Hilbert Space Theory Research Articles

An RKHS framework for functional data analysis

Linear combinations of random variables play a crucial role in multivariate analysis. Two extension of this concept are considered for functional data and shown to coincide using the Loève–Parzen reproducing kernel Hilbert space representation of a stochastic process. This theory is then used to provide an extension of the multivariate concept of canonical correlation. A solution to the regression problem of best linear unbiased prediction is obtained from this abstract canonical correlation formulation. The classical identities of Lawley and Rao that lead to canonical factor analysis are also generalized to the functional data setting. Finally, the relationship between Fisher's linear discriminant analysis and canonical correlation analysis for random vectors is extended to include situations with function-valued random elements. This allows for classification using the canonical Y scores and related distance measures.

The three-body Coulomb scattering problem in a discrete Hilbert-space basis representation

We propose modified Faddeev-Merkuriev integral equations for solving the 2→2, 3 quantum three-body Coulomb scattering problem. We show that the solution of these equations can be obtained using a discrete Hilbert-space basis and that the error in the scattering amplitudes due to truncating the basis can be made arbitrarily small. The Coulomb Green’s function is also confined to the two-body sector of the three-body configuration space by this truncation and can be constructed in the leading order using convolution integrals of two-body Green’s functions. To evaluate the convolution integral, we propose an integration contour that is applicable for all energies including bound-state energies and scattering energies below and above the three-body breakup threshold.

Adaptive Kernel-Based Image Denoising Employing Semi-Parametric Regularization

The main contribution of this paper is the development of a novel approach, based on the theory of Reproducing Kernel Hilbert Spaces (RKHS), for the problem of noise removal in the spatial domain. The proposed methodology has the advantage that it is able to remove any kind of additive noise (impulse, gaussian, uniform, etc.) from any digital image, in contrast to the most commonly used denoising techniques, which are noise dependent. The problem is cast as an optimization task in a RKHS, by taking advantage of the celebrated Representer Theorem in its semi-parametric formulation. The semi-parametric formulation, although known in theory, has so far found limited, to our knowledge, application. However, in the image denoising problem, its use is dictated by the nature of the problem itself. The need for edge preservation naturally leads to such a modeling. Examples verify that in the presence of gaussian noise the proposed methodology performs well compared to wavelet based technics and outperforms them significantly in the presence of impulse or mixed noise.

Existence of zerofree functions N-associated to a de Branges Pontryagin space

In the theory of de Branges Hilbert spaces of entire functions, so-called ‘functions associated to a space’ play an important role. In the present paper we deal with a generalization of this notion in two directions, namely with functions N-associated \(({N \in\mathbb {Z}})\) to a de Branges Pontryagin space. Let a de Branges Pontryagin space \({\mathcal {P}}\) and \({N \in \mathbb {Z}}\) be given. Our aim is to characterize whether there exists a real and zerofree function N-associated to \({\mathcal {P}}\) in terms of Kreĭn’s Q-function associated with the multiplication operator in \({\mathcal {P}}\) . The conditions which appear in this characterization involve the asymptotic distribution of the poles of the Q-function plus a summability condition. Although this question may seem rather abstract, its answer has a variety of nontrivial consequences. We use it to answer two questions arising in the theory of general (indefinite) canonical systems. Namely, to characterize whether a given generalized Nevanlinna function is the intermediate Weyl-coefficient of some system in terms of its poles and residues, and to characterize whether a given general Hamiltonian ends with a specified number of indivisible intervals in terms of the Weyl-coefficient associated to the system. In addition, we present some applications, e.g., dealing with admissible majorants in de Branges spaces or the continuation problem for hermitian indefinite functions.

