Infinite-dimensional Hilbert Space Research Articles

Forcing Strong Convergence of a Mann-Based Iteration for Nonexpansive and Monotone Operators in a Hilbert Space

Mann iteration is weakly convergent in infinite dimensional spaces. We, in this paper, use the nearest point projection to force the strong convergence of a Mann-based iteration for nonexpansive and monotone operators. A strong convergence theorem of common elements is obtained in an infinite dimensional Hilbert space. No compact conditions are needed.

On solving variational inequality problems involving quasimonotone operators via modified Tseng's extragradient methods with convergence analysis

The main objective of this research is to find the numerical solution of variational inequalities involving quasimonotone operators in infinite-dimensional real Hilbert spaces. The main advantage of these iterative schemes is that they allow the uncomplicated calculation of step size rules that depend on the knowledge of an operator explanation instead of the Lipschitz constant or some other line search method. The proposed iterative schemes follow a monotone and non-monotone step size procedure based on mapping (operator) information as a replacement for its Lipschitz constant or some other line search method. The strong convergences are well proven, analogous to the proposed methods, and impose certain control specification conditions. Finally, to verify the effectiveness of the iterative methods, we present some numerical experiments.

Qubit Regularization and Qubit Embedding Algebras

Qubit regularization is a procedure to regularize the infinite dimensional local Hilbert space of bosonic fields to a finite dimensional one, which is a crucial step when trying to simulate lattice quantum field theories on a quantum computer. When the qubit-regularized lattice quantum fields preserve important symmetries of the original theory, qubit regularization naturally enforces certain algebraic structures on these quantum fields. We introduce the concept of qubit embedding algebras (QEAs) to characterize this algebraic structure associated with a qubit regularization scheme. We show a systematic procedure to derive QEAs for the O(N) lattice spin models and the SU(N) lattice gauge theories. While some of the QEAs we find were discovered earlier in the context of the D-theory approach, our method shows that QEAs are far richer. A more complete understanding of the QEAs could be helpful in recovering the fixed points of the desired quantum field theories.

Optimal convergence rates for Galerkin approximation of operator Riccati equations

AbstractIn this paper we consider the problem of determining optimal convergence rates of Galerkin approximations to infinite dimensional operator Riccati equations (OREs). Optimal rates are obtained for a class of abstract distributed parameter systems evolving in an infinite dimensional Hilbert space. These general results are then applied to systems modeled by partial differential equations that generate compact and analytic semigroups. The estimates apply to distributed control and observation of classical parabolic equations and to certain vibration problems with sufficiently strong damping. The ORE is formulated as an equivalent operator‐valued Bochner integral equation and the Brezzi–Rappaz–Raviart theorem is used to obtain convergence rates. First we establish smoothing property and bounds for the solutions of the infinite dimensional ORE. Then it is shown that, under sui\ assumptions on the coefficients and domain geometry, the hp‐finite element approximations of the classical solution converges on the order of . Furthermore, these optimal error bounds are shown to hold for the functional gains that define observer and control gain operators. We provide numerical examples that corroborate the theoretical convergence rates.

New approach to property (ω) for functions of operators

Let be an infinite dimensional complex Hilbert space and be the algebra of all bounded linear operators on . For , we say T satisfies property (ω) if σa (T) \\ σea (T) = π 00(T), where π 00(T) = {λ ∈ isoσ(T) : 0 < n(T − λI) < ∞}. In this paper, we research on the property (ω) for functions of operators by using the new spectrum σvea (T) which is a variant of the Weyl essential approximate point spectrum. At the same time, the stability of SVEP as well as the relationships between the two parts is also given.

On simplicity of Lie algebras of compact operators: A direct approach

We investigate an algebraic variant of the Wojtyński problem on the simplicity of Lie algebras of compact operators on a separable infinite-dimensional complex Hilbert space. We prove the non-simplicity of Lie algebras of compact operators under a mild softness condition using the notion of soft-edged operator ideals introduced by Kaftal and Weiss. We believe our study offers a new perspective on the investigation of the simplicity of Lie algebras by relating it to the study of operator ideals.

