pinecone/core-2020-05-10-deduplication huggingface dataset:
The Distribution Function Inequality and Products of Toeplitz Operators and Hankel Operators

AbstractIn this paper we study Hankel operators and Toeplitz operators through a distribution function inequality on the Lusin area integral function and the Littlewood–Paley theory. A sufficient condition and a necessary condition are obtained for the boundedness of the product of two Hankel operators. They lead to a way to approach Sarason's conjecture on products of Toeplitz operators and shed light on the compactness of the product of Hankel operators. An elementary necessary and sufficient condition for the product of two Toeplitz operators to be a compact perturbation of a Toeplitz operator is obtained. Moreover, a necessary condition is given for the product of Hankel operators to be in the commutator ideal of the algebra generated by the Toeplitz operators with symbols in a Sarason algebra
Hausdorff Dimension of Limit Sets for Spherical CR Manifolds

AbstractLetM2n+1(n⩾1) be a compact, spherical CR manifold. SupposeM2n+1is its universal cover andΦ:M2n+1→S2n+1is on injective CR developing map, whereS2n+1is the standard unit sphere in the complex (n+1)-spaceCn+1, thenM2n+1is of the quotient formΩ/Λ, whereΩis a simply connected open set inS2n+1, andΛis a complex Klein group acting onΩproperly discontinuously. In this paper, we show that if the CR Yamabe invariant ofM2n+1is positive, then the Carnot Hausdorff dimension of the limit set ofΛis bounded above byn·s(M2n+1), wheres(M2n+1)⩽1 and is a CR invariant. The method that we adopt is analysis of the CR invariant Laplacian. We also explain the geometric origin of this question
Sobolev Estimates for Fractional and Singular Radon Transforms

AbstractWe prove Sobolev inequalities for singular and fractional Radon transforms which are defined as in a paper of Phong and Stein under certain hypothesis on the corresponding Lagrangian ((N*C)′) which does not necessarily have to be a canonical graph. In the proof we use oscillatory integrals, the Cotlar–Stein almost orthogonality theorem, a sort of Littlewood–Paley decomposition for a certain operator, some basic facts about Fourier integral operators and pseudodifferential operators. The main ideas come from papers by Phong and Stein (Acta Math.157(1986), 99–157) and Sogge and Stein (J. Analyse Math.54(1990), 165–188)
The Range of a Structural Projection

AbstractLetAbe a JBW*-triple. A linear subspaceJofAis called aninner idealinAprovided that the subspace {JAJ} is contained inJ. A subtripleBinAis said to becomplementedifA=B⊕Ker(B), where Ker(B)={a∈A:{BaB}=0}. A complemented subtriple inAis a weak*-closed inner ideal. A linear projection onAis said to bestructuralif, for all elementsa,bandcinA,[formula]The range of a structural projection is a complemented subtriple and, conversely, a complemented subtriple is the range of a unique structural projection. We analyze the structure of the weak*-closed inner ideal generated by two arbitrary tripotents in a JBW*-triple in terms of the simultaneous Peirce spaces of three suitably chosen pairwise compatible tripotents. This result is then used to show that every weak* closed inner idealJin a JBW*-tripleAis a complemented subtriple inAand therefore the range of a unique structural projection onA. As an application structural projections on W*-algebras are considered
Capacitary Estimates for Dirichlet Eigenvalues

AbstractLetHbe the non-negative definite selfadjoint operator associated to a regular irreducible Dirichlet form onL2(X, m). Assume thatHhas discrete spectrum. We study perturbations of this operator which arise through the imposition of Dirichlet boundary conditions on a compact subset ofX. The eigenvalues of the perturbed operator are of course shifted to the right. Under an ultracontractivity condition, we show that the magnitude of this shift can be estimated by the capacity. We also obtain a capacitary lower bound for the ground-state shift under suitable conditions. An application to the “crushed ice” problem is described
A Complete Invariant forADAlgebras with Real Rank Zero and Bounded Torsion inK1

AbstractFor each integerp>1, we consider an algebraic invariant forC*-algebras. The invariant consists ofK0,K1, theK0-group with Z/pcoefficients, the order structures these groups possess, and the natural maps between the three groups. We prove that this invariant is complete for the class ofADalgebras of real rank zero ifpannihilates every torsion element ofK1. TheADalgebras areC*-algebras which are inductive limits of finite direct sums ofC*-algebras of continuous matrix-valued functions over the circle, or over the interval, where in the interval case we require that the value over the endpoints is a scalar multiple of the unit matrix. Examples show that the condition on the torsion inK1is necessary when only one modp K0-group is considered. Applying a theorem of Dădărlat, the result also applies to a subclass of theAHalgebras
Radon–Penrose Transform for D-Modules

