Nonnegative Operator Research Articles

On the Birman problem in the theory of non-negative symmetric operators with compact inverse

Указаны широкие классы неотрицательных операторов Шредингера в $\mathbb{R}^2$ и $\mathbb{R}^3$, обладающих следующими свойствами: 1. Подходящее множество нулевой меры в $\mathbb{R}^2(\mathbb{R}^3)$ определяет сужение каждого из таких операторов, являющееся неотрицательным симметрическим оператором (задачи Дирихле) с компактной пререзольвентой. 2. При некоторых дополнительных условиях на потенциал расширение Фридрихса такого сужения имеет непрерывный (иногда абсолютно непрерывный) спектр, заполняющий положительную полуось. Приведенные результаты дают решение проблемы М. С. Бирмана.

On a nonlocal problem for the first-order differential-operator equations

In this work, we study the spaces of generalised elements identified with formal Fourier series and constructed via a non-negative self-adjoint operator in Hilbert space. The spectrum of this operator is purely discrete. For a differential-operator equation of the first order, we formulate a nonlocal multipoint by time problem if the corresponding condition is satisfied in a positive or negative space that is constructed via such operator; such problem can be treated as a generalisation of an abstract Cauchy problem for the specified differential-operator equation. The correct solvability of the aforementioned problem is proven, a fundamental solution is constructed, and its structure and properties are studied. The solution is represented as an abstract convolution of a fundamental solution with a boundary element. This boundary element is used to formulate a multipoint condition, and it is a linear continuous functional defined in the space of main elements. Furthermore, this solution satisfies multipoint condition in a negative space that is adjoint with a corresponding positive space of elements.

Linearized Boltzmann Collision Operator: I. Polyatomic Molecules Modeled by a Discrete Internal Energy Variable and Multicomponent Mixtures

The linearized Boltzmann collision operator appears in many important applications of the Boltzmann equation. Therefore, knowing its main properties is of great interest. This work extends some classical results for the linearized Boltzmann collision operator for monatomic single species to the case of polyatomic single species, while also reviewing corresponding results for multicomponent mixtures of monatomic species. The polyatomicity is modeled by a discrete internal energy variable, that can take a finite number of (given) different values. Results concerning the linearized Boltzmann collision operator being a nonnegative symmetric operator with a finite-dimensional kernel are reviewed.A compactness result, saying that the linearized operator can be decomposed into a sum of a positive multiplication operator, the collision frequency, and a compact operator, bringing e.g., self-adjointness, is extended from the classical result for monatomic single species, under reasonable assumptions on the collision kernel. With a probabilistic formulation of the collision operator as a starting point, the compactness property is shown by a splitting, such that the terms can be shown to be, or be the uniform limit of, Hilbert-Schmidt integral operators and as such being compact operators. Moreover, bounds on - including coercivity of - the collision frequency are obtained for a hard sphere like model, from which Fredholmness of the linearized collision operator follows, as well as its domain.

Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities

We establish boundedness estimates for solutions of generalized porous medium equations of the form∂tu+(−L)[um]=0in RN×(0,T), where m≥1 and −L is a linear, symmetric, and nonnegative operator. The wide class of operators we consider includes, but is not limited to, Lévy operators. Our quantitative estimates take the form of precise L1–L∞-smoothing effects and absolute bounds, and their proofs are based on the interplay between a dual formulation of the problem and estimates on the Green function of −L and I−L.In the linear case m=1, it is well-known that the L1–L∞-smoothing effect, or ultracontractivity, is equivalent to Nash inequalities. This is also equivalent to heat kernel estimates, which imply the Green function estimates that represent a key ingredient in our techniques.We establish a similar scenario in the nonlinear setting m>1. First, we can show that operators for which ultracontractivity holds, also provide L1–L∞-smoothing effects in the nonlinear case. The converse implication is not true in general. A counterexample is given by 0-order Lévy operators like −L=I−J⁎. They do not regularize when m=1, but we show that surprisingly enough they do so when m>1, due to the convex nonlinearity. This reveals a striking property of nonlinear equations: the nonlinearity allows for better regularizing properties, almost independently of the linear operator.Finally, we show that smoothing effects, both linear and nonlinear, imply families of inequalities of Gagliardo-Nirenberg-Sobolev type, and we explore equivalences both in the linear and nonlinear settings through the application of the Moser iteration.

