Krein Space Research Articles

On a class of non-Hermitian matrices with positive definite Schur complements

Given a positive definite nXn matrix A and a Hermitian mXm matrix D, we characterize under which conditions there exists a strictly contractive matrix K such that the non-Hermitian block-matrix with the enties A and -AK in the first row and K^*A and D in the second has a positive definite Schur complement with respect to its submatrix A. Additionally, we show that K can be chosen such that diagonalizability of the block-matrix is guaranteed and we compute its spectrum. Moreover, we show a connection to the recently developed frame theory for Krein spaces.

More on geometry of Krein space C-numerical range

For n × n complex matrices A, C and H, where H is non-singular Hermitian, the Krein space C-numerical range of A induced by H is the subset of the complex plane given by {Tr(CU[*]AU):U−1=U[*]} with U[*]=H−1U*H the H-adjoint matrix of U. We revisit several results on the geometry of Krein space C-numerical range of A and in particular we obtain a condition for the Krein space C-numerical range to be a subset of the real line.

On $J$ -frames related to maximal definite subspaces

We propose a definition of frames in Krein spaces which generalizes the concept of J-frames defined relatively recently by Giribet, Maestripieri, Martínez-Pería, and Massey. The difference consists in the fact that a J-frame is related to maximal definite subspaces M± which are not assumed to be uniformly definite. The latter allows us to extend the set of J-frames. In particular, some J-orthogonal Schauder bases can be interpreted as J-frames.

On the K-theoretic classification of dynamically stable systems

This paper deals with the construction of a suitable topological [Formula: see text]-theory capable of classifying topological phases of dynamically stable systems described by gapped [Formula: see text]-self-adjoint operators on a Krein space with indefinite metric [Formula: see text].

On hyponormal operators in Krein spaces

In this paper the hyponormal operators on Krein spaces are introduced. We state conditions for the hyponormality of bounded operators focusing, in particular, on those operators $T$ for which there exists a fundamental decomposition $\mathbb{K}= \mathbb{K}^{+} \oplus \mathbb{K}^{-}$ of the Krein space $\mathbb{K}$ with $\mathbb{K}^{+}$ and $\mathbb{K}^{-}$ invariant under $T$.

Dynamics of finite dimensional non-hermitian systems with indefinite metric

We discuss the time evolution of physical finite dimensional systems which are modelled by non-hermitian Hamiltonians. We address both general non-hermitian Hamiltonians and pseudo-hermitian ones. We apply the theory of Krein Spaces to construct metric operators and well-defined inner products. As an application, we study the stationary behavior of dissipative one axis twisting Hamiltonians. We discuss the effect of decoherence under different coupling schemes.

<inline-formula> <tex-math notation="LaTeX">$H_{\infty}$ </tex-math> </inline-formula> Fault Estimation for 2-D Linear Discrete Time-Varying Systems Based on Krein Space Method

This paper addresses the finite horizon ${H_\infty }$ fault estimation problem for 2-D linear discrete time-varying systems with bounded unknown input and measurement noise. The main contribution of this paper is the ${H_\infty }$ fault estimator for 2-D systems with a necessary and sufficient existence condition. By introducing a partially equivalent stochastic dynamic system in Krein space, the necessary and sufficient condition for the existence of the ${H_\infty }$ fault estimator is derived based on innovation analysis and projection formula in Krein space. Then, the solution of the estimator is achieved by means of a Riccati-like difference equation for 2-D systems. Finally, a thermal process example is given to demonstrate the effectiveness of the proposed method.

Rigged de Branges–Pontryagin spaces and their application to extensions and embedding

De Branges–Pontryagin spaces B(E) with negative index κ of entire p×1 vector valued functions based on an entire p×2p entire matrix valued function E(λ) (called the de Branges matrix) are studied. An explicit description of these spaces and an explicit formula for the indefinite inner product are presented. A characterization of those spaces B(E) that are invariant under the generalized backward shift operator that extends known results when κ=0 is given. The theory of rigged de Branges–Pontryagin spaces is developed and then applied to obtain an embedding of de Branges matrices with negative squares in generalized J-inner matrices and selfadjoint extensions of the multiplication operator in B(E). A formula for factoring an arbitrary generalized J-inner matrix valued function into the product of a singular factor and a perfect factor is found analogous to the known factorization formulas for J-inner matrix valued functions.