AbstractLet[formula]be a correspondence of complex analytic manifolds,Fbe a sheaf onX, and M be a coherent DX-module. Consider the associated sheaf theoretical and D-module integral transforms given byΦSF=Rg!f−1F[d] andΦSM=g!f−1M, whereRg!andf−1(resp.gandf−1) denote the direct and inverse image functors for sheaves (resp. for D-modules), andd=dS−dYis the difference of dimension betweenSandY. In this paper, assuming thatfis smooth,gis proper, and (f, g) is a closed embedding, we prove some general adjunction formulas for the functorsΦSandΦS. Moreover, under an additional geometrical hypothesis, we show that the transformationΦSestablishes an equivalence of categories between coherent DX-modules, modulo flat connections, and coherent DY-modules with regular singularities along an involutive manifoldV, modulo flat connections (hereVis determined by the geometry of the correspondence). Applications are given to the case of Penrose's twistor correspondence
Injectivity Sets for the Radon Transform over Circles and Complete Systems of Radial Functions

AbstractA necessary and sufficient characterization is given that specifies which sets of sums of translations of radial functions are dense in the set of continuous functions in the plane. This problem is shown to be equivalent to inversion for the Radon transform on circles centered on restricted subsets of the plane. The proofs rest on the geometry of zero sets for harmonic polynomials and the microlocal analysis of this circular Radon transform. A characterization of nodal sets for the heat and wave equation in the plane are consequences of our theorems, and questions of Pinkus and Ehrenpreis are answered
Amenability of Hopf von Neumann Algebras and Kac Algebras

AbstractLet (M,Γ) be a Hopf von Neumann algebra. The operator predualM*ofMis a completely contractive Banach algebra with multiplicationm=Γ*:M*⊗M*→M*. We call (M,Γ)operator amenableif the completely contractive Banach algebraM*isoperator amenable, i.e., for every operatorM*-bimoduleV, every completely bounded derivation fromM*into the dualM*-bimoduleV* is inner. There is a weaker notion of amenability introduced by D. Voiculescu. We say that a Hopf von Neumann algebra (M,Γ) isVoiculescu amenableif there exists a left invariant mean onM. We show that if a Hopf von Neumann algebra (M,Γ) is operator amenable, then it is Voiculescu amenable.For Kac algebras, there is astrong Voiculescu amenability. We show that for discrete Kac algebras, these amenabilities are all equivalent. In fact, if we letK=(M,Γ,κ,ϕ) be a discrete Kac algebra and letK=(M,Γ,κ,ϕ) be its (compact) dual Kac algebra, then the following are equivalent: (1)Kis operator amenable; (2)Kis Voiculescu amenable; (3) The von Neumann algebraMis hyperfinite; (4)Kis strong Voiculescu amenable; (5)Kis operator amenable; (6)M*has a bounded approximate identity
Interpolation of Operator Spaces

AbstractWe develop the real interpolation theory for operator spaces. We show that the main theorems in the classical real interpolation theory (for Banach spaces) hold in our operator space setting. These include the equivalence, duality and reiteration theorems. As an application of the duality theorem, we prove that the real interpolation operator space of parameters (1/2, 2) between an operator space and its antidual is completely isomorphic to an operator Hilbert spaceOH(I), discovered recently by G. Pisier. We also present an outline of a general interpolation theory in the category of operator spaces. In particular, we construct the operator space version of the orbit and coorbit functors, and extend to operator spaces the classical Aronzsajn–Gagliardo theorem. Our development complements some recent works of Pisier
Quasisimilarity of Invariant Subspaces for Uniform Jordan Operators of Infinite Multiplicity

AbstractLetTbe a bounded linear operator on a separable Hilbert spaceH, and letMandNbe two invariant subspaces forT. We will say thatMandNare quasisimilar if there exist quasiaffinities (i.e. bounded one-to-one operators with dense ranges)XandY, commuting withT, such that (XM)−=Nand (YN)−=M. Recall that operatorsTandT′ are quasisimilar if there exist quasiaffinitiesX:H→H′ andY:H′→Hsuch thatXT=T′XandYT′=TY. We show that if T is a uniform Jordan operator of infinite multiplicity, thenMis quasisimilar toNif and only ifT∣Mis quasisimilar toT∣Nand (T*∣M⊥)* is quasisimilar to (T*∣N⊥)*
Hurewicz's Theorems and Renormings of Banach Spaces