Manifolds with $$4\frac{1}{2}$$-Positive Curvature Operator of the Second Kind

We show that a closed four-manifold with \(4\frac{1}{2}\)-positive curvature operator of the second kind is diffeomorphic to a spherical space form. The curvature assumption is sharp as both \({\mathbb{CP}\mathbb{}}^2\) and \({\mathbb {S}}^3 \times {\mathbb {S}}^1\) have \(4\frac{1}{2}\)-nonnegative curvature operator of the second kind. In higher dimensions \(n\ge 5\), we show that closed Riemannian manifolds with \(4\frac{1}{2}\)-positive curvature operator of the second kind are homeomorphic to spherical space forms. These results are proved by showing that \(4\frac{1}{2}\)-positive curvature operator of the second kind implies both positive isotropic curvature and positive Ricci curvature. Rigidity results for \(4\frac{1}{2}\)-nonnegative curvature operator of the second kind are also obtained.

On the domain of non-symmetric and, possibly, degenerate Ornstein–Uhlenbeck operators in separable Banach spaces

Let X be a separable Banach space and let X^* be its topological dual. Let Q:X^*\rightarrow X be a linear, bounded, non-negative, and symmetric operator and let A:D(A)\subseteq X\rightarrow X be the infinitesimal generator of a strongly continuous semigroup of contractions on X . We consider the abstract Wiener space (X,\mu_\infty,H_\infty) , where \mu_\infty is a centered non-degenerate Gaussian measure on X with covariance operator defined, at least formally, as Q_\infty=\int_0^{+\infty} e^{sA}Qe^{sA^*}\,ds, and H_\infty is the Cameron–Martin space associated to \mu_\infty . Let H be the reproducing kernel Hilbert space associated with Q with inner product [\cdot,\cdot]_H . We assume that the operator Q_\infty A^*:D(A^*)\subseteq X^*\rightarrow X extends to a bounded linear operator B\in\mathcal{L}(H) which satisfies B+B^*=-\mathrm{Id}_H , where \mathrm{Id}_H denotes the identity operator on H . Let D and D^2 be the first and second order Fréchet derivative operators. We denote by D_H and (D_H,D^2_H) the closure in L^2(X,\mu_\infty) of the operators QD and (QD,QD^2) , respectively, defined on smooth cylindrical functions, and by W^{1,2}_H(X,\mu_\infty) and W^{2,2}_H(X,\mu_\infty) , respectively, their domains in L^2(X,\mu_\infty) . Furthermore, we denote by D_{A_\infty} the closure of the operator Q_\infty A^*D in L^2(X,\mu_\infty) defined on smooth cylindrical functions, and by W^{1,2}_{A_\infty}(X,\mu_\infty) the domain of D_{A_\infty} in L^2(X,\mu_\infty) . We characterize the domain of the operator L , associated to the bilinear form (u,v)\mapsto-\int_{X}[BD_Hu,D_Hv]_H\,d\mu_\infty, \quad u,v\in W^{1,2}_H(X,\mu_\infty), in L^2(X,\mu_\infty) . More precisely, we prove that D(L) coincides, up to an equivalent renorming, with a subspace of W^{2,2}_H(X,\mu_\infty)\cap W^{1,2}_{A_\infty}(X,\mu_\infty) . We stress that we are able to treat the case when L is degenerate and non-symmetric.

Sharp Estimates for Schrödinger Groups on Hardy Spaces for 0

Let X be a space of homogeneous type with the doubling order n. Let L be a nonnegative self-adjoint operator on L^2(X) and suppose that the kernel of e^{-tL} satisfies a Gaussian upper bound. This paper shows that for 0<ple 1 and s=n(1/p-1/2), ‖(I+L)-seitLf‖HLp(X)≲(1+|t|)s‖f‖HLp(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned}\\Vert (I+L)^{-s}e^{itL}f\\Vert _{H^p_L(X)} \\lesssim (1+|t|)^{s}\\Vert f\\Vert _{H^p_L(X)} \\end{aligned}$$\\end{document}for all tin {mathbb {R}}, where H^p_L(X) is the Hardy space associated to L. This recovers the classical results in the particular case when L=-Delta and extends a number of known results.