Poisson measures on semi-direct products of infinite-dimensional Hilbert spaces

Poisson measures on semi-direct products of infinite-dimensional Hilbert spaces

Stochastic differential equations in infinite dimensional Hilbert space and its optimal control problem with Lévy processes

&lt;abstract&gt;&lt;p&gt;The paper is concerned with a class of stochastic differential equations in infinite dimensional Hilbert space with random coefficients driven by Teugels martingales which are more general processes and the corresponding optimal control problems. Here Teugels martingales are a family of pairwise strongly orthonormal martingales associated with Lévy processes (see Nualart and Schoutens &lt;sup&gt;[&lt;xref ref-type="bibr" rid="b21"&gt;21&lt;/xref&gt;]&lt;/sup&gt;). There are three major ingredients. The first is to prove the existence and uniqueness of the solutions by continuous dependence theorem of solutions combining with the parameter extension method. The second is to establish the stochastic maximum principle and verification theorem for our optimal control problem by the classic convex variation method and dual techniques. The third is to represent an example of a Cauchy problem for a controlled stochastic partial differential equation driven by Teugels martingales which our theoretical results can solve.&lt;/p&gt;&lt;/abstract&gt;

Infinite multidimensional scaling for metric measure spaces

For a given metric measure space (X, d,μ) we consider finite samples of points, calculate the matrix of distances between them and then reconstruct the points in some finite-dimensional space using the multidimensional scaling (MDS) algorithm with this distance matrix as an input. We show that this procedure gives a natural limit as the number of points in the samples grows to infinity and the density of points approaches the measure μ. This limit can be viewed as “infinite MDS” embedding of the original space, now not anymore into a finite-dimensional space but rather into an infinitedimensional Hilbert space. We further show that this embedding is stable with respect to the natural convergence of metric measure spaces. However, contrary to what is usually believed in applications, we show that in many cases it does not preserve distances, nor is even bi-Lipschitz, but may provide snowflake (Assouad-type) embeddings of the original space to a Hilbert space (this is, for instance, the case of a sphere and a flat torus equipped with their geodesic distances).

Dictionary Learning of Symmetric Positive Definite Data Based on Riemannian Manifold Tangent Spaces and Local Homeomorphism

Symmetric Positive Definite (SPD) data are increasingly prevalent in dictionary learning recently. SPD data are the typical non-Euclidean data and cannot constitute a Euclidean space. Therefore, many dictionary learning algorithms cannot be directly adopted on SPD data. Reproducing Kernel Hilbert Spaces (RKHS) is now commonly used to deal with this problem. However, RKHS is an infinite-dimensional Hilbert space, rather than a Euclidean space, resulting in the inability of the dictionary learning to be directly used on SPD data. In this paper, we propose a novel dictionary learning algorithm for SPD data, which is based on the Riemannian Manifold Tangent Space (RMTS). Since RMTS is based on a finite-dimensional Hilbert space, i.e., Euclidean space, most machine learning algorithms developed on Euclidean space can be directly applied to RMTS. The introduction of RMTS provides a better mathematical platform for machine learning of non-Euclidean data. The proposed RMTS method first selects a point on a Riemannian manifold as an anchor point. It transforms the other points on the Riemannian manifold into tangent vectors of the geodesics between these points and the anchor point, with the tangent vectors at the anchor point. We set the length of the tangent vector equal to the length of the geodesic. As a result, such tangent vectors have an explicit geometric meaning, such as direction information, while the RKHS method may cause some geometric meaning to be lost in the original data during the mapping process. In addition, the proposed algorithm adds a regular term of local homomorphism between SPD data and its RMTS dictionary coding coefficients, so that the similarity of SPD data and its RMTS dictionary coding coefficients are as close as possible. Experimental results on four public datasets demonstrate that the proposed algorithm significantly outperforms five state-of-the-art algorithms. This work opens a new pathway towards SPD data dictionary learning methods.

