Self-adjoint Linear Operator Research Articles

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.

Efficient representation of the linear density-density response function.

We present a thorough derivation of the mathematical foundations of the representation of the molecular linear electronic density-density response function in terms of a computationally highly efficient moment expansion. Our new representation avoids the necessities of computing and storing numerous eigenfunctions of the response kernel by means of a considerable dimensionality reduction about from 103 to 101 . As the scheme is applicable to any compact, self-adjoint, and positive definite linear operator, we present a general formulation, which can be transferred to other applications with little effort. We also present an explicit application, which illustrates the actual procedure for applying the moment expansion of the linear density-density response function to a water molecule that is subject to a varying external perturbation potential. © 2019 The Authors. Journal of Computational Chemistry published by Wiley Periodicals, Inc.

Strong convergence of the linear implicit Euler method for the finite element discretization of semilinear non-autonomous SPDEs driven by multiplicative or additive noise

This paper aims to investigate the numerical approximation of semilinear non-autonomous stochastic partial differential equations (SPDEs) driven by multiplicative or additive noise. Such equations are more realistic than the autonomous equations when modelling real world phenomena. Such equations find applications in many fields such as transport in porous media, quantum fields theory, electromagnetism and nuclear physics. Numerical approximations of autonomous SPDEs are thoroughly investigated in the literature, while the non-autonomous case are not yet well understood. Here, a non-autonomous SPDE is discretized in space by the finite element method and in time by the linear implicit Euler method. We break the complexity in the analysis of the time depending, not necessarily self-adjoint linear operators with the corresponding semi group and provide the strong convergence result of the fully discrete scheme toward the mild solution. The results indicate how the converge orders depend on the regularity of the initial solution and the noise. Additionally, for additive noise we achieve convergence order in time approximately 1 under less regularity assumptions on the nonlinear drift term than required in the current literature, even in the autonomous case. Numerical simulations motivated from realistic porous media flow are provided to illustrate our theoretical finding.

On the generalized principal eigenvalue of quasilinear operator: definitions and qualitative properties

The notions of generalized principal eigenvalue for linear second order elliptic operators in general domains introduced by Berestycki et al. (Commun Pure Appl Math 47:47–92, 1994) and Berestycki and Rossi (J Eur Math Soc (JEMS) 8:195–215, 2006, Commun Pure Appl Math 68:1014–1065, 2015) have become a very useful and important tool in analysis of partial differential equations. This motivates us for our study of various concepts of eigenvalue for quasilinear operator of the form $$\begin{aligned} {\mathcal {K}}_V[u]:=-\,\Delta _p u +Vu^{p-1},\quad u \ge 0. \end{aligned}$$ This operator is a natural generalization of self-adjoint linear operators. If $$\Omega $$ is a smooth bounded domain, we already proved in Nguyen and Vo (J Funct Anal 269:3120–3146, 2015) that the generalized principal eigenvalue coincides with the (classical) first eigenvalue of $${\mathcal {K}}_V$$ . Here we investigate the relation between three types of the generalized principal eigenvalue for $${\mathcal {K}}_V$$ on general smooth domain (possibly unbounded), which plays an important role in the investigation of their limits with respect to the parameters. We also derive a nice simple condition for the simplicity of the generalized principal eigenvalue and the spectrum of $${\mathcal {K}}_V$$ in $$\mathbb {R}^N$$ . To these aims, we employ new ideas to overcome fundamental difficulties originated from the nonlinearity of p-Laplacian. We also discuss applications of the notions by examining some examples.

On the estimation of a rational approximation to the functions of linear self-adjoint operator pencils

We suggest an approximate method for evaluating functions of a linear pencil ℒ(λ) = λA — B with self-adjoint operator coefficients A and B. It is based on the calculation of a special rational function of ℒ. We assume that the operator A is bounded and positive definite, and the operator B can be unbounded. The estimates of the approximation error are obtained. As an application, an approximate method for calculating the impulse response of a linear differential equation Ax′ = Bx + g(t) is given. The suggested approach can be used in remodeling of the complicated systems.