Imaginary power operators on Hardy spaces

Let ( X , d , μ ) (X,d, \mu ) be an Ahlfors n n -regular metric measure space. Let L \mathcal {L} be a non-negative self-adjoint operator on L 2 ( X ) L^2(X) with heat kernel satisfying Gaussian estimate. Assume that the kernels of the spectral multiplier operators F ( L ) F(\mathcal {L}) satisfy an appropriate weighted L 2 L^2 estimate. By the spectral theory, we can define the imaginary power operator L i s , s ∈ R \mathcal {L}^{is}, s\in \mathbb R , which is bounded on L 2 ( X ) L^2(X) . The main aim of this paper is to prove that for any p ∈ ( 0 , ∞ ) p \in (0,\infty ) , ‖ L i s f ‖ H L p ( X ) ≤ C ( 1 + | s | ) n | 1 / p − 1 / 2 | ‖ f ‖ H L p ( X ) , s ∈ R , \begin{equation*} \big \|\mathcal {L}^{is} f\big \|_{H^p_{\mathcal {L}}(X)} \leq C (1+|s|)^{n|1/p-1/2|} \|f\|_{H^p_{\mathcal {L}}(X)}, \quad s \in \mathbb {R}, \end{equation*} where H L p ( X ) H^p_\mathcal {L}(X) is the Hardy space associated to L \mathcal {L} , and C C is a constant independent of s s . Our result applies to sub-Laplaicans on stratified Lie groups and Hermite operators on R n \mathbb {R}^n with n ≥ 2 n\ge 2 .

Almost everywhere convergence of spectral sums for self-adjoint operators

Let L L be a non-negative self-adjoint operator acting on the space L 2 ( X ) L^2(X) , where X X is a positive measure space. Let L = ∫ 0 ∞ λ d E L ( λ ) { L}=\int _0^{\infty } \lambda dE_{ L}({\lambda }) be the spectral resolution of L L and S R ( L ) f = ∫ 0 R d E L ( λ ) f S_R({ L})f=\int _0^R dE_{ L}(\lambda ) f denote the spherical partial sums in terms of the resolution of L { L} . In this article we give a sufficient condition on L L such that lim R → ∞ S R ( L ) f ( x ) = f ( x ) , a.e. \begin{equation*} \lim _{R\rightarrow \infty } S_R({ L})f(x) =f(x),\ \ \text {a.e.} \end{equation*} for any f f such that log ⁡ ( 2 + L ) f ∈ L 2 ( X ) \operatorname {log}(2+L) f\in L^2(X) . These results are applicable to large classes of operators including Dirichlet operators on smooth bounded domains, the Hermite operator and Schrödinger operators with inverse square potentials.

Generalized boundary triples, II. Some applications of generalized boundary triples and form domain invariant Nevanlinna functions

AbstractThe paper is a continuation of Part I and contains several further results on generalized boundary triples, the corresponding Weyl functions, and applications of this technique to ordinary and partial differential operators. We establish a connection between Post's theory of boundary pairs of closed nonnegative forms on the one hand and the theory of generalized boundary triples of nonnegative symmetric operators on the other hand. Applications to the Laplacian operator on bounded domains with smooth, Lipschitz, and even rough boundary, as well as to mixed boundary value problem for the Laplacian are given. Other applications concern with the momentum, Schrödinger, and Dirac operators with local point interactions. These operators demonstrate natural occurrence of ‐generalized boundary triples with domain invariant Weyl functions and essentially selfadjoint reference operators A0.

Oracle inequalities and upper bounds for kernel density estimators on manifolds and more general metric spaces

We prove oracle inequalities and upper bounds for kernel density estimators on a very broad class of metric spaces. Precisely we consider the setting of a doubling measure metric space in the presence of a non-negative self-adjoint operator whose heat kernel enjoys Gaussian regularity. Many classical settings like Euclidean spaces, spheres, balls, cubes as well as general Riemannian manifolds, are contained in our framework. Moreover the rate of convergence we achieve is the optimal one in these special cases. Finally we provide the general methodology of constructing the proper kernels when the manifold under study is given and we give precise examples for the case of the sphere.