AbstractLetN(X) be the set of all equivalent norms on a separable Banach spaceX, equipped with the topology of uniform convergence on bounded subsets ofX. We show that ifXis infinite dimensional, the set of all locally uniformly rotund norms onXreduces every coanalytic set and, thus, is in particular non-Borel. Dually, we show the same result for the set of all continuously differentiable norms onX, under the assumptionX* is separable. This provides an analogue to a classical result of Mazurkiewicz within convex analysis
Covariance Inequalities Induced by Biorthogonal Operators

AbstractThe logarithmic derivative of a weakly second order probability measureμon a general vector spaceXis an example of an operator fromXintoL2(μ) which preserves the pairing ofXandX′ under the inner product ofL2(μ). This biorthogonality property is exploited to obtain, by means of comparison, a lower bound onμ's covariance operator. The logarithmic derivative is then explicitly evaluated for probability measures induced by the solutions to stochastic ordinary and partial differential equations
Lattices of Intermediate Subfactors

AbstractLetNbe an irreducible subfactor of a typeII1factorM. If the Jones index [M:N] is finite, then the set Lat(N⊂M) of the intermediate subfactors for the inclusionN⊂Mforms afinitelattice. The commuting and co-commuting square conditions for intermediate subfactors are related to the modular identity in the lattice Lat(N⊂N). In particular, simplicity of a finite groupGis characterized in terms of commuting square conditions of intermediate subfactors forN⊂M=N⋊G. We investigate the question of which finite lattices can be realized as intermediate subfactor lattices
On the Korteweg–de Vries Equation: Convergent Birkhoff Normal Form

AbstractThe Korteweg–de Vries equation (KdV)[formula]is a completely integrable Hamiltonian system of infinite dimension with phase space the Sobolev spaceHN(S1; R), (N⩾1), Hamiltonian H(q):=∫S1(12(∂xq(x))2+q(x)3)dx, and Poisson structure ∂/∂x. The functionq≡0 is an elliptic fixed point. We prove that for anyN⩾1, the Korteweg–de Vries equation (and thus the entire KdV-hierarchy) admits globally defined real analytic action-angle variables. As a consequence it follows that in a neighborhood ofq≡0 inH1(S1; R), the KdV-Hamiltonian H (and similarly any Hamiltonian in the KdV-hierarchy) admits a convergent Birkhoff normal form; to the best of our knowledge this is the first such example in infinite dimension. Moreover, using the constructed action-angle variables, we analyze the regularity properties of the Hamiltonian vectorfield of KdV
The Central Limit Theorem for Free Additive Convolution

AbstractLet {Xn}∞n=1be a sequence of free, identically distributed random variables with common distributionμ. Then there exist sequences {Bn}∞n=1and {An}∞n=1of positive and real numbers, respectively, such that sequence of random variables[formula]converges in distribution to the semicircle law if and only if the function[formula]is slowly varying in Karamata's sense. In other words, the free domain of attraction of the semicircle law coincides with the classical domain of attraction of the Gaussian. We prove an analogous result for normal domains of attraction in the sense of Linnik
Logarithmic Sobolev Inequalities for Pinned Loop Groups

AbstractLetGbe a connected compact type Lie group equipped with anAdG-invariant inner product on the Lie algebra g ofG. Given this data there is a well known left invariant “H1-Riemannian structure” on L=L(G)—the infinite dimensional group of continuous based loops inG. Using this Riemannian structure, we define and construct a “heat kernel”νT(g0, ·) associated to the Laplace–Beltrami operator on L(G). HereT>0,g0∈L(G), andνT(g0,·) is a certain probability measure on L(G). For fixedg0∈L(G) andT>0, we use the measureνT(g0,·) and the Riemannian structure on L(G) to construct a “classical” pre-Dirichlet form. The main theorem of this paper asserts that this pre-Dirichlet form admits a logarithmic Sobolev inequality
On Some Subfactors of Integer Index Arising from Vertex Models

AbstractThis paper is devoted to the study of subfactors arising out of commuting squares constructed out of the simplest possible vertex models. After setting up the necessary general background, we start with two classes of commuting squares and compute the principal graphs of the resulting subfactors, one of them being related to the group dual of a suitable (closed) subgroup ofU(N), and the other to the Cayley graph of a (non-closed) group, modulo scalars, generated byNelements ofU(N). In the last section, we “classify” the commuting squares in dimension 2, and identify the possible resulting principal graphs asA(1)(2n−1), 1⩽n⩽
Bezoutians of Rational Matrix Functions