Regularized functional matching pursuit for the spherical multiple-shell electro-magnetoencephalography problem

Reconstruction of the neuronal current inside the human brain from non-invasive measurements of the magnetic flux density via magnetoencephalography (MEG) or of electric potential differences via electroencephalography (EEG) is an invaluable tool for neuroscientific research, as it provides measures of activity in the brain. However, it is also a severely ill-posed inverse problem. Assuming spherical geometries, we consider the spherical multiple-shell model for the inverse MEG and EEG problem and apply the regularized functional matching pursuit algorithm (RFMP) for its solution. We present a new convergence proof for the RFMP for operators between two infinite-dimensional Hilbert spaces. Moreover, we utilize the complementarity of EEG and MEG data to combine inversions of simultaneous electric and magnetic measurements. Finally, we test the algorithm numerically on synthetic data using several Sobolev norms as penalty term and apply it to real data.

Convergence of the conjugate gradient method with unbounded operators

In the framework of inverse linear problems on infinite-dimensional Hilbert space, we prove the convergence of the conjugate gradient iterates to an exact solution to the inverse problem in the most general case where the self-adjoint, non-negative operator is unbounded and with minimal, technically unavoidable assumptions on the initial guess of the iterative algorithm. The convergence is proved to always hold in the Hilbert space norm (error convergence), as well as at other levels of regularity (energy norm, residual, etc.) depending on the regularity of the iterates. We also discuss, both analytically and through a selection of numerical tests, the main features and differences of our convergence result as compared to the case, already available in the literature, where the operator is bounded.

On the space of Schwartz operators in the symmetric Fock space and its dual

A long-standing problem that arises when constructing the mathematical apparatus of quantum mechanics is the need to work with unbounded operators. Since the space of nuclear operators is preconjugate for the algebra of all bounded operators, we can consider the states of a quantum system as nuclear operators, and consider bounded operators as observables. In this case, taking the trace for the product of a nuclear operator (a quantum state) and a bounded operator (a quantum observable) gives the average value of a quantum observable in a fixed state. The existence of such an average for unbounded operators is not guaranteed. If we want to define a space of observables that includes such naturally occurring unbounded operators as the position and momentum operators, for which average values are always determined, we should consider a space of states smaller than all nuclear operators. Recently, this approach has been mathematically correct implemented in the Hilbert space H = L^2(R^N). The so-called space of Schwarz operators, equipped with a system of semi-norms and being a Frechet space, was chosen as the space of states. Schwarz operators appeared to be integral operators whose kernels are functions belonging to the usual Schwarz space. The dual space to the space of Schwarz operators should be considered as the space of quantum observables and it really includes such standard observables as polynomials from the products of the position and momentum operators. In the present work we transfer this approach to the symmetric Fock space H = F(H) over an infinite dimensional separable Hilbert space H. We introduce the space of Schwartz operators in F(H) over an infinite dimensional separable space H and investigate which of the standard operators of quantum white noise belong to the space dual to the space of Schwartz operators.

Self-adjoint perturbations of left (right)Weyl spectrum for upper triangular operator matrices

Let H be a separable infinite-dimensional Hilbert space. Given the operators A ? B(H) and B ? B(H), we define MX := [A X 0 B] where X ? S(H) is a self-adjoint operator. In this paper, a necessary and sufficient condition is given for MX to be a left (right) Weyl operator for some X ? S(H). Moreover, it is shown that ? X?S(H) ?*(MX) = ? X?S(H)?Inv(H) ?*(MX) = ? X?B(H) ?*(MX) ? ?, where ?* is the left (right)Weyl spectrum. Finally, we further characterize the perturbation of the left (right) Weyl spectrum for Hamiltonian operators.

Vector Symbolic Architectures for Context-Free Grammars

Vector symbolic architectures (VSA) are a viable approach for the hyperdimensional representation of symbolic data, such as documents, syntactic structures, or semantic frames. We present a rigorous mathematical framework for the representation of phrase structure trees and parse trees of context-free grammars (CFG) in Fock space, i.e. infinite-dimensional Hilbert space as being used in quantum field theory. We define a novel normal form for CFG by means of term algebras. Using a recently developed software toolbox, called FockBox, we construct Fock space representations for the trees built up by a CFG left-corner (LC) parser. We prove a universal representation theorem for CFG term algebras in Fock space and illustrate our findings through a low-dimensional principal component projection of the LC parser state. Our approach could leverage the development of VSA for explainable artificial intelligence (XAI) by means of hyperdimensional deep neural computation.

