Self-adjoint Linear Operator Research Articles

English

In this work, we present the study of the regularity of the solutions of the abstract system (1) that includes the Euler-Bernoulli and Kirchhoff-Love thermoelastic plates, we consider for both fractional couplings given by and where is a strictly positive and self-adjoint linear operator and the parameter Our research stems from the work of [1], [4], and [8]. Our contribution was to directly determine the Gevrey sharp classes: for and when and respectively. And for case when This work also contains direct proofs of the analyticity of the corresponding semigroups In the case the analyticity of the semigroup occurs when and for the case the semigroup is analytic for the parameter The abstract system is given by: (1) where

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.

Relative Morse Index Theory Without Compactness Assumption

We develop the relative Morse index theory for linear self-adjoint operator equation without compactness assumption and give the relationship with the index defined in [J. Differential Equations, 2016, 260(4): 3749–3784].

On completeness of weak eigenfunctions for multi-interval Sturm-Liouville equations with boundary-interface conditions

Abstract The goal of this study is to analyse the eigenvalues and weak eigenfunctions of a new type of multi-interval Sturm-Liouville problem (MISLP) which differs from the standard Sturm-Liouville problems (SLPs) in that the Strum-Liouville equation is defined on a finite number of non-intersecting subintervals and the boundary conditions are set not only at the endpoints but also at finite number internal points of interaction. For the self-adjoint treatment of the considered MISLP, we introduced some self-adjoint linear operators in such a way that the considered multi-interval SLPs can be interpreted as operator-pencil equation. First, we defined a concept of weak solutions (eigenfunctions) for MISLPs with interface conditions at the common ends of the subintervals. Then, we found some important properties of eigenvalues and corresponding weak eigenfunctions. In particular, we proved that the spectrum is discrete and the system of weak eigenfunctions forms a Riesz basis in appropriate Hilbert space.

Primal-dual splittings as fixed point iterations in the range of linear operators

In this paper we study the convergence of the relaxed primal-dual algorithm with critical preconditioners for solving composite monotone inclusions in real Hilbert spaces. We prove that this algorithm define Krasnosel’skiĭ-Mann (KM) iterations in the range of a particular monotone self-adjoint linear operator with non-trivial kernel. Our convergence result generalizes (Condat in J Optim Theory Appl 158: 460–479, 2013, Theorem 3.3) and follows from that of KM iterations defined in the range of linear operators, which is a real Hilbert subspace under suitable conditions. The Douglas–Rachford splitting (DRS) with a non-standard metric is written as a particular instance of the primal-dual algorithm with critical preconditioners and we recover classical results from this new perspective. We implement the algorithm in total variation reconstruction, verifying the advantages of using critical preconditioners and relaxation steps.

Riesz representation theorems for positive linear operators

We generalise the Riesz representation theorems for positive linear functionals on \(\text {C}_{\text {c}}(X)\) and \(\text {C}_{\text {0}}(X)\), where X is a locally compact Hausdorff space, to positive linear operators from these spaces into a partially ordered vector space E. The representing measures are defined on the Borel \(\sigma \)-algebra of X and take their values in the extended positive cone of E. The corresponding integrals are order integrals. We give explicit formulas for the values of the representing measures at open and at compact subsets of X. Results are included where the space E need not be a vector lattice, nor a normed space. Representing measures exist, for example, for positive linear operators into Banach lattices with order continuous norms, into the regular operators on KB-spaces, into the self-adjoint linear operators on complex Hilbert spaces, and into JBW-algebras.

On unitary equivalence to a self-adjoint or doubly–positive Hankel operator

Abstract Let A be a bounded, injective and self-adjoint linear operator on a complex separable Hilbert space. We prove that there is a pure isometry, V, so that AV > 0 and A is Hankel with respect to V, i.e. V * A = AV, if and only if A is not invertible. The isometry V can be chosen to be isomorphic to N ∈ ℕ ∪ {+∞} copies of the unilateral shift if A has spectral multiplicity at most N. We further show that the set of all isometries, V, so that A is Hankel with respect to V, are in bijection with the set of all closed, symmetric restrictions of A −1.

Some iterative methods for solving operator equations by using fusion frames

In this paper, two iterative methods are constructed to solve the operator equation Lu = f where L : H ? H is a bounded, invertible and self-adjoint linear operator on a separable Hilbert space H. By using the concept of fusion frames, which is a generalization of frame theory, we design some algorithms based on Chebyshev polynomials and adaptive one according to conjugate gradient iterative method, and accordingly, we then investigate their convergence via their correspond convergence rates.

