pinecone/core-2020-05-10-deduplication huggingface dataset:
On the Local Lifting Property for Operator Spaces

AbstractWe study the local lifting property for operator spaces. This is a natural non-commutative analogue of the Banach space local lifting property, but is very different from the local lifting property studied in C*-algebra theory. We show that an operator space has the λ-local lifting property if and only if it is an LΓ1, λ space. These operator space are λ-completely isomorphic to the operator subspaces of the operator preduals of von Neumann algebras, and thus λ-locally reflexive. Moreover, we show that an operator space V has the λ-local lifting property if and only if its operator space dual V* is λ-injective
Inverse Problem for Periodic “Weighted” Operators

AbstractDefine the periodic weighted operator Ty=−ρ−2(ρ2y′)′ in L2(R, ρ(x)2dx). Suppose a function ρ∈W21(R/Z) is 1-periodic real positive, ρ(0)=1, and let q=ρ′/ρ∈L2(0, 1). The spectrum of T consists of intervals σn=[λ+n−1, λ−n] separated by gaps γn=(λ−n, λ+n), n⩾1, with the lengths |γn| and λ+0=0. Let m2n, n⩾1, be the Dirichlet eigenvalue of the equation −y″−2qy′=z2y, y(0)=y(1)=0 where mn>0. Introduce the Lyapunov function Δ(z, q) for T and note that Δz(zn, q)=0 for some zn∈[λ−n, λ+n]. Let ϕ(x, z, q) be the solution of the equation −ϕ″−2qϕ′=z2ϕ, z∈C, satisfying ϕ(0, z, q)=0, ϕ′(0, z, q)=1. Introduce the vector hn=(hcn, hsn)∈R2, with components hcn=−log[(−1)nϕ′(1, mn, q)], hsn=||hn|2−(hcn)2|1/2sign(zn−mn), where |hn| is defined by the equation cosh|hn|=(−1)nΔ(zn, q)⩾1 and coincides with the euclidien norm of the vector hn. Using nonlinear functional analysis in Hilbert space, we prove that the mapping h:q→h(q)={hn}∞1 is a real analytic isomorphism
Removable Singularities for Lu=Ψ(u) and Orlicz Capacities

AbstractSuppose L is a second order elliptic differential operator in Rd and let α>1. Baras and Pierre have proved in 1984 that Γ is removable for Lu=uα if and only if its Bessel capacity Cap2, α′(Γ)=0. We extend this result to a general equation Lu=Ψ(u) where Ψ(u) is an increasing convex function subject to Δ2 and ∇2 conditions. Namely, we prove that Γ is removable for Lu=Ψ(u) if and only if its Orlicz capacity is zero, that is, the integral ∫BdxΨ(∫Γ|x−y|2−dν(dy)) is equal to 0 or ∞ for every measure ν concentrated on Γ, where B stands for any ball containing Γ
White Noise Driven Korteweg–de Vries Equation

AbstractWe consider a stochastic Korteweg–de Vries equation on the real line. The noise is additive. We use function spaces similar to those introduced by Bourgain to prove well posedness results for the Korteweg–de Vries equation in L2(R). We are able to handle a noise which is locally white in space and time. More precisely, it is a space-time white noise multiplied by an L2-function of the space variable. Due to the lack of a priori estimates, we can only get a local existence result in time. However, we obtain the global existence of L2(R) solutions when the covariance operator of the noise is Hilbert–Schmidt in L2(R)
Embedding the Abstract Wiener Space in a Probability Space

AbstractIn the first part, of this paper it is pointed out that for certain applications of the stochastic calculus of variations it is useful to replace the classical domain of definition—the Wiener space—with a general probability space in which the Wiener space is embedded. This yields a certain “conditional Malliavin calculus” and is applicable to “signal” and “noise” problems. In a somewhat analogous way, it is pointed out in the second part of the paper that formulating the Itô calculus in a setup of an abstract Wiener space embedded in a general probability space endowed with a filtration has certain useful applications. In particular it enables the formulation and derivation of a dimension-free form of the Girsanov theorem as well as a dimension free form of the representation of Lp-Wiener functionals as Itô integrals
On Space-Time Regularity for the Stochastic Heat Equation on Lie Groups