Decision theory with prospect interference and entanglement

We present a novel variant of decision making based on the mathematical theory of separable Hilbert spaces. This mathematical structure captures the effect of superposition of composite prospects, including many incorporated intentions, which allows us to describe a variety of interesting fallacies and anomalies that have been reported to particularize the decision making of real human beings. The theory characterizes entangled decision making, non-commutativity of subsequent decisions, and intention interference. We demonstrate how the violation of the Savage’s sure-thing principle, known as the disjunction effect, can be explained quantitatively as a result of the interference of intentions, when making decisions under uncertainty. The disjunction effects, observed in experiments, are accurately predicted using a theorem on interference alternation that we derive, which connects aversion-to-uncertainty to the appearance of negative interference terms suppressing the probability of actions. The conjunction fallacy is also explained by the presence of the interference terms. A series of experiments are analyzed and shown to be in excellent agreement with a priori evaluation of interference effects. The conjunction fallacy is also shown to be a sufficient condition for the disjunction effect, and novel experiments testing the combined interplay between the two effects are suggested.

A Hilbert Space Representation of Generalized Observables and Measurement Processes in the ESR Model

The extended semantic realism (ESR) model recently worked out by one of the authors embodies the mathematical formalism of standard (Hilbert space) quantum mechanics in a noncontextual framework, reinterpreting quantum probabilities as conditional instead of absolute. We provide here a Hilbert space representation of the generalized observables introduced by the ESR model that satisfy a simple physical condition, propose a generalization of the projection postulate, and suggest a possible mathematical description of the measurement process in terms of evolution of the compound system made up of the measured system and the measuring apparatus.

The classical basis for the κ-Poincaré Hopf algebra and doubly special relativity theories

Several issues concerning the quantum κ-Poincaré algebra are discussed and reconsidered here. We propose two different formulations of the κ-Poincaré quantum algebra. Firstly we present a complete Hopf algebra formula of κ-Poincaré in classical Poincaré basis. Further by adding one extra generator, which modifies the classical structure of the Poincaré algebra, we eliminate nonpolynomial functions in the κ-parameter. Hilbert space representations of such algebras make doubly special relativity (DSR) similar to Stueckelberg's version of (proper-time) relativistic quantum mechanics.

Processing Information in Quantum Decision Theory

A survey is given summarizing the state of the art of describing information processing in Quantum Decision Theory, which has been recently advanced as a novel variant of decision making, based on the mathematical theory of separable Hilbert spaces. This mathematical structure captures the effect of superposition of composite prospects, including many incorporated intended actions. The theory characterizes entangled decision making, non-commutativity of subsequent decisions, and intention interference. The self-consistent procedure of decision making, in the frame of the quantum decision theory, takes into account both the available objective information as well as subjective contextual effects. This quantum approach avoids any paradox typical of classical decision theory. Conditional maximization of entropy, equivalent to the minimization of an information functional, makes it possible to connect the quantum and classical decision theories, showing that the latter is the limit of the former under vanishing interference terms.

Weak amenability of CAT(0)-cubical groups

We prove that if G is a discrete group that admits a metrically proper action on a finite-dimensional CAT(0) cube complex X, then G is weakly amenable. We do this by constructing uniformly bounded Hilbert space representations π z for which the quantities z ℓ (g) are matrix coefficients. Here ℓ is a length function on G obtained from the combinatorial distance function on the complex X.

Classical and quantum mechanics from the universal Poisson-Rinehart algebra of a manifold

The Lie and module (Rinehart) algebraic structure of vector fields of compact support over C infinity functions on a (connected) manifold M define a unique universal non-commutative Poisson * algebra. For a compact manifold, a (antihermitian) variable Z, central with respect to both the product and the Lie product, relates commutators and Poisson brackets; in the non-compact case, sequences of locally central variables allow for the addition of an element with the same role. Quotients with respect to the (positive) values taken by Z* Z define classical Poisson algebras and quantum observable algebras, with the Planck constant given by -iZ. Under standard regularity conditions, the corresponding states and Hilbert space representations uniquely give rise to classical and quantum mechanics on M.