THE SPECTRAL DECOMPOSITION OF CAUCHY PROBLEM’S SOLUTION FOR LAPLACE EQUATION

The spectral decomposition of Cauchy problem for Laplace equation is obtained in Krein space, and is made a regularization of a problem, using the resolvent of the corresponding operator.

The factorization of generalized Nevanlinna functions and the invariant subspace property

The well-known invariant subspace property of selfadjoint relations (multi-valued operators) in Pontryagin spaces is shown to be equivalent to the factorization property of (scalar) generalized Nevanlinna functions. This connection is established by a new realization for generalized Nevanlinna functions explicitly reflecting this connection. Combining this result with the new function-theoretic proof for the factorization property of generalized Nevanlinna functions contained in Wietsma (2018) immediately yields a new proof for the invariant subspace property of selfadjoint relations in Pontryagin spaces.

Superunitary Representations of Heisenberg Supergroups

Abstract Numerous Lie supergroups do not admit superunitary representations (SURs) except the trivial one, for example, Heisenberg and orthosymplectic supergroups in mixed signature. To avoid this situation, we introduce in this paper a broader definition of SUR, relying on a new definition of Hilbert superspace. The latter is inspired by the notion of Krein space and was developed initially for noncommutative supergeometry. For Heisenberg supergroups, this new approach yields a smooth generalization, whatever the signature, of the unitary representation theory of the classical Heisenberg group. First, we obtain Schrödinger-like representations by quantizing generic coadjoint orbits. They satisfy the new definition of irreducible SURs and serve as ground to the main result of this paper: a generalized Stone–von Neumann theorem. Then, we obtain the superunitary dual and build a group Fourier transformation, satisfying Parseval theorem. We eventually show that metaplectic representations, which extend Schrödinger-like representations to metaplectic supergroups, also fit into this definition of SURs.

Space and time dimensions of algebras with application to Lorentzian noncommutative geometry and quantum electrodynamics

An analogy with real Clifford algebras on even-dimensional vector spaces suggests assigning an ordered pair (s, t) of space and time dimensions (or equivalently an ordered pair (m, n) of metric and KO dimensions) modulo 8 to any algebraic structure (that we call CPT corepresentation) represented over a Hilbert space by two self-adjoint involutions and an anti-unitary operator having specific commutation relations. It is shown that this assignment is compatible with the tensor product: the space and time dimensions of the tensor product of two CPT corepresentations are the sums of the space and time dimensions of its factors, and the same holds for the metric and KO dimensions. This could provide an interpretation of the presence of such algebras in PT-symmetric Hamiltonians or the description of topological matter. This construction is used to build an indefinite (i.e., pseudo-Riemannian) version of the spectral triple of noncommutative geometry, defined over a Krein space and classified by the pair (m, n) instead of the KO dimension only. Within this framework, we can express the Lagrangian (both bosonic and fermionic) of a Lorentzian almost-commutative spectral triple. We exhibit a space of physical states that solves the fermion-doubling problem. The example of quantum electrodynamics is described.

Families of spectral triples and foliations of space(time)

We study a noncommutative analog of a spacetime foliated by spacelike hypersurfaces, in both Riemannian and Lorentzian signatures. First, in the classical commutative case, we show that the canonical Dirac operator on the total spacetime can be reconstructed from the family of Dirac operators on the hypersurfaces. Second, in the noncommutative case, the same construction continues to make sense for an abstract family of spectral triples. In the case of Riemannian signature, we prove that the construction yields in fact a spectral triple, which we call a product spectral triple. In the case of Lorentzian signature, we correspondingly obtain a “Lorentzian spectral triple,” which can also be viewed as the “reverse Wick rotation” of a product spectral triple. This construction of “Lorentzian spectral triples” fits well into the Krein space approach to noncommutative Lorentzian geometry.

