Unbounded Operators Research Articles

Spectral theorem approach to commutative C⁎-algebras generated by Toeplitz operators on the unit ball: Quasi-elliptic related cases

We consider commutative C⁎-algebras of Toeplitz operators in the weighted Bergman space on the unit ball in Cn. For the algebras of elliptic type we find a new representation, namely as the algebra of operators which are functions of certain collections of commuting unbounded self-adjoint operators in the Bergman space.

Hypocoercivity for non-linear infinite-dimensional degenerate stochastic differential equations

The aim of this article is to construct solutions to second order in time stochastic partial differential equations and to show hypocoercivity of the corresponding transition semigroups. More generally, we analyze non-linear infinite-dimensional degenerate stochastic differential equations in terms of their infinitesimal generators. In the first part of this article we use resolvent methods developed by Beznea, Boboc and Röckner to construct diffusion processes with infinite lifetime and explicit invariant measures. The processes provide weak solutions to infinite-dimensional Langevin dynamics. The second part deals with a general abstract Hilbert space hypocoercivity method, developed by Dolbeaut, Mouhot and Schmeiser. In order to treat stochastic (partial) differential equations, Grothaus and Stilgenbauer translated these concepts to the Kolmogorov backwards setting taking domain issues into account. To apply these concepts in the context of infinite-dimensional Langevin dynamics we use an essential m-dissipativity result for infinite-dimensional Ornstein–Uhlenbeck operators, perturbed by the gradient of a potential. We allow unbounded diffusion operators as coefficients and apply corresponding regularity estimates. Furthermore, essential m-dissipativity of a non-sectorial Kolmogorov backward operator associated to the dynamic is needed. Poincaré inequalities for measures with densities w.r.t. infinite-dimensional non-degenerate Gaussian measures are studied. Deriving a stochastic representation of the semigroup generated by the Kolmogorov backward operator as the transition semigroup of a diffusion process enables us to show an L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-exponential ergodicity result for the weak solution. Finally, we apply our results to explicit infinite-dimensional degenerate diffusion equations.

Maximal sectorial operators and invariant operator ranges

For unbounded maximal sectorial operators we establish necessary and sufficient conditions for the domain equality domA=domA⁎ and for the equality ReA=AR of operator real part ReA and form real part AR. Here ReA=12(A+A⁎) is half of the operator sum defined on domA∩domA⁎, whereas AR=12(A+˙A⁎) is the self-adjoint operator given by half of the form-sum of A and A⁎ so that, in general, ReA⊆AR. The natural question posed in [6], whether for a maximal sectorial operator A the equalitydomA=domA⁎implies the equalityReA=AR, is answered negatively in this paper. We construct families of unbounded coercive m-sectorial operators A such that domA=domA⁎ for which ReA is a closed symmetric non-selfadjoint operator or a non-closed essentially selfadjoint operator. Moreover, we show that the domain equalities domA=domA⁎ and domReA=domAR are equivalent to problems of invariant operator ranges of bounded selfadjoint or unitary operators as well as to the existence of bounded operators with specific operator range properties.

On an inverse problem for a linearized system of Navier–Stokes equations with a final overdetermination condition

Abstract The theory of inverse problems is an actively studied area of modern differential equation theory. This paper studies the solvability of the inverse problem for a linearized system of Navier–Stokes equations in a cylindrical domain with a final overdetermination condition. Our approach is to reduce the inverse problem to a direct problem for a loaded equation. In contrast to the well-known works in this field, our approach is to find an equation for a loaded term whose solvability condition provides the solvability of the original inverse problem. At the same time, the classical theory of spectral decomposition of unbounded self-adjoint operators is actively used. Concrete examples demonstrate that the assertions of our theorems naturally develop and complement the known results on inverse problems. Various cases are considered when the known coefficient on the right-hand side of the equation depends only on time or both on time and a spatial variable. Theorems establishing new sufficient conditions for the unique solvability of the inverse problem under consideration are proved.