Unitarizable representations and fixed points of groups of biholomorphic transformations of operator balls

We show that the open unit ball of the space of operators from a finite-dimensional Hilbert space into a separable Hilbert space (we call it “operator ball”) has a restricted form of normal structure if we endow it with a hyperbolic metric (which is an analogue of the standard hyperbolic metric on the unit disc in the complex plane). We use this result to get a fixed point theorem for groups of biholomorphic automorphisms of the operator ball. The fixed point theorem is used to show that a bounded representation in a separable Hilbert space which has an invariant indefinite quadratic form with finitely many negative squares is unitarizable (equivalent to a unitary representation). We apply this result to find dual pairs of invariant subspaces in Pontryagin spaces. In Appendix A we present results of Itai Shafrir about hyperbolic metrics on the operator ball.

Estimation of Improper Complex-Valued Random Signals in Colored Noise by Using the Hilbert Space Theory

In this paper, the problem of estimating an improper complex-valued random signal in colored noise with an additive white part is addressed. We tackle the problem from a mathematical perspective and emphasize the advantages of this rigorous treatment. The formulation considered is very general in the sense that it permits us to estimate any functional of the signal of interest. Finally, the superiority of the widely linear estimation with respect to the conventional estimation techniques is both theoretically and experimentally illustrated.

Fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces. I. Abstract theory

N. Dunford and J.T. Schwartz (1963) striking Hilbert space theory about completeness of a system of root vectors (generalized eigenvectors) of an unbounded operator has been generalized by J. Burgoyne (1995) to the Banach spaces framework. We use the Burgoyne's theorem and prove n-fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces. The theory will allow to consider, in application, boundary value problems for ODEs and elliptic PDEs which polynomially depend on the spectral parameter in both the equation and the boundary conditions.

A quantum probability explanation for violations of ‘rational’ decision theory

Two experimental tasks in psychology, the two-stage gambling game and the Prisoner's Dilemma game, show that people violate the sure thing principle of decision theory. These paradoxical findings have resisted explanation by classical decision theory for over a decade. A quantum probability model, based on a Hilbert space representation and Schrödinger's equation, provides a simple and elegant explanation for this behaviour. The quantum model is compared with an equivalent Markov model and it is shown that the latter is unable to account for violations of the sure thing principle. Accordingly, it is argued that quantum probability provides a better framework for modelling human decision-making.

On the commutant of [formula omitted] in [formula omitted]-crossed products by [formula omitted] and their representations

For the C ∗ -crossed product C ∗ ( Σ ) associated with an arbitrary topological dynamical system Σ = ( X , σ ) , we provide a detailed analysis of the commutant, in C ∗ ( Σ ) , of C ( X ) and the commutant of the image of C ( X ) under an arbitrary Hilbert space representation π ˜ of C ∗ ( Σ ) . In particular, we give a concrete description of these commutants, and also determine their spectra. We show that, regardless of the system Σ, the commutant of C ( X ) has non-zero intersection with every non-zero, not necessarily closed or self-adjoint, ideal of C ∗ ( Σ ) . We also show that the corresponding statement holds true for the commutant of π ˜ ( C ( X ) ) under the assumption that a certain family of pure states of π ˜ ( C ∗ ( Σ ) ) is total. Furthermore we establish that, if C ( X ) ⊊ C ( X ) ′ , there exist both a C ∗ -subalgebra properly between C ( X ) and C ( X ) ′ which has the aforementioned intersection property, and such a C ∗ -subalgebra which does not have this property. We also discuss existence of a projection of norm one from C ∗ ( Σ ) onto the commutant of C ( X ) .

