Hilbert Space Research Articles

Two novel two-step inertial algorithms for a class of bilevel variational inequalities

This paper introduces two novel two-step inertial and self-adaptive step sizes algorithms for solving a strongly monotone variational inequality over the solution set of a split convex feasibility problem with multiple output sets in real Hilbert spaces. We establish strong convergence properties of the proposed method under mild conditions and employ it to solve monotone variational inequalities over the solution set of the split feasibility problem. The method is further applied to the image classification problem via the support vector machine learning. Numerical results are given to demonstrate the accelerating behaviors of our method over other related methods in the literature.

Berezin radius type inequalities for functional Hilbert space operators

Berezin radius type inequalities for functional Hilbert space operators

Block‐iterative schemes for the split common fixed point problem and applications

In this paper, based on the extrapolated Landweber‐type operators, we present new strongly convergent block‐iterative schemes for solving the split common fixed point problem (SCFPP) with demiclosed strongly quasi‐nonexpansive operators on Hilbert spaces. The strong convergence is proved without the additional assumptions such as the boundedly regular condition and the closedness property of the range of the transformation operator, assumed recently in the literature for the problem. A necessary and sufficient condition that ensures that a th iterate is a solution is given. An application of our results to solve the multiple‐sets split convex feasibility problem (MSSCFP) is showed with computational experiments for illustration.

Hamiltonian Computational Chemistry: Geometrical Structures in Chemical Dynamics and Kinetics.

The common geometrical (symplectic) structures of classical mechanics, quantum mechanics, and classical thermodynamics are unveiled with three pictures. These cardinal theories, mainly at the non-relativistic approximation, are the cornerstones for studying chemical dynamics and chemical kinetics. Working in extended phase spaces, we show that the physical states of integrable dynamical systems are depicted by Lagrangian submanifolds embedded in phase space. Observable quantities are calculated by properly transforming the extended phase space onto a reduced space, and trajectories are integrated by solving Hamilton's equations of motion. After defining a Riemannian metric, we can also estimate the length between two states. Local constants of motion are investigated by integrating Jacobi fields and solving the variational linear equations. Diagonalizing the symplectic fundamental matrix, eigenvalues equal to one reveal the number of constants of motion. For conservative systems, geometrical quantum mechanics has proved that solving the Schrödinger equation in extended Hilbert space, which incorporates the quantum phase, is equivalent to solving Hamilton's equations in the projective Hilbert space. In classical thermodynamics, we take entropy and energy as canonical variables to construct the extended phase space and to represent the Lagrangian submanifold. Hamilton's and variational equations are written and solved in the same fashion as in classical mechanics. Solvers based on high-order finite differences for numerically solving Hamilton's, variational, and Schrödinger equations are described. Employing the Hénon-Heiles two-dimensional nonlinear model, representative results for time-dependent, quantum, and dissipative macroscopic systems are shown to illustrate concepts and methods. High-order finite-difference algorithms, despite their accuracy in low-dimensional systems, require substantial computer resources when they are applied to systems with many degrees of freedom, such as polyatomic molecules. We discuss recent research progress in employing Hamiltonian neural networks for solving Hamilton's equations. It turns out that Hamiltonian geometry, shared with all physical theories, yields the necessary and sufficient conditions for the mutual assistance of humans and machines in deep-learning processes.

Approximating quantum channels by completely positive maps with small Kraus rank

We study the problem of approximating a quantum channel by one with as few Kraus operators as possible (in the sense that, for any input state, the output states of the two channels should be close to one another). Our main result is that any quantum channel mapping states on some input Hilbert space A to states on some output Hilbert space B can be compressed into one with order dlog⁡d Kraus operators, where d=max(|A|,|B|), hence much less than |A||B|. In the case where the channel's outputs are all very mixed, this can be improved to order d. We discuss the optimality of this result as well as some consequences.

Double inertial extragradient algorithms for solving variational inequality problems with convergence analysis

In this paper, we introduce a novel dual inertial Tseng's extragradient method for solving variational inequality problems in real Hilbert spaces, particularly those involving pseudomonotone and Lipschitz continuous operators. Our secondary method incorporates variable step‐size, updated at each iteration based on some previous iterates. A notable advantage of these algorithms is their ability to operate without prior knowledge of Lipschitz‐type constants and without the need for any line‐search procedure. We establish the convergence theorem of the proposed algorithms under mild assumptions. To illustrate the numerical behavior of the algorithms and to make comparisons with other methods, we conduct several numerical experiments. The results of these evaluations are showcased and thoroughly examined to exemplify the practical significance and effectiveness of the proposed methods.

Half-quantized Hall states with C=3/2 exited by chiral photons

The topology of Hilbert space in semiconductor quantum wells (QW) can become non-trivial when excited by an electromagnetic wave. It means the quasi-energy bands acquire a nonzero integer Floquet-Chern number even if QW is trivial in the equilibrium. We analyze electron-photon interaction in the Floquet state and identify dynamical quantum phases characterized by half-quantized Floquet-Chern numbers. The non-integer topology depends on which equilibrium state is exposed to illumination. Trivial III-V semiconductor QW's such as AlGaAs and AlGaN, being resonantly illuminated, generate the dynamical index |CF|=1/2, similar to a gapless quantum anomalous semimetal (QAS) in equilibrium. Excitation of 2D topological insulator CdHgTe reveals |CF|=3/2 - the index stemmed exclusively from interaction with chiral photons. Comparing results obtained in continuous and full-Brillouin zone (BZ) models, we conclude that high-momentum regions in BZ contribute to the dynamical half-quantized topology. Results indicate that by manipulating parameters of external force, one can bring inverted-gap QW into a non-equilibrium state with a high topological non-integer charge |CF|=3/2. We present the phase diagram, which includes all dynamical states of non-trivial topology.