Higher-order asymptotic expansion for abstract linear second-order differential equations with time-dependent coefficients

This paper is concerned with the asymptotic expansion of solutions to the initial-value problem of u″(t)+Au(t)+b(t)u′(t)=0 in a Hilbert space with a nonnegative selfadjoint operator A and a coefficient b(t)∼(1+t)−β(−1<β<1). In the case b(t)≡1, it is known that the higher-order asymptotic profiles are determined via a family of first-order differential equations of the form v′(t)+Av(t)=Fn(t) (Sobajima (2021) [10]). For the time-dependent case, it is only known that the asymptotic behavior of such a solution is given by the one of b(t)v′(t)+Av(t)=0. The result of this paper is to find the equations for all higher-order asymptotic profiles. It is worth noticing that the equation for n-th order profile u˜n is given via v′(t)+mn(t)Av(t)=Fn(t) which coefficient mn (time-scale) differs each other.

Cwikel estimates and negative eigenvalues of Schrödinger operators on noncommutative tori

In this paper, we establish Cwikel-type estimates for noncommutative tori for any dimension n ≥ 2. We use them to derive Cwikel–Lieb–Rozenblum inequalities and Lieb–Thirring inequalities for negative eigenvalues of fractional Schrödinger operators on noncommutative tori. The latter leads to a Sobolev inequality for noncommutative tori. On the way, we establish a “borderline version” of the abstract Birman–Schwinger principle for the number of negative eigenvalues of relatively compact form perturbations of a non-negative semi-bounded operator with isolated 0-eigenvalue.

Schrödinger Harmonic Functions with Morrey Traces on Dirichlet Metric Measure Spaces

Assume that (X,d,μ) is a metric measure space that satisfies a Q-doubling condition with Q&gt;1 and supports an L2-Poincaré inequality. Let 𝓛 be a nonnegative operator generalized by a Dirichlet form E and V be a Muckenhoupt weight belonging to a reverse Hölder class RHq(X) for some q≥(Q+1)/2. In this paper, we consider the Dirichlet problem for the Schrödinger equation −∂t2u+𝓛u+Vu=0 on the upper half-space X×R+, which has f as its the boundary value on X. We show that a solution u of the Schrödinger equation satisfies the Carleson type condition if and only if there exists a square Morrey function f such that u can be expressed by the Poisson integral of f. This extends the results of Song-Tian-Yan [Acta Math. Sin. (Engl. Ser.) 34 (2018), 787-800] from the Euclidean space RQ to the metric measure space X and improves the reverse Hölder index from q≥Q to q≥(Q+1)/2.

Tachibana-type theorems and special Holonomy

We prove rigidity results for compact Riemannian manifolds in the spirit of Tachibana. For example, we observe that manifolds with divergence-free Weyl tensors and -nonnegative curvature operators are locally symmetric or conformally equivalent to a quotient of the sphere. The main focus of the paper is to prove similar results for manifolds with special holonomy. In particular, we consider Kähler manifolds with divergence-free Bochner tensors. For quaternion Kähler manifolds, we obtain a partial result towards the LeBrun–Salamon conjecture.

Littlewood–Paley Characterization for Musielak–Orlicz–Hardy Spaces Associated with Self-Adjoint Operators

Let X , d , μ be a metric measure space endowed with a metric d and a non-negative Borel doubling measure μ . Let L be a non-negative self-adjoint operator on L 2 X . Assume that the (heat) kernel associated to the semigroup e − t L satisfies a Gaussian upper bound. In this paper, we prove that the Musielak–Orlicz–Hardy space H φ , L X associated with L in terms of the Lusin-area function and the Musielak–Orlicz–Hardy space H L , G , φ X associated with L in terms of the Littlewood–Paley function coincide and their norms are equivalent. To do this, we first establish the discrete characterization of these two spaces. It improves the known results in the literature.