AbstractWe consider the stochastic heat equations on Lie groups, that is, equations of the form ∂tu=Δxu+b(u)+F(u)W on R+×G, where G is a compact Lie group, Δ is the Laplace–Beltrami operator on G, b and F are Lipschitz coefficients, and where W is a Gaussian space-correlated noise, which is white-noise in time. We find necessary and sufficient conditions on the space correlation of W such that u is an L2 or Hölder-continuous function in the spatial variable x, using some basic tools of stochastic analysis and harmonic analysis on the Lie group G
Products of Hankel and Toeplitz Operators on the Bergman Space

AbstractWe consider the question for which square integrable analytic functions f and g on the unit disk the densely defined products TfTg are bounded on the Bergman space. We prove results analogous to those obtained by the second author [17] for such Toeplitz products on the Hardy space. We furthermore obtain similar results for Hankel products HfH*g, where f and g are square integrable on the unit disk, and for the mixed Haplitz products HfTg and TgH*f, where f and g are square integrable on the unit disk and g is analytic
Modules over Operator Algebras, and the Maximal C*-Dilation

AbstractWe continue our study of the general theory of possibly nonselfadjoint algebras of operators on a Hilbert space, and modules over such algebras, developing a little more technology to connect “nonselfadjoint operator algebra” with the C*-algebraic framework. More particularly, we make use of the universal, or maximal, C*-algebra generated by an operator algebra, and C*-dilations. This technology is quite general, however it was developed to solve some problems arising in the theory of Morita equivalence of operator algebras, and as a result most of the applications given here (and in a companion paper) are to that subject. Other applications given here are to extension problems for module maps, and characterizations of C*-algebras
The Spectrum of Kleinian Manifolds

AbstractWe obtain the Plancherel theorem for L2(Γ\G), where G is a classical simple Lie group of real rank one and Γ⊂G is convex–cocompact discrete subgroup, and deduce its consequences for the spectrum of locally invariant differential operators on bundles over Kleinian manifolds. As the main tool, we develop a geometric version of scattering theory which, in particular, contains the meromorphic continuation of the Eisenstein series for this situation. The central role played by invariant distribution sections supported on the limit set is emphasized
Domain Perturbations, Shift of Eigenvalues and Capacity

AbstractThe notion of capacity of a subspace which was introduced in [16] is used to prove new estimates on the shift of the eigenvalues which arises if the form domain of a self-adjoint and semibounded operator is restricted to a smaller subspace. The upper bound on the shift of the spectral bound given in [16] is improved and another lower bound is proved which leads to a generalization of Thirring's inequality if the underlying Hilbert space is an L2-space. Moreover we prove a similar capacitary upper bound for the second eigenvalue. The results are applied to elliptic constant coefficient differential operators of arbitrary order. Finally it is given a capacitary characterization for the shift of the spectral bound being positive which works for operators with spectral bound of arbitrary type
Kernels of Toeplitz Operators via Bourgain's Factorization Theorem

AbstractUsing Bourgain's factorization theorem, we characterize the subspaces of Hp, 1⩽p⩽∞, that coincide with the kernels of Toeplitz operators. This is related to (but entirely independent of) earlier work of E. Hayashi. One consequence of our characterization is that, for some inner functions θ, the class of uniqueness sets for Kpθ depends on p; here Kpθ=defHp∩θzHp is the star-invariant subspace in Hp generated by θ. Moreover, a construction is provided of Blaschke products B and b for which the dimension function p↦dim(KpB∩bHp) has prescribed jumps at prescribed points
Brownian Motion Penetrating Fractals An Application of the Trace Theorem of Besov Spaces