Minimum energy with infinite horizon: From stationary to non-stationary states

We study a non-standard infinite horizon, infinite dimensional linear–quadratic control problem arising in the physics of non-stationary states (see e.g. Bertini et al. (2004, 2005)): finding the minimum energy to drive a given stationary state x̄=0 (at time t=−∞) into an arbitrary non-stationary state x (at time t=0). This is the opposite to what is commonly studied in the literature on null controllability (where one drives a generic state x into the equilibrium state x̄=0). Consequently, the Algebraic Riccati Equation (ARE) associated with this problem is non-standard since the sign of the linear part is opposite to the usual one and since its solution is intrinsically unbounded. Hence the standard theory of AREs does not apply. The analogous finite horizon problem has been studied in the companion paper (Acquistapace and Gozzi, 2017). Here, similarly to such paper, we prove that the linear selfadjoint operator associated with the value function is a solution of the above mentioned ARE. Moreover, differently to Acquistapace and Gozzi (2017), we prove that such solution is the maximal one. The first main result (Theorem 5.8) is proved by approximating the problem with suitable auxiliary finite horizon problems (which are different from the one studied in Acquistapace and Gozzi (2017)). Finally in the special case where the involved operators commute we characterize all solutions of the ARE (Theorem 6.5) and we apply this to the Landau–Ginzburg model.

Hyponormal operators in soft sets

In this paper, we introduce the notion of hyponormal operators in soft Hilbert spaces, some properties of these operators are studied, and in a similar way to [6], a hyponormal soft operator is built from a family of operators. In addition, some results are obtained for self-adjoint, invertible, unitary, unit equivalent and normal soft linear operators that relate the properties of these soft operators with a family of operators in certain Hilbert spaces.

Analysis of the spectral symbol associated to discretization schemes of linear self-adjoint differential operators

Given a linear self-adjoint differential operator mathscr {L} along with a discretization scheme (like Finite Differences, Finite Elements, Galerkin Isogeometric Analysis, etc.), in many numerical applications it is crucial to understand how good the (relative) approximation of the whole spectrum of the discretized operator mathscr {L},^{(n)} is, compared to the spectrum of the continuous operator mathscr {L}. The theory of Generalized Locally Toeplitz sequences allows to compute the spectral symbol function omega associated to the discrete matrix mathscr {L},^{(n)}. Inspired by a recent work by T. J. R. Hughes and coauthors, we prove that the symbol omega can measure, asymptotically, the maximum spectral relative error mathscr {E}ge 0. It measures how the scheme is far from a good relative approximation of the whole spectrum of mathscr {L}, and it suggests a suitable (possibly non-uniform) grid such that, if coupled to an increasing refinement of the order of accuracy of the scheme, guarantees mathscr {E}=0.

Using Frames in Steepest Descent-Based Iteration Method for Solving Operator Equations

In this paper, by using the concept of frames, two iterative methods are constructed to solve the operator equation $ Lu=f $ where $ L:Hrightarrow H $ is a bounded, invertible and self-adjoint linear operator on a separable Hilbert space $ H $. These schemes are analogous with steepest descent method which is applied on a preconditioned equation obtained by frames instead. We then investigate their convergence via corresponding convergence rates, which are formed by the frame bounds. We also investigate the optimal case, which leads to the exact solution of the equation. The first scheme refers to the case where $H$ is a real separable Hilbert space, but in the second scheme, we drop this assumption.

Existence of Three Solutions for Nonlinear Operator Equations and Applications to Second-Order Differential Equations

The existence of three solutions for nonlinear operator equations is established via index theory for linear self-adjoint operator equations, critical point reduction method, and three critical points theorems obtained by Brezis-Nirenberg, Ricceri, and Averna-Bonanno. Applying the results to second-order Hamiltonian systems satisfying generalized periodic boundary conditions or Sturm-Liouville boundary conditions and elliptic partial differential equations satisfying Dirichlet boundary value conditions, we obtain some new theorems concerning the existence of three solutions.

Unifying the treatment of indefinite and semidefinite perturbations in the subspace perturbation problem

The variation of spectral subspaces for linear self-adjoint operators under an additive bounded perturbation is considered. The objective is to estimate the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators. Recent results for semidefinite and general, not necessarily semidefinite, perturbations are unified to statements that cover both types of perturbations and, at the same time, also allow for certain perturbations that were not covered before.