On the left (right) invertibility of operator matrices

Let H be a complex separable infinite-dimensional Hilbert space. Given the operators A ∈ B ( H ) and B ∈ B ( H ) , we define M X := [ A X 0 B ] where X ∈ S ( H ) is a self-adjoint operator. In this paper, a necessary and sufficient condition is given for M X to be a left (right) invertible operator for some X ∈ S ( H ) . Moreover, it is shown that ⋂ X ∈ S ( H ) σ ∗ ( M X ) = ⋂ X ∈ B ( H ) σ ∗ ( M X ) ∪ Δ , where σ ∗ is the left (right) spectrum. Finally, we further characterize the perturbation of the left (right) spectrum for Hamiltonian operators.

Spectrum of Discrete Sturm-Liouville Equation with Self-adjoint Operator Coefficients on the Half-line

We investigate the spectrum of the Sturm-Liouville difference equation on the half-line with self-adjoint operator coefficients in an infinite dimensional Hilbert space together with the Dirichlet boundary condition. We find the Jost solution and examine its analytical and asymptotical properties. Using these properties, we obtain the continuous and point spectrum of the discrete operator generated by the Sturm-Liouville difference equation with self-adjoint operator coefficients. We also show that this operator has a finite number of eigenvalues with finite multiplicities under a certain condition on the operator coefficients.

The space of contractive C0-semigroups is a Baire space

Working over infinite dimensional separable Hilbert spaces, residual results have been achieved for the space of contractive C0-semigroups under the topology of uniform weak operator convergence on compact subsets of R+. Eisner and Serény raised in [6] and [3] the open problem: Does this space constitute a Baire space? Observing that the subspace of unitary semigroups is completely metrisable and appealing to known density results, we solve this problem positively by showing that certain topological properties can in general be transferred from dense subspaces to larger spaces. The transfer result in turn relies upon classification of topological properties via infinite games. Our approach is sufficiently general and can be applied to other contexts, e.g. the space of contractions under the ▪-topology.

Positive answer to the invariant and hyperinvariant subspaces problems for hyponormal operators

Abstract The question whether every operator on infinite-dimensional Hilbert space 𝐻 has a nontrivial invariant subspace or a nontrivial hyperinvariant subspace is one of the most difficult problems in operator theory. This problem is open for more than half a century. A subnormal operator has a nontrivial invariant subspace, but the existence of nontrivial invariant subspace for a hyponormal operator 𝑇 still open. In this paper we give an affirmative answer of the existence of a nontrivial hyperinvariant subspace for a hyponormal operator. More generally, we show that a large classes of operators containing the class of hyponormal operators have nontrivial hyperinvariant subspaces. Finally, every generalized scalar operator on a Banach space 𝑋 has a nontrivial invariant subspace.

On restricted diagonalization

Let H be a separable infinite-dimensional complex Hilbert space, B(H) the algebra of bounded linear operators acting on H and J a proper two-sided ideal of B(H). Denote by UJ(H) the group of all unitary operators of the form I+J. Recall that an operator A∈B(H) is diagonalizable if there exists a unitary operator U such that UAU⁎ is diagonal with respect to some orthonormal basis. A more restrictive notion of diagonalization can be formulated with respect to a fixed orthonormal basis e={en}n≥1 and a proper operator ideal J as follows: A∈B(H) is called restrictedly diagonalizable if there exists U∈UJ(H) such that UAU⁎ is diagonal with respect to e. In this work we give a sufficient condition for a diagonalizable operator to be restrictedly diagonalizable. This condition becomes a characterization when the ideal is arithmetic mean closed. Then we obtain results on the structure of the set of all restrictedly diagonalizable operators. In this way we answer several open problems recently raised by Beltiţă, Patnaik and Weiss.