AbstractFor a closed connected set F in Rn, assume that there is a local regular Dirichlet form (a symmetric diffusion process) on F whose domain is included in a Lipschitz space or a Besov space on F. Under some condition for the order of the space and the Newtonian 1-capacity of F, we prove that there exists a symmetric diffusion process on Rn which moves like the process on F and like Brownian motion on Rn outside F. As an application, we will show that when F is a nested fractal or a Sierpinski carpet whose Hausdorff dimension is greater than n−2, we can construct Brownian motion penetrating the fractal. For the proof, we apply the technique developed in the theory of Besov spaces
Weak Type Interpolation Near “Endpoint” Spaces

AbstractLet T be an operator of weak types (a, b) and (p, q), where a<p and b<q. The Marcinkiewicz interpolation theorem and its generalizations due to Boyd, Krein–Semenov and others show that T maps certain rearrangement invariant spaces E which are “not too close” to La or Lp into certain spaces F. In this paper we consider analogous results for such an operator T in the case where, on the contrary, E is close to Lp in the sense that its fundamental function is t1/p. For example, E can be a Lorentz space Lp, r for 1⩽r⩽∞. The corresponding range spaces F are explicitly described and shown to be optimal. When E=Lp, r, then in some cases F is a member of the class of Lorentz–Zygmund spaces Lq, r(logL)s which were introduced and studied by Bennett and Rudnick. But in general F is strictly smaller than the corresponding Lorentz–Zygmund space and belongs to an apparently new class of r.i. spaces. Necessary and sufficient conditiones are given for the members of this new class to coincide with Lorentz–Zygmund spaces. Certain results of this paper can be applied to give an alternative proof and generalization of the optimal form of the limiting case of the Sobolev embedding theorem due to Hansson and Brezis–Wainger
Heat Kernel Analysis and Cameron–Martin Subgroup for Infinite Dimensional Groups

AbstractThe heat kernel measure μt is constructed on GL(H), the group of invertible operators on a complex Hilbert space H. This measure is determined by an infinite dimensional Lie algebra g and a Hermitian inner product on it. The Cameron–Martin subgroup GCM is defined and its properties are discussed. In particular, there is an isometry from the L2μt-closure of holomorphic polynomials into a space Ht(GCM) of functions holomorphic on GCM. This means that any element from this L2μt-closure of holomorphic polynomials has a version holomorphic on GCM. In addition, there is an isometry from Ht(GCM) into a Hilbert space associated with the tensor algebra over g. The latter isometry is an infinite dimensional analog of the Taylor expansion. As examples we discuss a complex orthogonal group and a complex symplectic group
On Form-Sum Approximations of Singularly Perturbed Positive Self-adjoint Operators

AbstractWe discuss singular perturbations of a self-adjoint positive operator A in Hilbert space H formally given by AT=A+T, where T is a singular positive operator (singularity means that KerT is dense in H). We prove the following result: if T is strongly singular with respect to A in the sense that KerT is dense in the Hilbert space H1(A)=D(A1/2) equipped by the graph-norm, then any suitable approximation by positive operators, Tn→T, gives a trivial result, i.e., ATn→A in the strong resolvent sense, where ATn is defined as a form-sum of A and Tn. A corresponding statement is true for operators T, Tn of finite rank which are not necessarily positive. This can be considered as an abstract version of the well known result for the perturbation by a point interaction of the Laplace operator in L2(R3). In the more general case, where the singular operator T has a nontrivial regular component Tr in H1(A), we prove that ATn→ATr in the strong resolvent sense. We give applications to the case of perturbations of the Laplace operator by a positive Radon measure with a nontrivial singular component
Lp−Lp Estimates for the Schrödinger Equation on the Line and Inverse Scattering for the Nonlinear Schrödinger Equation with a Potential

AbstractIn this paper I prove a Lp−Lp estimate for the solutions to the one-dimensional Schrödinger equation with a potential in L1γ where in the generic caseγ>3/2 and in the exceptional case (i.e., when there is a half-bound state of zero energy) γ>5/2. I use this estimate to construct the scattering operator for the nonlinear Schrödinger equation with a potential. I prove moreover, that the low-energy limit of the scattering operator uniquely determines the potential and the coupling constant of the nonlinearity using a method that allows as well for the reconstruction of the potential and of the nonlinearity
Optimal Sobolev Imbeddings Involving Rearrangement-Invariant Quasinorms