The Lie–Rinehart Universal Poisson Algebra of Classical and Quantum Mechanics

The Lie–Rinehart algebra of a (connected) manifold \({\mathcal {M}}\) , defined by the Lie structure of the vector fields, their action and their module structure over \({C^\infty({\mathcal {M}})}\) , is a common, diffeomorphism invariant, algebra for both classical and quantum mechanics. Its (noncommutative) Poisson universal enveloping algebra \({\Lambda_{R}({\mathcal {M}})}\) , with the Lie–Rinehart product identified with the symmetric product, contains a central variable (a central sequence for non-compact \({{\mathcal {M}}}\)) \({Z}\) which relates the commutators to the Lie products. Classical and quantum mechanics are its only factorial realizations, corresponding to Z = iz, z = 0 and \({z = \hbar}\) , respectively; canonical quantization uniquely follows from such a general geometrical structure. For \({z =\hbar \neq 0}\) , the regular factorial Hilbert space representations of \({\Lambda_{R}({\mathcal{M}})}\) describe quantum mechanics on \({{\mathcal {M}}}\) . For z = 0, if Diff(\({{\mathcal {M}}}\)) is unitarily implemented, they are unitarily equivalent, up to multiplicity, to the representation defined by classical mechanics on \({{\mathcal {M}}}\) .

Appell polynomials and their relatives III. Conditionally free theory

We extend to the multivariate noncommutative context the descriptions of a “once-stripped” probability measure in terms of Jacobi parameters, orthogonal polynomials, and the moment generating function. The corresponding map $Φ$ on states was introduced previously by Belinschi and Nica. We then relate these constructions to the c-free probability theory, which is a version of free probability for algebras with two states, introduced by Bozejko, Leinert, and Speicher. This theory includes the free and Boolean probability theories as extreme cases. The main objects in the paper are the analogs of the Appell polynomial families in the two state context. They arise as fixed points of the transformation which takes a polynomial family to the associated polynomial family (in several variables), and their orthogonality is also related to the map $Φ$ above. In addition, we prove recursions, generating functions, and factorization and martingale properties for these polynomials, and describe the c-free version of the Kailath–Segall polynomials, their combinatorics, and Hilbert space representations.

On perturbations of Hilbert spaces and probability algebras with a generic automorphism

We prove that $IHS_A$, the theory of infinite dimensional Hilbert spaces equipped with a generic automorphism, is $\aleph_0$-stable up to perturbation of the automorphism, and admits prime models up to perturbation over any set. Similarly, $APr_A$, the theory of atomless probability algebras equipped with a generic automorphism is $\aleph_0$-stable up to perturbation. However, not allowing perturbation it is not even superstable.

Nonexpansive retractions in Hilbert spaces

Let H be a Hilbert space and C H be closed and convex. The mapping P : H ! C known as the nearest point projection is nonexpansive (1-lipschitzian). We observed that, the natural question: Are there nonex- pansive projections Q : H! C other than P? is neglected in the literature. Also, the answer is not often present in the folklore of the Hilbert space theory. We provide here the answer and discuss some facts connected with the subject.

The Lie–Rinehart Universal Poisson Algebra of Classical and Quantum Mechanics

The Lie–Rinehart algebra of a (connected) manifold $${\mathcal {M}}$$ , defined by the Lie structure of the vector fields, their action and their module structure over $${C^\infty({\mathcal {M}})}$$ , is a common, diffeomorphism invariant, algebra for both classical and quantum mechanics. Its (noncommutative) Poisson universal enveloping algebra $${\Lambda_{R}({\mathcal {M}})}$$ , with the Lie–Rinehart product identified with the symmetric product, contains a central variable (a central sequence for non-compact $${{\mathcal {M}}}$$ ) $${Z}$$ which relates the commutators to the Lie products. Classical and quantum mechanics are its only factorial realizations, corresponding to Z = i z, z = 0 and $${z = \hbar}$$ , respectively; canonical quantization uniquely follows from such a general geometrical structure. For $${z =\hbar \neq 0}$$ , the regular factorial Hilbert space representations of $${\Lambda_{R}({\mathcal{M}})}$$ describe quantum mechanics on $${{\mathcal {M}}}$$ . For z = 0, if Diff( $${{\mathcal {M}}}$$ ) is unitarily implemented, they are unitarily equivalent, up to multiplicity, to the representation defined by classical mechanics on $${{\mathcal {M}}}$$ .