AbstractVarious concepts of a Bezoutian of two rational matrix functions are introduced, thereby extending this concept (previously studied in the framework of matrix and operator polynomials and analystic functions) beyond the class of analytic functions. Basic properties of the Bezoutians are established. The Bezoutian based on realizations of the functions in question turns out to be most adequate and is studied in depth. In particular, its kernel is discribed in terms of the realizations. The result serves as a crucial ingredient in the proof of the key theorem of the paper, providing a characterization of the common zero data of two square-size rational matrix functions in terms of the kernel of their Bezoutian
Asymptotic Behaviour and Self-Similarity for the Three Dimensional Vlasov–Poisson–Fokker–Planck System

AbstractThe aim of this work is to study the asymptotic behaviour of global in time solutions of the Vlasov–Poisson–Fokker–Planck system in three dimensions. We consider both cases, with gravitational and electrostatic interaction, but disregard friction. It is proved that the distribution of particles tends for large time to the fundamental solution of the linear operator inL1norm, which means that the effect of the interaction potential vanishes comparatively att→∞. In quantitative terms the result assures that the total nonlinear interaction force decays for large time with a decay rate of ordert−3and the potential energy behaves likeO(t−3/2). The asymptotic result is independent of the repulsive or attractive character of the interaction field. The main idea is to use the self-similarity of the fundamental solution of the linear part of the equation and the regularity of the Fokker–Planck operator in order to study the large-time distribution of particles
First Order Perturbations of Dirichlet Operators: Existence and Uniqueness

AbstractWe study perturbations of typeB·∇ of Dirichlet operators (L0, D(L0)) associated with Dirichlet forms of typeE0(u,v)=12∫〈∇u,∇v〉HdμonL2(E,μ) whereEis a finite or infinite dimensional Banach space. HereHdenotes a Hilbert space densely and continuously embedded inE. Assuming quasi-regularity of (E0,D(E0)) we show that there always exists a closed extension ofLu:=L0u+〈B,∇u〉Hthat generates a sub-MarkovianC0-semigroup of contractions onL2(E,μ) (resp.L1(E,μ)), ifB∈L2(E;H,μ) and ∫〈B,∇u〉Hdμ⩽0,u⩾0. IfDis an appropriate core for (L0,D(L0)) we show that there is only one closed extension of (L,D) inL1(E,μ) generating a strongly continuous semigroup. In particular we apply our results to operators of typeΔH+B·∇, whereΔHdenotes the Gross–Laplacian on an abstract Wiener space (E,H,γ) andB=−idE+v, wherevtakes values in the Cameron–Martin spaceH
Singular Continuous Limiting Eigenvalue Distributions for Schrödinger Operators on a 2-Sphere

AbstractLetH=−Δ+Vbe a Schrödinger operator acting inL2(S), withSthe two-dimensional unit sphere,Δthe spherical Laplacian, andVa continuous potential. As is well known, the eigenvalues ofHin thelth cluster, i.e., those eigenvalues within a radius sup|V| ofl(l+1), thelth eigenvalue of −Δ, have a limiting distribution;l→∞. We provide an alternative self-contained proof of this fact. We then exhibit Hölder continuous potentialsV, both axially- and nonaxially-symmetric, for which the limiting distributions are singular continuous
Local Extrema of Traces of Heat Kernels onS2

AbstractOnS2we consider metrics conformal to the standard round metricgand of area 4π. We show that among such metrics the trace of the heat kernelTr(etΔ) is locally minimized atg, for any givent>0. The local condition is expressed in terms of anL∞neighborhood of the set of conformal factors ofgof the form |τ′|, withτa Möbius transformation. To prove this result we use a power series expansion of the trace in terms of its conformal variations. We derive a combinatorial formula for the second variation and prove that it is positive definite except along the first eigenspace, where it vanishes. We estimate the higher variations and use a center of mass argument to complete the proof
Hamburger Moment Problem: Parseval Equality andA-Singularity

AbstractIn one of the author's previous papers, the “Parseval Equality” was established for anyA-regular scattering operator function (i.e., for one associated with the Generalized Bitangential Schur–Nevanlinna–Pick Problem). The infinite Hamburger Moment Problem gives an example of anA-singular scattering operator-function with the “Parseval Equality” property. This leads to a negative answer to a question due to V. M. Adamjan, D. Z. Arov, and M. G. Krein, and (what is the same) produces a counter-example to D. Sarason's conjecture concerning exposed points inH1
Folner Conditions, Nuclearity, and Subexponential Growth inC*-Algebras