Operator Hypergeometric Functions

We consider operator hypergeometric functions 1F2(∙) and 2F3(∙) constructed by an unbounded operator. Using these functions, we solve Cauchy problems for singular integro-differential equations. A new pair of similar operators is given.

Exponentially Convergent Numerical Method for Abstract Cauchy Problem with Fractional Derivative of Caputo Type

We present an exponentially convergent numerical method to approximate the solution of the Cauchy problem for the inhomogeneous fractional differential equation with an unbounded operator coefficient and Caputo fractional derivative in time. The numerical method is based on the newly obtained solution formula that consolidates the mild solution representations of sub-parabolic, parabolic and sub-hyperbolic equations with sectorial operator coefficient A and non-zero initial data. The involved integral operators are approximated using the sinc-quadrature formulas that are tailored to the spectral parameters of A, fractional order α and the smoothness of the first initial condition, as well as to the properties of the equation’s right-hand side f(t). The resulting method possesses exponential convergence for positive sectorial A, any finite t, including t=0 and the whole range α∈(0,2). It is suitable for a practically important case, when no knowledge of f(t) is available outside the considered interval t∈[0,T]. The algorithm of the method is capable of multi-level parallelism. We provide numerical examples that confirm the theoretical error estimates.

The free field: Realization via unbounded operators and Atiyah property

Let X1,…,Xn be operators in a finite von Neumann algebra and consider their division closure in the affiliated unbounded operators. We address the question when this division closure is a skew field (aka division ring) and when it is the free skew field. We show that the first property is equivalent to the strong Atiyah property and that the second property can be characterized in terms of the noncommutative distribution of X1,…,Xn. More precisely, X1,…,Xn generate the free skew field if and only if there exist no non-zero finite rank operators T1,…,Tn such that ∑i[Ti,Xi]=0. Sufficient conditions for this are the maximality of the free entropy dimension or the existence of a dual system of X1,…,Xn. Our general theory is not restricted to selfadjoint operators and thus does also include and recover the result of Linnell that the generators of the free group give the free skew field.We give also consequences of our result for the question of atoms in the distribution of rational functions in free variables or in the asymptotic eigenvalue distribution of matrices over polynomials in asymptotically free random matrices. This solves in particular a conjecture of Charlesworth and Shlyakhtenko.

Stability analysis for a coupled Schrödinger system with one boundary damping

In this paper, we study the stability of a Schrödinger system with one boundary damping, which consists of two constant coefficients Schrödinger equations coupled through zero‐order terms. First, we show that when the one‐dimensional Schrödinger system is not exponentially stable by the asymptotic expansions of eigenvalues. Then, by the frequency domain approach and the multiplier method, we show that the energy decay rate of the multidimensional Schrödinger system is for sufficiently smooth initial data when is sufficiently small, and the boundary of domain satisfies suitable geometric assumption. Next, by solving the characteristic equation of unbounded operator, we show that the strong stability of the one‐dimensional Schrödinger system is completely determined by and and give the necessary and sufficient condition that and satisfy. Finally, by solving the resolvent equation of unbounded operator and using the frequency domain approach, we show that when and is small enough, the energy of the one‐dimensional Schrödinger system decays polynomially and the decay rate depends on the arithmetic property of

Convergence Rates for Learning Linear Operators from Noisy Data

This paper studies the learning of linear operators between infinite-dimensional Hilbert spaces. The training data comprises pairs of random input vectors in a Hilbert space and their noisy images under an unknown self-adjoint linear operator. Assuming that the operator is diagonalizable in a known basis, this work solves the equivalent inverse problem of estimating the operator's eigenvalues given the data. Adopting a Bayesian approach, the theoretical analysis establishes posterior contraction rates in the infinite data limit with Gaussian priors that are not directly linked to the forward map of the inverse problem. The main results also include learning-theoretic generalization error guarantees for a wide range of distribution shifts. These convergence rates quantify the effects of data smoothness and true eigenvalue decay or growth, for compact or unbounded operators, respectively, on sample complexity. Numerical evidence supports the theory in diagonal and non-diagonal settings.