Semidefinite perturbations in the subspace perturbation problem

The variation of spectral subspaces for linear self-adjoint operators under an additive bounded semidefinite perturbation is considered. A variant of the Davis-Kahan sin2Θ theorem adapted to this situation is proved. Under a certain additional geometric assumption on the separation of the spectrum of the unperturbed operator, this leads to a sharp estimate on the norm of the difference of the spectral projections associated with isolated components of the spectrum of the perturbed and unperturbed operators, respectively. Without this additional geometric assumption on the isolated components of the spectrum of the unperturbed operator, a corresponding estimate is obtained by transferring a known optimization approach for general perturbations to the present situation.

A Note on a Paper by Nieminen

A criteria due to Toivo Nieminen concerning selfadjoint linear operators in complex Hilbert spaces is extended to the case of linear relations.

Second-order difference approximations for Volterra equations with the completely monotonic kernels

The second-order difference type methods are studied for the solution of the problem $$u^{\prime}(t)+{{\int}_{0}^{t}} (t-\tau)u (\tau)d\tau = 0 , t>0, u(0) = u_{0}, $$ with $ L(t)=\sum \limits _{j = 1}^{n} a_{j}(t) L_{j} $ . The operators Lj are densely defined positive self-adjoint linear operator on a Hilbert space H and have spectral decompositions with respect to a common resolution of the identity {Eλ} in H. Here, the kernel functions aj(t), 1 ≤ j ≤ n, are completely monotonic on (0, ∞) and locally integrable, but not constant. The convergence properties of the time discretization are proven in the weighted $ l^{1}(\rho ;0,\infty ; \mathbf {H} ) $ and $ l^{\infty }(\rho ; 0, \infty ; \mathbf {H} ) $ norm, where ρ is a given weighted function.

Complex symmetric differential operators on Fock space

The space of entire functions which are integrable with respect to the Gaussian weight, known also as the Fock space, is one of the preferred functional Hilbert spaces for modeling and experimenting harmonic analysis, quantum mechanics or spectral analysis phenomena. This space of entire functions carries a three parameter family of canonical isometric involutions. We characterize the linear differential operators acting on Fock space which are complex symmetric with respect to these conjugations. In parallel, as a basis of comparison, we discuss the structure of self-adjoint linear differential operators. The computation of the point spectrum of some of these operators is carried out in detail.

On an estimate in the subspace perturbation problem

The problem of variation of spectral subspaces for linear self-adjoint operators under an additive bounded perturbation is considered. The aim is to find the best possible upper bound on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators. In the approach presented here, a constrained optimization problem on a specific set of parameters is formulated, whose solution yields an estimate on the arcsine of the norm of the difference of the corresponding spectral projections. The problem is solved explicitly. This optimizes the approach by Albeverio and Motovilov in [Complex Anal. Oper. Theory 7 (2013), 1389--1416]. In particular, the resulting estimate is stronger than the one obtained there.

Optimization of Mayer Problem with Sturm–Liouville-Type Differential Inclusions

The present paper studies a new class of problems of optimal control theory with Sturm–Liouville-type differential inclusions involving second-order linear self-adjoint differential operators. Our main goal is to derive the optimality conditions of Mayer problem for differential inclusions with initial point constraints. By using the discretization method guaranteeing transition to continuous problem, the discrete and discrete-approximation inclusions are investigated. Necessary and sufficient conditions, containing both the Euler–Lagrange and Hamiltonian-type inclusions and “transversality” conditions are derived. The idea for obtaining optimality conditions of Mayer problem is based on applying locally adjoint mappings. This approach provides several important equivalence results concerning locally adjoint mappings to Sturm–Liouville-type set-valued mappings. The result strengthens and generalizes to the problem with a second-order non-self-adjoint differential operator; a suitable choice of coefficients then transforms this operator to the desired Sturm–Liouville-type problem. In particular, if a positive-valued, scalar function specific to Sturm–Liouville differential inclusions is identically equal to one, we have immediately the optimality conditions for the second-order discrete and differential inclusions. Furthermore, practical applications of these results are demonstrated by optimization of some “linear” optimal control problems for which the Weierstrass–Pontryagin maximum condition is obtained.

Pompeiu-Čebyšev type inequalities for selfadjoint operators in Hilbert spaces

In this work, generalizations of some inequalities for continuous $h$-synchronous ($h$-asynchronous) functions of selfadjoint linear operators in Hilbert spaces are proved.