On the convergence of gradient descent for robust functional linear regression

Functional data analysis offers a set of statistical methods concerned with extracting insights from intrinsically infinite-dimensional data and has attracted considerable amount of attentions in the past few decades. In this paper, we study robust functional linear regression model with a scalar response and a functional predictor in the framework of reproducing kernel Hilbert spaces. A gradient descent algorithm with early stopping is introduced to solve the corresponding empirical risk minimization problem associated with robust loss functions. By appropriately selecting the early stopping rule and the scaling parameter of the robust losses, the convergence of the proposed algorithm is established when the response variable is bounded or satisfies a moment condition. Explicit learning rates with respect to both estimation and prediction error are provided in terms of regularity of the regression function and eigenvalue decay rate of the integral operator induced by the reproducing kernel and covariance function.

Hidden variable model for quantum computation with magic states on qudits of any dimension

It was recently shown that a hidden variable model can be constructed for universal quantum computation with magic states on qubits. Here we show that this result can be extended, and a hidden variable model can be defined for quantum computation with magic states on qudits with any Hilbert space dimension. This model leads to a classical simulation algorithm for universal quantum computation.

Existence of a Class of Doubly Perturbed Stochastic Functional Differential Equations with Poisson Jumps

In this paper, we use successive approximations and Picard iterative method to establish the existence and uniqueness of mild solution for a class of doubly perturbed impulsive neutral stochastic functional differential equations with Poisson jumps in Hilbert spaces. An example of a doubly perturbed stochastic differential equation with delays is given to illustrate our main results.

New bounds for the Euclidean operator radius of two Hilbert space operators with applications

New bounds for the Euclidean operator radius of two Hilbert space operators with applications

Upper bounds for the norm of the sum and Kronecker product of Hilbert space operators

Upper bounds for the norm of the sum and Kronecker product of Hilbert space operators

On some operator norm inequalities

In this paper, we use known classical inequalities to establish some operator norm inequalities for a pair of norm-attainable operators on a complex Hilbert space.

Some properties of a complex symmetric operator

A square matrix with complex entries which is equal to its transpose is called a complex symmetric matrix. Such matrices are found in many areas of pure and applied mathematics. A conjugation on H is defined as a conjugate linear isometric involution C on a Hilbert Space H. A bounded linear operator T : H → H has a complex symmetric matrix with regard to an orthonormal basis if, for every conjugation C on H, T = CT *C. Such an operator is called a Complex symmetric operator. Several characteristics of Complex Symmetric matrices and associated operators are discussed in this article.

Stability of Krein-von Neumann self-adjoint operator extension under unbounded perturbations

We have considered a fourth order difference operator defined on the Hilbert space of square summable sequences on N. We investigated the stability of existence of Krein-von Neumann self-adjoint extension of difference operators under bounded and unbounded coefficients. Using asymptotic summation based on discretised Levinson's theorem and appropriate smoothness and decay conditions, we have shown that unlike the case of deficiency indices and discrete spectrum, the existence of positive self-adjoint operator extensions is stable under unbounded perturbations. These results now exhaustively characterise the spectral and structural properties of difference operators with unbounded coefficients.

On the Berezin number of operators on the reproducing kernel of Hilbert space and related questions

UDC 517.956 We obtain some new inequalities for the Berezin number of operators via the Cauchy–Schwarz-type inequalities. Some other related questions are also discussed.

A note on 𝐶^{1,𝛼}-smooth approximation of Lipschitz functions

We show that on super-reflexive spaces a Moreau-Yosida type of regularisation by infimal convolution together with a known insertion-type theorem (a variant of Ilmanen’s lemma) easily give an approximation of a Lipschitz function by a C 1 , α C^{1,\alpha } -smooth Lipschitz function with the same Lipschitz constant. This is a generalisation of the well-known theorem of J.-M. Lasry and P.-L. Lions from Hilbert spaces. It also gives a new self-contained and probably simpler proof of the Lasry-Lions theorem.

Weighted 1MP and MP1 inverses for operators

Abstract The main aim of this paper is to extend the concepts of the 1MP and MP1 inverses defined for rectangular complex matrices. We present the weighted 1MP and MP1 inverses for a bounded linear operator between two Hilbert spaces as two new kinds of generalized inverses. The notions of the weighted 1MP and MP1 inverses are new in the context of rectangular complex matrices too. We establish a number of characterizations and some representations of the weighted 1MP and MP1 inverses. Several operator equations are solved applying the weighted 1MP and MP1 inverses. A special case of one of these equations is the normal equation which is related with the least-squares solution. As consequences of our results, we obtain new properties of the 1MP and MP1 inverses.

Operator Bohr-type inequalities

Abstract The classical Bohr inequality for scalars was extended to the non-commutative case of Hilbert space operators in the literature. The sole goal of this article is to discuss the operator Bohr inequality and present some of its new variants. This includes fresh reverses and refinements of this inequality with applications towards an operator’s real and imaginary parts, not to forget the new discussion of different domains of the parameters. One further application towards the operator Dunkl-Williams inequality will be presented too. While the new results are interesting, we emphasize that the approach used to explore these inequalities differs from the existing literature methods for this context.

Asymptotic normality of spectral means of Hilbert space valued random processes

A variety of statistics for functional time series allows for a representation as weighted average of corresponding periodogram operators over the frequency domain. We study consistency and asymptotic normality of such spectral mean estimators under mild assumptions. We show that weak convergence of these estimators can be deduced from the (joint) weak convergence of the sample autocovariance operators. The latter is established for a large class of weakly dependent functional time series, which admit expansions as Bernoulli shifts and the weak dependence is quantified by the condition of L4-m-approximability.