Gleason’s problem, rational functions and spaces of left-regular functions: The split-quaternion setting

We study Gleason’s problem, rational functions and spaces of regular functions in the setting of split-quaternions. There are two natural symmetries in the algebra of split-quaternions. The first symmetry allows to define positive matrices with split-quaternionic entries, and also reproducing Hilbert spaces of regular functions. The second leads to reproducing kernel Krein spaces.

Operator Least Squares Problems and Moore–Penrose Inverses in Krein Spaces

Fil: Contino, Maximiliano. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Saavedra 15. Instituto Argentino de Matematica Alberto Calderon; Argentina. Universidad de Buenos Aires. Facultad de Ingenieria. Departamento de Matematicas; Argentina

Krein-space based robust [formula omitted] fault estimation for two-dimensional uncertain linear discrete time-varying systems

In this study, the robust H∞ fault estimation problem for two-dimensional linear time-varying systems with norm-bounded unknown input, measurement noise, and time-varying process uncertainty is investigated. By introducing an equivalent auxiliary system and a new certain indefinite quadratic form performance function, the system uncertainty can be appropriately considered into the new performance function and the fault estimator design is converted to the minimization problem of a quadratic form. Based on the partially equivalence property between the deterministic quadratic form problem and the Krein space estimation theory, the two-dimensional H∞ fault estimation problem can be solved via signal deconvolution in Krein space. Through employing projection operation and Riccati-like difference equation in two dimensions, both the recursive form fault estimator and the explicit condition for existence of the estimator are derived. One Darboux equation example is provided to illustrate the effectiveness of the proposed fault estimator.

Topological edge states for disordered bosonic systems

Quadratic bosonic Hamiltonians over a one-particle Hilbert space can be described by a Bogoliubov-de Gennes (BdG) Hamiltonian on a particle-hole Hilbert space. In general, the BdG Hamiltonian is not self-adjoint, but only J-self-adjoint on the particle-hole space viewed as a Krein space. Nevertheless, its energy bands can have non-trivial topological invariants like Chern numbers or winding numbers. By a thorough analysis for tight-binding models, it is proved that these invariants lead to bosonic edge modes which are robust to a large class of possibly disordered perturbations. Furthermore, general scenarios are presented for these edge states to be dynamically unstable even though the bulk modes are stable.

Lorentz signature and twisted spectral triples

We show how twisting the spectral triple of the Standard Model of elementary particles naturally yields the Krein space associated with the Lorentzian signature of spacetime. We discuss the associated spectral action, both for fermions and bosons. What emerges is a tight link between twists and Wick rotation.

$\alpha $-positive/$\alpha $-negative definite functions on groups

In this paper, we introduce the notions of an $\alpha $-positive/$\alpha $-negative definite function on a (discrete) group. We first construct the Naimark-GNS type representation associated to an $\alpha $-positive definite function and prove the Schoenberg type theorem for a matricially bounded $\alpha $-negative definite function. Using a $J$-representation on a Krein space $(\mathcal{K} ,J)$ associated to a nonnegative normalized $\alpha $-negative definite function, we also construct a $J$-cocycle associated to a $J$-representation. Using a $J$-cocycle, we show that there exist two sequences of $\alpha $-positive definite functions and proper $(\alpha ,J)$-actions on a Krein space $(\mathcal{K} ,J)$ corresponding to a proper matricially bounded $\alpha $-negative definite function.

Multiplicity of self-adjoint realisations of the (2+1)-fermionic model of Ter-Martirosyan—Skornyakov type

We reconstruct the whole family of self-adjoint Hamiltonians of Ter-Martirosyan—Skornyakov type for a system of two identical fermions coupled with a third particle of different nature through an interaction of zero range. We proceed through an operator-theoretic approach based on the self-adjoint extension theory of Krein, Visik, and Birman. We identify the explicit ‘Krein—Visik-Birman extension parameter’ as an operator on the ‘space of charges’ for this model (the ‘Krein space’) and we come to formulate a sharp conjecture on the dimensionality of its kernel. Based on our conjecture, for which we also discuss an amount of evidence, we explain the emergence of a multiplicity of extensions in a suitable regime of masses and we reproduce for the first time the previous partial constructions obtained by means of an alternative quadratic form approach.