Separable Hilbert Space Research Articles

Some Results on Oblique Dual and Approximate Oblique Dual Hilbert-Schmidt Frames

The concept of Hilbert-Schmidt frames (HS-frames) is more general than that of g-frames. In this paper, we investigate the oblique dual (OD) and ε -approximate oblique dual ( ε -AOD) HS-frames on closed subspaces W and V of a separable Hilbert space H . We first introduce the concepts of OD and ε -AOD HS-frames. Then we present some conditions for a pair of OD and ε -AOD HS-frames to be symmetric. We also obtain the algebraic formula for all OD and ε -AOD HS-frames on V for a given HS-frame in W . As application, we get the canonical OD HS-frame has a minimum norm among all the representations of elements in W . We also discuss the perturbations of ε -AOD HS-frames. Finally, applying our results, we not only recover some known results but also derive a new result in the classical Hilbert space frame setting.

Square-mean S-Asymptotically $$\omega $$-Periodic Solutions for Some Stochastic Delayed Integrodifferential Inclusions

AbstractIn this work, by the resolvent operator theory, stochastic analysis theory and the fixed-point technique in Fréchet spaces, we present sufficient conditions for the existence of square-mean S-asymptotically $$\omega $$ ω -periodic mild solutions of abstract stochastic integrodifferential inclusions with infinite delay in separable Hilbert spaces and provide an example to illustrate the abstract results.

Higher order $\mathcal{S}^{p}$-differentiability: The unitary case

Consider the set of unitary operators on a complex separable Hilbert space \mathcal{H} , denoted as \mathcal{U}(\mathcal{H}) . Consider 1<p<\infty . We establish that a function f defined on the unit circle \mathbb{T} is n times continuously Fréchet \mathcal{S}^{p} -differentiable at every point in \mathcal{U}(\mathcal{H}) if and only if f\in C^{n}(\mathbb{T}) . Take a function U \colon\R\rightarrow\mathcal{U}(\mathcal{H}) such that the function t\in\R\mapsto U(t)-U(0) takes values in \mathcal{S}^{p} and is n times continuously \mathcal{S}^{p} -differentiable on \R . Consequently, for f\in C^{n}(\mathbb{T}) , we prove that f is n times continuously Gâteaux \mathcal{S}^{p} -differentiable at U(t) . We provide explicit expressions for both types of derivatives of f in terms of multiple operator integrals. In the domain of unitary operators, these results closely follow the n -th order successes for self-adjoint operators achieved by the second author, Le Merdy, Skripka, and Sukochev. Furthermore, as for application, we derive a formula and \mathcal{S}^{p} -estimates for operator Taylor remainders for a broader class of functions. Our results extend those of Peller, Potapov, Skripka, Sukochev, and Tomskova.

On Some Dynamics in Conceptual Spaces

AbstractIn this text, we consider Gärdenfors’ conceptual spaces that are separable Hilbert spaces. In particular, the results we obtained apply to finite-dimensional Euclidean spaces. Our main contribution can be formulated as a combination of the theory of opinion dynamics with the theory of conceptual spaces. This combination, in turn, leads us to propose a new model for the time evolution of conceptual spaces. To achieve this goal, we propose some extension of the multidimensional opinion dynamics model of Parsegov, Proskurnikov, Tempo and Friedkin to opinions with values in Hilbert spaces.

Existence of a mild solution and approximate controllability for fractional random integro-differential inclusions with non-instantaneous impulses

This paper investigates the existence and approximate controllability (ACA) of fractional neutral-type stochastic differential inclusions (NTSDIs) characterized by non-instantaneous impulses within a separable Hilbert space (HS) framework. Employing the Atangana–Baleanu–Caputo (ABC) derivative, we transform the system into an equivalent fixed-point (FP) problem through an integral operator. Subsequently, the Bohnenblust–Karlin FP theorem is leveraged to establish existence results. By assuming ACA of the corresponding linear system, we derive sufficient conditions for the ACA of the nonlinear stochastic impulsive control system. Our analysis relies on concepts from stochastic analysis, fractional calculus, FP theory, semigroup theory, and the theory of multivalued maps (MVMs). The theoretical findings are illustrated through a concrete example.