Hardy spaces for semigroups with Gaussian bounds

Let T_t=e^{-tL} be a semigroup of self-adjoint linear operators acting on L^2(X,mu ), where (X,d,mu ) is a space of homogeneous type. We assume that T_t has an integral kernel T_t(x,y) which satisfies the upper and lower Gaussian bounds: C1μ(B(x,t))exp-c1d(x,y)2/t≤Tt(x,y)≤C2μ(B(x,t))exp-c2d(x,y)2/t.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\frac{C_1}{\\mu (B(x,\\sqrt{t}))} \\exp \\left( {-\\,c_1d(x,y)^2/ t} \\right) \\le T_t(x,y)\\le \\frac{C_2}{\\mu (B(x,\\sqrt{t}))} \\exp \\left( {-\\,c_2 d(x,y)^2/ t} \\right) . \\end{aligned}$$\\end{document}By definition, f belongs to H^1(L) if Vert fVert _{{H^1(L)}}=Vert sup _{t>0}|T_tf(x)|Vert _{L^1(X,mu )} <infty . We prove that there is a function omega (x), 0<cle omega (x)le C, such that H^1(L) admits an atomic decomposition with atoms satisfying: {mathrm{supp}}, asubset B, Vert aVert _{L^infty } le mu (B)^{-1}, and the weighted cancelation condition int a(x)omega (x)hbox {d}mu (x)=0.

Minimum energy for linear systems with finite horizon: a non-standard Riccati equation

This paper deals with a non-standard infinite dimensional linear quadratic control problem arising in the physics of non-stationary states (see, for example, Bertini et al. J Statist Phys 116:831–841, 2004): finding the minimum energy to drive a fixed stationary state $$\bar{x}=0$$ into an arbitrary non-stationary state x. The Riccati equation (RE) associated with this problem is not standard since the sign of the linear part is opposite to the usual one, thus preventing the use of the known theory. Here we consider the finite horizon case when the leading semigroup is exponentially stable. We prove that the linear selfadjoint operator P(t), associated with the value function, solves the above-mentioned RE (Theorem 4.12). Uniqueness does not hold in general, but we are able to prove a partial uniqueness result in the class of invertible operators (Theorem 4.13). In the special case where the involved operators commute, a more detailed analysis of the set of solutions is given (Theorems 4.14, 4.15 and 4.16). Examples of applications are given.

Analytic definition of spin structure

We work on a parallelizable time-orientable Lorentzian 4-manifold and prove that in this case, the notion of spin structure can be equivalently defined in a purely analytic fashion. Our analytic definition relies on the use of the concept of a non-degenerate two-by-two formally self-adjoint first order linear differential operator and gauge transformations of such operators. We also give an analytic definition of spin structure for the 3-dimensional Riemannian case.

Maps on states preserving generalized entropy of convex combinations

Let S(H) be the set of all linear positive-semidefinite self-adjoint Trace-one operators (states) on H where H is an at least two-dimensional finite-dimensional real or complex Hilbert space or at least three-dimensional left quaternionic Hilbert space of dimension n. Given a strictly convex function f:[0,1]↦R, for any ρ∈S(H) we define F(ρ)=∑if(λi), where λ1,λ2,…,λn are the eigenvalues of ρ counted with multiplicities. In this note, we completely describe maps ϕ:S(H)→S(H) having the property F(tρ+(1−t)σ)=F(tϕ(ρ)+(1−t)ϕ(σ)) for all t∈[0,1] and every ρ,σ∈S(H). It turns out that ϕ(ρ)=UρU⁎, ρ∈S(H), where U is a real-linear isometry of H. Note that there is no surjectivity assumption and that our result in particular improves the description of maps preserving the von Neumann entropy of convex combinations of states in the complex Hilbert space. It can as well be applied to preserving Schatten or some other strictly convex norms of convex combinations of states.

Self-adjoint Fuzzy Operator in Fuzzy Hilbert Space and its Properties

In this work, we focus our study on adjoint Fuzzy linear operator and self-adjoint Fuzzy linear operator acting on a Fuzzy Hilbert space (FH-Space). We have given several definitions, theorems and discuss in details, The properties of the adjoint and self-adjoint Fuzzy operators in a FH-adjoint Fuzzy operators in a FH-space.

