlinna families in a parameter space. Recently it was shown by V.A. Derkach, S. Hassi, M.M. Malamud and H.S.V. de Snoo that these parameter families can be interpreted as so-called Weyl families of boundary relations, and a new proof of the Krein-Naimark formula in the Hilbert space setting was given with the help of a coupling method. The main objective of this paper is to adapt the notion of boundary relations and their Weyl families to the Krein space case and to prove some variants of the Krein-Naimark formula in an indefinite setting.

A Sturm–Liouville problem depending rationally on the eigenvalue parameter

AbstractWe consider the Sturm–Liouville problem (1.1) and (1.2) with a potential depending rationally on the eigenvalue parameter. With these equations a λ ‐linear eigenvalue problem is associated in such a way that L2‐solutions of (1.1), (1.2) correspond to eigenvectors of a linear operator. If the functions q and u are real and satisfy some additional conditions, the corresponding linear operator is a definitizable self‐adjoint operator in some Krein space. Moreover we consider the problem (1.1) and (1.3) on the positive half‐axis. Here we use results on the absense of positive eigenvalues for Sturm–Liouville operators to exclude critical points of the associated definitizable operator. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On finite rank perturbations of definitizable operators

It was shown by P. Jonas and H. Langer that a selfadjoint definitizable operator A in a Krein space remains definitizable after a finite rank perturbation in resolvent sense if the perturbed operator B is selfadjoint and the resolvent set ρ ( B ) is nonempty. It is the aim of this note to prove a more general variant of this perturbation result where the assumption on ρ ( B ) is dropped. As an application a class of singular ordinary differential operators with indefinite weight functions is studied.

Noncommuting Domination in Krein Spaces Via Commutators of Block Operator Matrices

The commutators of 2 × 2 block operator matrices with (unbounded) operator entries are investigated. The matrix representation of a symmetric operator in a Krein space is exploited. As a consequence, the domination result due to Cichon, Stochel and Szafraniec is extended to the case of Krein spaces.

Schur Complements in Krein Spaces

The aim of this work is to generalize the notions of Schur complements and shorted operators to Krein spaces. Given a (bounded) J-selfadjoint operator A (with the unique factorization property) acting on a Krein space \({\mathcal{H}}\) and a suitable closed subspace \({\mathcal{S}}\) of \({\mathcal{H}}\), the Schur complement \(A_{/[\mathcal{S}]}\) of A to \({\mathcal{S}}\) is defined. The basic properties of \(A_{/[\mathcal{S}]}\) are developed and different characterizations are given, most of them resembling those of the shorted of (bounded) positive operators on a Hilbert space.

Design of power controller in CDMA system with power and SIR error minimization

In this paper, an uplink power control problem is considered for code division multiple access (CDMA) systems. A distributed algorithm is proposed based on linear quadratic optimal control theory. The proposed scheme minimizes the sum of the power and the error of signal-to-interference ratio (SIR). A power controller is designed by constructing an optimization problem of a stochastic linear quadratic type in Krein space and solving a Kalman filter problem.

Zero distribution of matrix polynomials

The theory of finite dimensional reproducing kernel Krein spaces is exploited to obtain matrix analogues of the Schur–Cohn theorem and the Hermite theorem on the distribution of zeros. Formulas for the number of zeros of the determinant of an m × m matrix polynomial N ( λ ) inside and outside the region Ω + in terms of the signature of an associated matrix (that is subsequently identified as a Bezoutian in the sense of Haimovici and Lerer for appropriately chosen realizations of the polynomials under consideration) are developed when det N ( λ ) ≠ 0 on the boundary of Ω + and Ω + is taken equal to either the open unit disk or the open upper half-plane. The proof is reasonably self contained and reasonably uniform for both choices of Ω + . The conditions imposed are less restrictive than those imposed in the papers [H. Dym, N. Young, A Schur–Cohn theorem for matrix polynomials, Proceedings of the Edinburgh Mathematical Society, vol. 33, 1990, pp. 337–366] and [H. Dym, A Hermite theorem for matrix polynomials, in: Operator Theory: Advances and Applications, vol. 50, Birkhäuser Verlag, Basel, 1991, pp. 191–214]. Comparisons with the Anderson–Jury Bezoutian and the Haimovici–Lerer Bezoutians referred to above are made, and an application to block Toeplitz matrices is given.

Dissipative operators in the Krein space. Invariant subspaces and properties of restrictions

We prove that a dissipative operator in the Krein space has a maximal nonnegative invariant subspace provided that the operator admits matrix representation with respect to the canonical decomposition of the space and the upper right operator in this representation is compact relative to the lower right operator. Under the additional assumption that the upper and lower left operators are bounded (the so-called Langer condition), this result was proved (in increasing order of generality) by Pontryagin, Krein, Langer, and Azizov. We relax the Langer condition essentially and prove under the new assumptions that a maximal dissipative operator in the Krein space has a maximal nonnegative invariant subspace such that the spectrum of its restriction to this subspace lies in the left half-plane. Sufficient conditions are found for this restriction to be the generator of a holomorphic semigroup or a C 0-semigroup.

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.

$H_{\infty}$ Control of Discrete-Time Systems With Multiple Input Delays

This paper is concerned with the finite horizon Hinfin full-information control for discrete-time systems with multiple control and exogenous input delays. We first establish a duality between the Hinfin full-information control and the H2 smoothing of a stochastic backward system in Krein space. Like the duality between the linear quadratic regulation (LQR) of linear systems without delays and the Kalman filtering, the established duality allows us to address complicated multiple input delay problems in a simple way. Indeed, by applying innovation analysis and standard projection in Krein space, in this paper we derive conditions under which the Hinfin full-information control is solvable. An explicit controller is constructed in terms of two standard Riccati difference equations of the same order as the original plant (ignoring the delays). As special cases, solutions to the Hinfin state feedback control problem for systems with delays only in control inputs and the Hinfin control with preview are obtained. An example is given to demonstrate the effectiveness of the proposed Hinfin control design

On the spectral theory of singular indefinite Sturm–Liouville operators

We consider a singular Sturm–Liouville differential expression with an indefinite weight function and we show that the corresponding self-adjoint differential operator in a Krein space locally has the same spectral properties as a definitizable operator.