Stochastic Control Problems with Unbounded Control Operators: Solutions Through Generalized Derivatives

This paper deals with a family of stochastic control problems in Hilbert spaces which arises in typical applications (such as boundary control and control of delay equations with delay in the control) and for which is difficult to apply the dynamic programming approach due to the unboudedness of the control operator and to the lack of regularity of the underlying transition semigroup. We introduce a specific concept of partial derivative, designed for this situation, and we develop a method to prove that the associated HJB equation has a solution with enough regularity to find optimal controls in feedback form.

Uniformly exponentially stable approximations for Timoshenko beams

In this note, Timoshenko beams with interior damping and boundary damping are studied from the viewpoints of control theory and numerical approximation. Especially, the uniform exponential stabilities of the beams are studied. The meaning of uniform exponential stability in this paper is two-fold: The first one is in the classical sense and also is concisely called exponential stability by many authors; The second one is that the semi-discretization systems, which are derived from an exponentially stable continuous beam by some semi-discretization schemes, are uniformly exponentially stable with respect to the discretized parameter. To investigate uniform exponential stability of continuous and discrete systems, five completely different methods, which are stability theory of port-Hamiltonian system, direct method of Lyapunov functional, perturbation theory of C0-semigroup, spectral analysis of unbounded operator and frequency standard of exponential stability for contractive semigroup, are involved. Especially, a new method, which is based on the frequency domain characteristics of uniform exponential stability of C0-semigroup of contractions, is established to verify the uniform exponential stability of semi-discretization systems derived from coupled system. The effectiveness of the numerical approximating algorithms is verified by numerical simulations.

Null Controllability of Nonlinear Mixed Integro-differential Systems

The present paper investigates the existence of mild solutions and provides sufficient conditions for the null controllability of nonlinear mixed integrodifferential systems with unbounded linear operators in Banach spaces. It is observed that, if the linear system is controllable, then the nonlinear equivalence is also controllable by subjecting certain smooth conditions on the perturbation function. The results are obtained using semi group of linear operators, fractional powers of operators and the Schauder fixed point theorem.

Perturbations of Weyl’s Theorems for Unbounded Class-A Operators

The significance of Weyl's spectrum is regional to the perturbation theory but recently it is related to operator theory in some theorems. The study of Weyl’s theorem has so far limited to the class of bounded operators, however, lately this study has been extended to some classes of unbounded operators such as hypanormal, posinormal and class- operators. The aim of this article is to continue study more spectral properties for class- operators. First, we use property to study property (b) and then show the equivalent of property (gb) with property (gw). Secondly, we start to study new spectral properties that was defined for bounded operators we start with and properties and show that for any operator that belongs to class- will possess them. Finally, we resuming our goal and give our last two properties and and prove them for , also we established the connection between these properties.

Local nets of unbounded operator algebras

Two rigorous mathematical foundations for relativistic quantum field theory are local nets of von Neumann algebras and Wightman field theory. We recall that local nets of von Neumann algebras are local nets of bounded operator algebras, whereas Wightman field theory makes use of unbounded linear operators. In this paper, we introduce local nets of unbounded operator algebras and show how these can be connected with Wightman field theory. This is motivated by the fact that observables in quantum mechanics are unbounded operators in general.

Weak convergence of the backward Euler method for stochastic Cahn–Hilliard equation with additive noise

We prove a weak rate of convergence of a fully discrete scheme for stochastic Cahn–Hilliard equation with additive noise, where the spectral Galerkin method is used in space and the backward Euler method is used in time. Compared with the Allen–Cahn type stochastic partial differential equation, the error analysis here is much more sophisticated due to the presence of the unbounded operator in front of the nonlinear term. To address such issues, a novel and direct approach has been exploited which does not rely on a Kolmogorov equation but on the integration by parts formula from Malliavin calculus. To the best of our knowledge, the rates of weak convergence are revealed in the stochastic Cahn–Hilliard equation setting for the first time.