AbstractAnswering a first question of Voiculescu (On the existence of quasi-central approximate units relative to normed ideals,J. Funct. Anal.91(1990), 1–36) Vaillant 8?and Kirchberg (OnC*-algebras having subexponential, polynomial and linear growth,Invent. Math.108(1992), 635–652) showed that subexponential growth implies nuclearity in theC*-context. Voiculescu suggested a Følner type condition forC*-algebras and asked about the relation to growth and nuclearity. In this work we clarify the relation among subexponential growth phenomena, this Følner condition suggested by Voiculescu, weak filtrability in the sense of Arveson and Bedos, and nuclearity in theC*-algebra context
Pointwise Functional Calculi

AbstractSupposeAis a (possibly unbounded) closed linear operator on a Banach spaceX,x∈X, and F is a Banach algebra of functions. We introduce apointwiseFfunctional calculus for A at x. This is a bounded linear map from F intoX, with the properties that one would expect from a mapf↦f(A)x, ifAhad a F functional calculus; howeverAmay not have such a functional calculus. We show that the existence of a pointwise F functional calculus is equivalent to the existence of a continuously embedded Banach subspace on whichAhas a (global) F functional calculus. We characterize being pointwise generalized scalar atxand give simple sufficient conditions. We also discuss the relationship between pointwise functional calculi and the many physical problems that may be modelled as an abstract Cauchy problem
Semigroup Kernels, Poisson Bounds, and Holomorphic Functional Calculus

AbstractLetLbe the generator of a continuous holomorphic semigroupSwhose action is determined by an integral kernelKon a scale of spacesLp(X; ρ). Under mild geometric assumptions on (X, ρ), we prove that ifLhas a boundedH∞-functional calculus onL2(X; ρ) andKsatisfies bounds typical for the Poisson kernel, thenLhas a boundedH∞-functional calculus onLp(X; ρ) for eachp∈⦠1, ∞⦔. Moreover, if (X, ρ) is of polynomial type andKsatisfies second-order Gaussian bounds, we establish criteria forLto have a bounded Hörmander functional calculus or a bounded Davies–Helffer–Sjöstrand functional calculus
Anticipating Flows on the Wiener Space Generated by Vector Fields of Low Regularity

AbstractSobolev spaces for Banach space valued Wiener functionals are constructed using the operator norm for theH-derivatives. The corresponding Sobolev norms are shown to be strictly weaker than the usual Malliavin Sobolev norms. Existing results on flows on an abstract Wiener space generated by a vector field are generalized to the case where the vector field is a sum of a not necessarily bounded skew-adjoint operator and a non-linear part whose regularity is measured by the new Sobolev norms. This is equivalent to a non-linear perturbation of a semi-group of rotations
A Smoothing Property for Fréchet Spaces

AbstractA smoothing property (SΩ)tfor Fréchet spaces is introduced generalizing the classical concept of smoothing operators which are important in the proof of Nash–Moser inverse function theorems. For Fréchet–Hilbert spaces property (Ω) in standard form in the sense of D. Vogt is shown to be sufficient for (SΩ)t. For instance, the spaces E(K) of infinitely differentiable functions in the sense of Whitney have property (SΩ)tfor an arbitrary compactK⊂Rn; applications to extensions of Whitney functions with estimates are included. In a forthcoming paper, an inverse function theorem will be proved for Fréchet spaces with properties (SΩ)tand (DN); this applies to E(K) if the compactK=K⊂Rnis subanalytic
Ideals in Some Continuous Nonselfadjoint Crossed Product Algebras

AbstractLet (Y, φt) be a locally compact dynamical system, where theφtdenotes a continuous, one parameter semigroup of maps onY. When eachφtis a homeomorphism, a nonselfadjoint crossed product algebra is defined as the subalgebra of the C*-crossed productC0(Y)⋊R supported ont>0; when eachφtis only continuous, the crossed productC0(Y)⋊R+can still be defined. The ideal structure of such an algebra is determined in the case where the semigroup action is the suspension of a discrete, free action on a smaller spaceX. A generalization of Effros–Hahn is given, whereby one may find a meet-irreducible ideal over any arc closure inY. The meet-irreducible ideals form a topological space in the hull-kernel topology, and there is a one-to-one correspondence between closed sets in this space and closed ideals in the algebra. A subset of this space is homeomorphic to the space of finite arcs in the subarc topology. The irrational flow algebra is considered as a special case
A Limit Evolution Problem for Time-Dependent Point Interactions

AbstractWe study the diffusion inR3of a particle interacting withNfixed points through point interactions whose strength varies in time. Under mild assumptions on the time dependence of the strengths, we prove existence for all times and uniqueness of the solution, for which we provide a rather explicit expression. We also prove that, under a suitable rescaling of the interaction strengths, the solution converges, whenN→∞, to the solution of a diffusion equation with a regular killing term (potential). We use properties of the local self-adjoint extensions of the Laplacian and results from the theory of fractional integrals and derivatives