On the Persistence of the Eigenvalues of a Perturbed Fredholm Operator of Index Zero under Nonsmooth Perturbations

Let $H$ be a real Hilbert space and denote by $S$ its unit sphere. Consider the nonlinear eigenvalue problem $Lx + \epsilon N(x) = \lambda x$, where $\epsilon, \lambda \in \mathbb R$, $L : H \to H$ is a bounded self-adjoint (linear) operator with nontrivial kernel and closed image, and $N : H \to H$ is a (possibly) nonlinear perturbation term. A unit eigenvector $\bar x \in S \cap \mathrm {Ker} L$ of $L$ (corresponding to the eigenvalue $\lambda = 0$) is said to be persistent if it is close to solutions $x \in S$ of the above equation for small values of the parameters $\epsilon \neq 0$ and $\lambda$. We give an affirmative answer to a conjecture formulated by R. Chiappinelli and the last two authors in an article published in 2008. Namely, we prove that, if $N$ is Lipschitz continuous and the eigenvalue $\lambda = 0$ has odd multiplicity, then the sphere $S\cap \mathrm {Ker} L$ contains at least one persistent eigenvector. We provide examples in which our results apply, as well as examples showing that, if the dimension of $\mathrm {Ker} L$ is even, then the persistence phenomenon may not occur.

Polynomial Solutions of the Boundary Value Problems for the Poisson Equation in a Layer

It is well known that the Dirichlet problem for the Laplace equation in a ball has a unique polynomial solution (harmonic polynomial) in the case if the given boundary value is the trace of an arbitrary polynomial on the sphere. S.M.Nikol'skii generalized this result in the case of a boundary value problem of the first kind for a linear differential self-adjoint operator of the order 2l with constant coefficients (in particular polyharmonic) and for a domain that is an ellipsoid in R n . For a polyharmonic equation in a ball (homogeneous and inhomogeneous), V.V. Karachik proposed the Almansi formula-based algorithm to construct a polynomial solution of the Dirichlet problem. The paper considers the Poisson equation with the polynomial right-hand side in a multidimensional infinite layer bounded by two hyper-planes. Shows that the Dirichlet boundary value problem and the mixed Dirichlet-Neumann boundary value problem with polynomial boundary conditions have a unique solution in the class of functions of polynomial growth, and this solution is a polynomial. Gives an algorithm for constructing this polynomial solution and considers examples. In particular, presents formulas to give exact values of certain integrals (including multi-dimensional ones) and sums of trigonometric series.

Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets

This article studies algebraic certificates of positivity for noncommutative (nc) operator-valued polynomials on matrix convex sets, such as the solution set DL, called a free Hilbert spectrahedron, of the linear operator inequality (LOI) L(X)=A0⊗I+∑j=1gAj⊗Xj⪰0, where Aj are self-adjoint linear operators on a separable Hilbert space, Xj matrices and I is an identity matrix. If Aj are matrices, then L(X)⪰0 is called a linear matrix inequality (LMI) and DL a free spectrahedron. For monic LMIs, i.e., A0=I, and nc matrix-valued polynomials the certificates of positivity were established by Helton, Klep and McCullough in a series of articles with the use of the theory of complete positivity from operator algebras and classical separation arguments from real algebraic geometry. Since the full strength of the theory of complete positivity is not restricted to finite dimensions, but works well also in the infinite-dimensional setting, we use it to tackle our problems. First we extend the characterization of the inclusion DL1⊆DL2 from monic LMIs to monic LOIsL1 and L2. As a corollary one immediately obtains the description of a polar dual of a free Hilbert spectrahedron DL and its projection, called a free Hilbert spectrahedrop. Further on, using this characterization in a separation argument, we obtain a certificate for multivariate matrix-valued nc polynomials F positive semidefinite on a free Hilbert spectrahedron defined by a monic LOI. Replacing the separation argument by an operator Fejér–Riesz theorem enables us to extend this certificate, in the univariate case, to operator-valued polynomials F. Finally, focusing on the algebraic description of the equality DL1=DL2, we remove the assumption of boundedness from the description in the LMIs case by an extended analysis. However, the description does not extend to LOIs case by counterexamples.