The Spherically Symmetric a2–dynamo and Some of its Spectral Peculiarities

A brief overview is given of recent results on the spectral properties of spherically symmetric MHD α2-dynamos. In particular, the spectra of sphere-confined fluid or plasma configurations with physically realistic boundary conditions (BCs) (surrounding vacuum) and with idealized BCs (super-conducting surrounding) are discussed. The subjects comprise third-order branch points of the spectrum, self-adjointness of the dynamo operator in a Krein space as well as the resonant unfolding of diabolical points. It is sketched how certain classes of dynamos with a strongly localized α-profile embedded in a conducting surrounding can be mode decoupled by a diagonalization ofthe dynamo operator matrix. A mapping of the dynamo eigenvalue problem to that of a quantum mechanical Hamiltonian with energy dependent potential is used to obtain qualitative information about the spectral behavior. Links to supersymmetric Quantum Mechanics and to the Dirac equation are indicated.

Two-pulse solutions in the fifth-order KdV equation: Rigorous theory and numerical approximations

We revisit existence and stability of two-pulse solutions in the fifth-order Korteweg–de Vries (KdV) equation with two new results. First, we modify the Petviashvili method of successive iterations for numerical (spectral) approximations of pulses and prove convergence of iterations in a neighborhood of two-pulse solutions. Second, we prove structural stability of embedded eigenvalues of negative Krein signature in a linearized KdV equation. Combined with stability analysis in Pontryagin spaces, this result completes the proof of spectral stability of the corresponding two-pulse solutions. Eigenvalues of the linearized problem are approximated numerically in exponentially weighted spaces where embedded eigenvalues are isolated from the continuous spectrum. Approximations of eigenvalues and full numerical simulations of the fifth-order KdV equation confirm stability of two-pulse solutions associated with the minima of the effective interaction potential and instability of two-pulse solutions associated with the maxima points.

Shapes of numerical ranges of operators on a 3-dimensional Krein space

Shapes of numerical ranges of operators on a 3-dimensional Krein space are investigated. Namely, a criterion for convexity is stated, an inclusion domain is obtained and properties like closedness are studied. Mathematics Subject Classification: 46C20; 47A12; 15A60

An Innovation Approach to <formula formulatype="inline"><tex>$H_{\infty }$</tex></formula> Fixed-Lag Smoothing for Descriptor Systems

The finite horizon H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> fixed-lag smoothing problem for linear descriptor systems is considered and a sufficient condition for the existence of an H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> smoother is derived. The key approach applied for deriving the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> fixed-lag smoother is the re-organization innovation analysis in Krein space. Under the Krein space, the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> fixed-lag smoothing is converted into an H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> estimation problem for the system with current and delayed measurements. By using innovation re-organization approach, we show that the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> estimation with delayed measurements is transformed into a problem of delay-free measurements and thus a simple solution to H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> smoothing is given. At last, an illustrative example shows the efficiency of the proposed estimator

On the Indefinite Trigonometric Moment Problem of I. S. Iohvidov and M. G. Kreîn

An indefinite trigonometric moment problem is solved using the one-step completion and the geometry involved in extending factorizations in Krein spaces. In addition, a description of all the solutions is obtained as well as a simple characterization of the case when there exists exactly one solution of the problem.

An analytic characterization of the eigenvalues of self-adjoint extensions

Let A ˜ be a self-adjoint extension in K ˜ of a fixed symmetric operator A in K ⊆ K ˜ . An analytic characterization of the eigenvalues of A ˜ is given in terms of the Q-function and the parameter function in the Krein–Naimark formula. Here K and K ˜ are Krein spaces and it is assumed that A ˜ locally has the same spectral properties as a self-adjoint operator in a Pontryagin space. The general results are applied to a class of boundary value problems with λ-dependent boundary conditions.

General aspects of -symmetric and -self-adjoint quantum theory in a Krein space

In our previous work, we proposed a mathematical framework for PT-symmetric quantum theory, and in particular constructed a Krein space in which PT-symmetric operators would naturally act. In this work, we explore and discuss various general consequences and aspects of the theory defined in the Krein space, not only spectral property and PT symmetry breaking but also several issues, crucial for the theory to be physically acceptable, such as time evolution of state vectors, probability interpretation, uncertainty relation, classical-quantum correspondence, completeness, existence of a basis, and so on. In particular, we show that for a given real classical system we can always construct the corresponding PT-symmetric quantum system, which indicates that PT-symmetric theory in the Krein space is another quantization scheme rather than a generalization of the traditional Hermitian one in the Hilbert space. We propose a postulate for an operator to be a physical observable in the framework.

A scheme to design power controller in wireless network systems

Abstract In this Letter, power control problem is firstly studied at an angle of LQG measurement-feedback control problem. A stochastic uplink power control problem is considered for CDMA systems. An effective distributed algorithm is proposed based on stochastic linear quadratic optimal control theory assuming SIR measurements contain white noise. The presented scheme minimizes the sum of the power and the error of SIR. A measurement-feedback power controller is designed by constructing an optimization problem of a stochastic linear quadratic type in Krein space and solving the Kalman filter problem for the systems.