Weyl Law on Asymptotically Euclidean Manifolds

We study the asymptotic behaviour of the eigenvalue counting function for self-adjoint elliptic linear operators defined through classical weighted symbols of order (1, 1), on an asymptotically Euclidean manifold. We first prove a two-term Weyl formula, improving previously known remainder estimates. Subsequently, we show that under a geometric assumption on the Hamiltonian flow at infinity, there is a refined Weyl asymptotics with three terms. The proof of the theorem uses a careful analysis of the flow behaviour in the corner component of the boundary of the double compactification of the cotangent bundle. Finally, we illustrate the results by analysing the operator Q=(1+|x|^2)(1-varDelta ) on mathbb {R}^d.

Existence of solutions for subquadratic convex operator equations at resonance and applications to Hamiltonian systems

This paper investigates the existence of solutions to subquadratic operator equations with convex nonlinearities and resonance by means of the index theory for self-adjoint linear operators developed by Dong and dual least action principle developed by Clarke and Ekeland. Applying the results to subquadratic convex Hamiltonian systems satisfying several boundary value conditions including Bolza boundary value conditions, generalized periodic boundary value conditions and Sturm–Liouville boundary value conditions yield some new theorems concerning the existence of solutions or nontrivial solutions. In particular, some famous results about solutions to subquadratic convex Hamiltonian systems by Mawhin and Willem and Ekeland are special cases of the theorems.

Guaranteed a posteriori bounds for eigenvalues and eigenvectors: Multiplicities and clusters

This paper presents a posteriori error estimates for conforming numerical approximations of eigenvalue clusters of second-order self-adjoint elliptic linear operators with compact resolvent. Given a cluster of eigenvalues, we estimate the error in the sum of the eigenvalues, as well as the error in the eigenvectors represented through the density matrix, i.e., the orthogonal projector on the associated eigenspace. This allows us to deal with degenerate (multiple) eigenvalues within the framework. All the bounds are valid under the only assumption that the cluster is separated from the surrounding smaller and larger eigenvalues; we show how this assumption can be numerically checked. Our bounds are guaranteed and converge with the same speed as the exact errors. They can be turned into fully computable bounds as soon as an estimate on the dual norm of the residual is available, which is presented in two particular cases: the Laplace eigenvalue problem discretized with conforming finite elements, and a Schrödinger operator with periodic boundary conditions of the form − Δ + V -\Delta + V discretized with planewaves. For these two cases, numerical illustrations are provided on a set of test problems.

Структура квантовой механики

Interest in quantum mechanics is growing in connection with new experiments with quantum particles and a deeper knowledge of nano- and subatomic principles. The practical use of quantum mechanic methods goes in parallel with the constant rethinking of its foundations, its ideological and epistemological role. A rigorous exposition of the foundations of quantum mechanics in abstract-Hilbert spaces was given in the book of J. Von Neumann in 1933. It is the first and only experiment that has been brought to the end, the presentation of the apparatus of quantum mechanics with the sequence and rigor which usually presented in a purely theory. Later in 1949 the group of French mathematicians under the pseudonym N. Bourbaki introduced the concept of mathematical into mathematics. To date, there is no narration of quantum mechanics, where rigor is combined with the concept of mathematical structure. In the framework of the approach formulated by the group of N. Burbaki, quantum mechanics is a composite structure consisting of three simple structures: the Hilbert space of complex-valued vectors, the space of linear self-adjoint operators and the structure of classical mechanics are showed in this article.

Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann\u2013Hilbert approach

In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms mathcal {H}_L:L^2([b_L,0])rightarrow L^2([0,b_R]) and mathcal {H}_R:L^2([0,b_R])rightarrow L^2([b_L,0]). These operators arise when one studies the interior problem of tomography. The diagonalization of mathcal {H}_R,mathcal {H}_L has been previously obtained, but only asymptotically when b_Lne -b_R. We implement a novel approach based on the method of matrix Riemann–Hilbert problems (RHP) which diagonalizes mathcal {H}_R,mathcal {H}_L explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates.

Generalized linking theorem with applications to nonautonomous Hamiltonian systems and Dirac equations on compact spin manifold

Let [Formula: see text] be a Riemannian manifold with finite volume and [Formula: see text] be a linear topological space. We consider the strongly indefinite superlinear problem [Formula: see text] where [Formula: see text] is a self-adjoint linear operator, [Formula: see text] is a real Hilbert space with the compact embedding [Formula: see text] if [Formula: see text] for some [Formula: see text], and [Formula: see text]. We obtain the existence of two solutions provided that [Formula: see text] and [Formula: see text] for a certain choice of [Formula: see text], [Formula: see text]. Moreover, we prove that, if [Formula: see text] and [Formula: see text] small enough, there exist prescribed number of nontrivial solutions. As applications, the corresponding results hold true for nonautonomous Hamiltonian systems and Dirac equations on compact spin manifold.