Hypercyclictty and Countable Hypercyclicity for Adjoint of Operators

Let be an infinite dimensional separable complex Hilbert space and let , where is the Banach algebra of all bounded linear operators on . In this paper we prove the following results. If is a operator, then 1. is a hypercyclic operator if and only if D and for every hyperinvariant subspace of . 2. If is a pure, then is a countably hypercyclic operator if and only if and for every hyperinvariant subspace of . 3. has a bounded set with dense orbit if and only if for every hyperinvariant subspace of , .

Hypocoercivity in Hilbert spaces

The concept of hypocoercivity for linear evolution equations with dissipation is discussed and equivalent characterizations that were developed for the finite-dimensional case are extended to separable Hilbert spaces. Using the concept of a hypocoercivity index, quantitative estimates on the short-time and long-time decay behavior of a hypocoercive system are derived. As a useful tool for analyzing the structural properties, an infinite-dimensional staircase form is also derived and connections to linear systems and control theory are presented. Several examples illustrate the new concepts and the results are applied to the Lorentz kinetic equation.

Weakly nonlinear hyperbolic differential equation of the second order in Hilbert space

We consider nonlinear perturbations of the hyperbolic equation in the Hilbert space. Necessary and sufficient conditions for the existence of solutions of boundary-value problem for the corresponding equation and iterative procedures for their finding are obtained in the case when the operator in linear part of the problem hasn't inverse and can have nonclosed set of values. As an application we consider boundary-value problem for van der Pol equation in a separable Hilbert space.

Mosco Convergence of Gradient Forms with Non-Convex Interaction Potential

This article provides a new approach to address Mosco convergence of gradient-type Dirichlet forms, EN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {E}}^N$$\\end{document} on L2(E,μN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(E,\\mu _N)$$\\end{document} for N∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\in {\\mathbb {N}}$$\\end{document}, in the framework of converging Hilbert spaces by K. Kuwae and T. Shioya. The basic assumption is weak measure convergence of the family (μN)N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${(\\mu _N)}_{N}$$\\end{document} on the state space E—either a separable Hilbert space or a locally convex topological vector space. Apart from that, the conditions on (μN)N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${(\\mu _N)}_{N}$$\\end{document} try to impose as little restrictions as possible. The problem has fully been solved if the family (μN)N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${(\\mu _N)}_{N}$$\\end{document} contain only log-concave measures, due to Ambrosio et al. (Probab Theory Relat. Fields 145:517–564, 2009). However, for a large class of convergence problems the assumption of log-concavity fails. The article suggests a way to overcome this hindrance, as it presents a new approach. Combining the theory of Dirichlet forms with methods from numerical analysis we find abstract criteria for Mosco convergence of standard gradient forms with varying reference measures. These include cases in which the measures are not log-concave. To demonstrate the accessibility of our abstract theory we discuss a first application, generalizing an approximation result by Bounebache and Zambotti (J Theor Probab 27:168–201, 2014).

The Invariant Subspace Problem for Separable Hilbert Spaces

In this paper, we prove that every bounded linear operator on a separable Hilbert space has a non-trivial invariant subspace. This answers the well-known invariant subspace problem.

Markovian lifting and asymptotic log-Harnack inequality for stochastic Volterra integral equations

We introduce a new framework of Markovian lifts of stochastic Volterra integral equations (SVIEs for short) with completely monotone kernels. We define the state space of the Markovian lift as a separable Hilbert space which incorporates the singularity or regularity of the kernel into the definition. We show that the solution of an SVIE is represented by the solution of a lifted stochastic evolution equation (SEE for short) defined on the Hilbert space and prove the existence, uniqueness and Markov property of the solution of the lifted SEE. Furthermore, we establish an asymptotic log-Harnack inequality and some consequent properties for the Markov semigroup associated with the Markovian lift via the asymptotic coupling method.