Certain properties involving the unbounded operators p(T), TT⁎, and T⁎T; and some applications to powers and nth roots of unbounded operators

In this paper, we are concerned with conditions under which [p(T)]⁎=p‾(T⁎), where p(z) is a one-variable complex polynomial, and T is an unbounded, densely defined, and linear operator. Then, we deal with the validity of the identities σ(AB)=σ(BA), where A and B are two unbounded operators. The equations (TT⁎)⁎=TT⁎ and (T⁎T)⁎=T⁎T, where T is a densely defined closable operator, are also studied. A particular interest will be paid to the equation T⁎T=p(T) and its variants. Then, we have certain results concerning nth roots of classes of normal and nonnormal (unbounded) operators. Some further consequences and counterexamples accompany our results.

Transferability of graph neural networks: An extended graphon approach

We study spectral graph convolutional neural networks (GCNNs), where filters are defined as continuous functions of the graph shift operator (GSO) through functional calculus. A spectral GCNN is not tailored to one specific graph and can be transferred between different graphs. It is hence important to study the GCNN transferability: the capacity of the network to have approximately the same repercussion on different graphs that represent the same phenomenon. Transferability ensures that GCNNs trained on certain graphs generalize if the graphs in the test set represent the same phenomena as the graphs in the training set. In this paper, we consider a model of transferability based on graphon analysis. Graphons are limit objects of graphs, and, in the graph paradigm, two graphs represent the same phenomenon if both approximate the same graphon. Our main contributions can be summarized as follows: 1) we prove that any fixed GCNN with continuous filters is transferable under graphs that approximate the same graphon, 2) we prove transferability for graphs that approximate unbounded graphon shift operators, which are defined in this paper, and, 3) we obtain non-asymptotic approximation results, proving linear stability of GCNNs. This extends current state-of-the-art results which show asymptotic transferability for polynomial filters under graphs that approximate bounded graphons.

Optimal feedback controls of stochastic linear quadratic control problems in infinite dimensions with random coefficients

It is a longstanding unsolved problem to characterize the optimal feedback controls for general linear quadratic optimal control problem of stochastic evolution equation with random coefficients. A solution to this problem is given in [22] under some assumptions which can be verified for interesting concrete models, such as controlled stochastic wave equations, controlled stochastic Schrödinger equations, etc. More precisely, the authors establish the equivalence between the existence of an optimal feedback operator and the solvability of the corresponding operator-valued, backward stochastic Riccati equations. However, their result cannot cover some important stochastic partial differential equations, such as stochastic heat equations, stochastic stokes equations, etc. A key contribution of the current work is to relax the C0-group assumption of unbounded linear operator A in [22] and using the contraction semigroup assumption instead. Therefore, our result can be well applicable in the linear quadratic problem of stochastic parabolic equations. To this end, we introduce a suitable notion to the aforementioned Riccati equation, and some delicate techniques which are even new in the finite dimensional case.

Inverse learning in Hilbert scales

We study linear ill-posed inverse problems with noisy data in the framework of statistical learning. The corresponding linear operator equation is assumed to fit a given Hilbert scale, generated by some unbounded self-adjoint operator. Approximate reconstructions from random noisy data are obtained with general regularization schemes in such a way that these belong to the domain of the generator. The analysis has thus to distinguish two cases, the regular one, when the true solution also belongs to the domain of the generator, and the ‘oversmoothing’ one, when this is not the case. Rates of convergence for the regularized solutions will be expressed in terms of certain distance functions. For solutions with smoothness given in terms of source conditions with respect to the scale generating operator, then the error bounds can then be made explicit in terms of the sample size.

On the Exact Controllability of a Semilinear Evolution Equation with an Unbounded Operator

For the Cauchy problem associated with a controlled semilinear evolution equation with an unbounded maximal monotone operator in a Hilbert space, sufficient conditions are obtained for exact controllability to a given final state. Here a generalization of the Browder–Minty theorem and results on the total global solvability of this equation obtained by the author earlier are used. As an example, a semilinear wave equation is considered.