Threshold approximations for the resolvent of a polynomial nonnegative operator pencil

In a Hilbert space H \mathfrak {H} , a family of operators A ( t ) A(t) , t ∈ R t\in \mathbb {R} , is treated admitting a factorization of the form A ( t ) = X ( t ) ∗ X ( t ) A(t) = X(t)^* X(t) , where X ( t ) = X 0 + X 1 t + ⋯ + X p t p X(t)=X_0+X_1t+\cdots +X_pt^p , p ≥ 2 p\ge 2 . It is assumed that the point λ 0 = 0 \lambda _0=0 is an isolated eigenvalue of finite multiplicity for A ( 0 ) A(0) . Let F ( t ) F(t) be the spectral projection of A ( t ) A(t) for the interval [ 0 , δ ] [0,\delta ] . For | t | ≤ t 0 |t| \le t^0 , approximation in the operator norm in H \mathfrak {H} for the projection F ( t ) F(t) with an error O ( t 2 p ) O(t^{2p}) is obtained as well as approximation for the operator A ( t ) F ( t ) A(t)F(t) with an error O ( t 4 p ) O(t^{4p}) (the so-called threshold approximations). The parameters δ \delta and t 0 t^0 are controlled explicitly. Using threshold approximations, approximation in the operator norm in H \mathfrak {H} is found for the resolvent ( A ( t ) + ε 2 p I ) − 1 (A(t)+\varepsilon ^{2p}I)^{-1} for | t | ≤ t 0 |t|\le t^0 and small ε &gt; 0 \varepsilon &gt;0 with an error O ( 1 ) O(1) . All approximations mentioned above are given in terms of the spectral characteristics of the operator A ( t ) A(t) near the bottom of the spectrum. The results are aimed at application to homogenization problems for periodic differential operators in the small period limit.

Dirichlet problem for Schrödinger equation with the boundary value in the BMO space

Let (X, d, μ) be a metric measure space satisfying a Q-doubling condition (Q > 1) and an L2-Poincaré inequality. Let $${\cal L} = {\cal L} + V$$ be a Schrödinger operator on X, where $${\cal L}$$ is a non-negative operator generalized by a Dirichlet form, and V is a non-negative Muckenhoupt weight that satisfies a reverse Hölder condition RHq for some q ⩾ (Q + 1)/2. We show that a solution to $$({\cal L} - \partial _t^2)u = 0$$ on X × ℝ+ satisfies the Carleson condition $$\mathop {{\rm{sub}}}\limits_{B({x_B},{r_B})} {1 \over {\mu (B({x_B},{r_B}))}}\int_0^{{r_B}} {\int_{B({x_B},{r_B})} {{{\left| {t\nabla u(x,t)} \right|}^2}{{d\mu dt} \over t} < \infty } } $$ if and only if u can be represented as the Poisson integral of the Schrödinger operator ℒ with the trace in the BMO (bounded mean oscillation) space associated with ℒ.

Continuous characterizations of inhomogeneous Besov and Triebel-Lizorkin spaces associated to non-negative self-adjoint operators

Let $$(M, \rho , \mu )$$ be a metric measure space satisfying the doubling, reverse doubling and noncollapsing conditions, and let $$\mathscr {L}$$ be a nonnegative self-adjoint operator acting on $$L^2 (M, d\mu )$$ whose heat kernel satisfies the small-time Gaussian upper bound, Hölder continuity and Markov property. In this paper, we establish new characterizations of the “classical” and “nonclassical” Besov and Triebel-Lizorkin spaces associated to $$\mathscr {L}$$ introduced by Kerkyacharian and Petrushev. More precisely, we obtain characterizations of these spaces in terms of continuous Littlewood-Paley and Lusin functions associated to the heat semigroup generated by $$\mathscr {L}$$ , for complete range of indices. This extends related known results in the classical Euclidean setting to our general setting, and extends corresponding results in (Trans Am Math Soc, 367:121–189, 2015) to complete range of indices.

Convergence of the conjugate gradient method with unbounded operators

In the framework of inverse linear problems on infinite-dimensional Hilbert space, we prove the convergence of the conjugate gradient iterates to an exact solution to the inverse problem in the most general case where the self-adjoint, non-negative operator is unbounded and with minimal, technically unavoidable assumptions on the initial guess of the iterative algorithm. The convergence is proved to always hold in the Hilbert space norm (error convergence), as well as at other levels of regularity (energy norm, residual, etc.) depending on the regularity of the iterates. We also discuss, both analytically and through a selection of numerical tests, the main features and differences of our convergence result as compared to the case, already available in the literature, where the operator is bounded.