From the Choi formalism in infinite dimensions to unique decompositions of generators of completely positive dynamical semigroups

Given any separable complex Hilbert space, any trace-class operator [Formula: see text] which does not have purely imaginary trace, and any generator [Formula: see text] of a norm-continuous one-parameter semigroup of completely positive maps we prove that there exists a unique bounded operator [Formula: see text] and a unique completely positive map [Formula: see text] such that (i) [Formula: see text], (ii) the superoperator [Formula: see text] is trace class and has vanishing trace, and (iii) [Formula: see text] is a real number. Central to our proof is a modified version of the Choi formalism which relates completely positive maps to positive semi-definite operators. We characterize when this correspondence is injective and surjective, respectively, which in turn explains why the proof idea of our main result cannot extend to non-separable Hilbert spaces. In particular, we find examples of positive semi-definite operators which have empty pre-image under the Choi formalism as soon as the underlying Hilbert space is infinite-dimensional.

Device-independent lower bounds on the conditional von Neumann entropy

The rates of several device-independent (DI) protocols, including quantum key-distribution (QKD) and randomness expansion (RE), can be computed via an optimization of the conditional von Neumann entropy over a particular class of quantum states. In this work we introduce a numerical method to compute lower bounds on such rates. We derive a sequence of optimization problems that converge to the conditional von Neumann entropy of systems defined on general separable Hilbert spaces. Using the Navascués-Pironio-Acín hierarchy we can then relax these problems to semidefinite programs, giving a computationally tractable method to compute lower bounds on the rates of DI protocols. Applying our method to compute the rates of DI-RE and DI-QKD protocols we find substantial improvements over all previous numerical techniques, demonstrating significantly higher rates for both DI-RE and DI-QKD. In particular, for DI-QKD we show a minimal detection efficiency threshold which is within the realm of current capabilities. Moreover, we demonstrate that our method is capable of converging rapidly by recovering all known tight analytical bounds up to several decimal places. Finally, we note that our method is compatible with the entropy accumulation theorem and can thus be used to compute rates of finite round protocols and subsequently prove their security.

Cross-ValidatedFunctional Generalized Partially Linear Single-Functional Index Model

In this paper, we have introduced a functional approach for approximating nonparametric functions and coefficients in the presence of multivariate and functional predictors. By utilizing the Fisher scoring algorithm and the cross-validation technique, we derived the necessary components that allow us to explain scalar responses, including the functional index, the nonlinear regression operator, the single-index component, and the systematic component. This approach effectively addresses the curse of dimensionality and can be applied to the analysis of multivariate and functional random variables in a separable Hilbert space. We employed an iterative Fisher scoring procedure with normalized B-splines to estimate the parameters, and both the theoretical and practical evaluations demonstrated its favorable performance. The results indicate that the nonparametric functions, the coefficients, and the regression operators can be estimated accurately, and our method exhibits strong predictive capabilities when applied to real or simulated data.

Fock structure of complete Boolean algebras of type I factors and of unital factorizations

The factorizable vectors of a complete Boolean algebra of type I factors, acting on a separable Hilbert space, are shown to be total, resolving a conjecture of Araki and Woods. En route, the spectral theory of noise-type Boolean algebras of Tsirelson is cast in the noncommutative language of “factorizations with unit” for which a multi-layered characterization of being “of Fock type” is provided.

Equivalence of Sobolev Norms with Respect to Weighted Gaussian Measures

AbstractWe consider the spaces $${\text {L}}^p(X,\nu ;V)$$ L p ( X , ν ; V ) , where X is a separable Banach space, $$\mu $$ μ is a centred non-degenerate Gaussian measure, $$\nu :=Ke^{-U}\mu $$ ν : = K e - U μ with normalizing factor K and V is a separable Hilbert space. In this paper we prove a vector-valued Poincaré inequality for functions $$F\in W^{1,p}(X,\nu ;V)$$ F ∈ W 1 , p ( X , ν ; V ) , which allows us to show that for every $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) and every $$k\in \mathbb {N}$$ k ∈ N the norm in $$W^{k,p}(X,\nu )$$ W k , p ( X , ν ) is equivalent to the graph norm of $$D_H^{k}$$ D H k (the k-th Malliavin derivative) in $${\text {L}}^p(X,\nu )$$ L p ( X , ν ) . To conclude, we show exponential decay estimates for the V-valued perturbed Ornstein-Uhlenbeck semigroup $$(T^V(t))_{t\ge 0}$$ ( T V ( t ) ) t ≥ 0 , defined in Section 2.6, as t goes to infinity. Useful tools are the study of the asymptotic behaviour of the scalar perturbed Ornstein-Uhlenbeck $$(T(t))_{t\ge 0}$$ ( T ( t ) ) t ≥ 0 , and pointwise estimates for $$|D_HT(t)f|_H^p$$ | D H T ( t ) f | H p by means of both $$T(t)|D_Hf|^p_H$$ T ( t ) | D H f | H p and $$T(t)|f|^p$$ T ( t ) | f | p .

On the relations between Auerbach or almost Auerbach Markushevich systems and Schauder bases

We establish that the summability of the series ∑εn is the necessary and sufficient criterion ensuring that every (1+εn)-bounded Markushevich basis in a separable Hilbert space is a Riesz basis. Further we show that if nεn→∞, then in ℓ2 there exists a (1+εn)-bounded Markushevich basis that under any permutation is non-equivalent to a Schauder basis. We extend this result to any separable Banach space. Finally we provide examples of Auerbach bases in 1-symmetric separable Banach spaces whose no permutations are equivalent to any Schauder basis or (depending on the space) any unconditional Schauder basis.

Consistent Sampling Approximations in Abstract Hilbert Spaces

This paper considers generalized consistent sampling and reconstruction processes in an abstract separable Hilbert space. Using an operator-theoretical approach, quasi-consistent and consistent approximations with optimal properties, such as possessing the minimum norm or being closest to the original vector, are derived. The results are illustrated with several examples.

Canonical Typicality for Other Ensembles than Micro-canonical

AbstractWe generalize Lévy’s lemma, a concentration-of-measure result for the uniform probability distribution on high-dimensional spheres, to a much more general class of measures, so-called GAP measures. For any given density matrix $$\rho $$ ρ on a separable Hilbert space $${\mathcal {H}}$$ H , $${\textrm{GAP}}(\rho )$$ GAP ( ρ ) is the most spread-out probability measure on the unit sphere of $${\mathcal {H}}$$ H that has density matrix $$\rho $$ ρ and thus forms the natural generalization of the uniform distribution. We prove concentration-of-measure whenever the largest eigenvalue $$\Vert \rho \Vert $$ ‖ ρ ‖ of $$\rho $$ ρ is small. We use this fact to generalize and improve well-known and important typicality results of quantum statistical mechanics to GAP measures, namely canonical typicality and dynamical typicality. Canonical typicality is the statement that for “most” pure states $$\psi $$ ψ of a given ensemble, the reduced density matrix of a sufficiently small subsystem is very close to a $$\psi $$ ψ -independent matrix. Dynamical typicality is the statement that for any observable and any unitary time evolution, for “most” pure states $$\psi $$ ψ from a given ensemble the (coarse-grained) Born distribution of that observable in the time-evolved state $$\psi _t$$ ψ t is very close to a $$\psi $$ ψ -independent distribution. So far, canonical typicality and dynamical typicality were known for the uniform distribution on finite-dimensional spheres, corresponding to the micro-canonical ensemble, and for rather special mean-value ensembles. Our result shows that these typicality results hold also for $${\textrm{GAP}}(\rho )$$ GAP ( ρ ) , provided the density matrix $$\rho $$ ρ has small eigenvalues. Since certain GAP measures are quantum analogs of the canonical ensemble of classical mechanics, our results can also be regarded as a version of equivalence of ensembles.

Exponential stability of non-instantaneous impulsive second-order fractional neutral stochastic differential equations with state-dependent delay

The aim of this manuscript mainly concerned with a new class of exponential stability for non-instantaneous impulsive second-order fractional neutral stochastic differential equations (NIIFNSDEs) with state-dependent delay driven by Poisson jump in a separable Hilbert spaces. Firstly, a more appropriate concept of mild solution is introduced. Secondly, sufficient conditions are derived for the existence of mild solution by means stochastic analysis, fractional calculus approach and Mönch fixed point theorem with appropriate hypotheses on non-linear continuous functions combined with solution operator. Finally, stability results are derived based on the pth moment exponential stable with the help of new impulsive integral inequality techniques. In addition, an example is provided to validate the theoretical results. This manuscript is the unique combination of the new theoretical investigation which is simulated numerically. Our work extends the works of Arthi et al. (2014), Das et al. (2016), Huang et al. (2018), Pandey et al. (2014), Wang et al. (2018), Yan and Jia (2016).