AbstractLet m and n be positive integers with n⩾2 and 1⩽m⩽n−1. We study rearrangement-invariant quasinorms ϱR and ϱD on functions f:(0, 1)→R such that to each bounded domain Ω in Rn, with Lebesgue measure |Ω|, there corresponds C=C(|Ω|)>0 for which one has the Sobolev imbedding inequality ϱR(u*(|Ω|t))⩽CϱD(|∇mu|*(|Ω|t)), u∈Cm0(Ω), involving the nonincreasing rearrangements of u and a certain mth order gradient of u. When m=1 we deal, in fact, with a closely related imbedding inequality of Talenti, in which ϱD need not be rearrangement-invariant, ϱR(u*(|Ω|t))⩽CϱD((d/dt)∫{x∈Rn:|u(x)|>u*(|Ω|t)}|(∇u)(x)|dx), u∈C10(Ω). In both cases we are especially interested in when the quasinorms are optimal, in the sense that ϱR cannot be replaced by an essentially larger quasinorm and ϱD cannot be replaced by an essentially smaller one. Our results yield best possible refinements of such (limiting) Sobolev inequalities as those of Trudinger, Strichartz, Hansson, Brézis, and Wainger
Finite Interpolation with Minimum Uniform Norm in Cn

AbstractGiven a finite sequence a≔{a1, …, aN} in a domain Ω⊂Cn, and complex scalars v≔{v1, …, vN}, consider the classical extremal problem of finding the smallest uniform norm of a holomorphic function verifying f(aj)=vj for all j. We show that the modulus of the solutions to this problem must approach its least upper bound along a subset of the boundary of the domain large enough so that its A(Ω)-hull contains a subset of the original a large enough to force the same minimum norm. Furthermore, all the solutions must agree on a variety which contains the hull (in an appropriate, weaker, sense) of a measure supported on the maximum modulus set. An example is given to show that the inclusions can be strict
Orthogonality Relations and Harmonic Forms for Semisimple Lie Groups

AbstractLet G be a connected semisimple Lie group with finite center, and suppose G contains a compact Cartan subgroup T. Certain irreducible unitary representations of G arise as spaces of harmonic forms associated to Dolbeault cohomology of line bundles over the complex homogeneous space G/T. In this work the unitary structures of these realizations are directly related to the orthogonality relations for the matrix coefficients of these representations. Using this connection, we exhibit unitary realizations of certain limits of discrete series representations of SU(2, 1) as spaces of harmonic forms
A Probabilistic Approach to the Equation Lu=−u2

AbstractLet L be a second order elliptic differential operator and let D be an arbitrary open subset of Rd. In [1] we introduced a class U1(D) of positive solutions of the equation Lu=−u2 which is in 1–1 correspondence with a convex class H1(D) of positive solutions of the equation Lu=0. In the present paper, we give a probabilistic characterization of U1(D) and a probabilistic representation of u∈U1(D) in terms of a superdiffusion. Similar results are obtained also for a parabolic equation u+Lu=−u2
Solutions of Semilinear Differential Equations Related to Harmonic Functions

AbstractThis is an attempt to establish a link between positive solutions of semilinear equations Lu=−ψ(u) and Lv=ψ(v) where L is a second order elliptic differential operator and ψ is a positive function. The equations were investigated separately by a number of authors. We try to link them via positive solutions of a linear equation Lu=0 (we call them L-harmonic functions). Let D be an arbitrary open subset of Rd and let U(D), V(D) and H(D) stand for the sets of all positive solutions in D for three equations mentioned above. We establish a 1–1 correspondence between certain subclasses of these classes. Similar results are obtained also for the corresponding parabolic equations. A probabilistic interpretation in terms of a superdiffusion is given in [1]
Functional Inequalities for Empty Essential Spectrum

AbstractIn terms of the equivalence of Poincaré inequality and the existence of spectral gap, the super-Poincaré inequality is suggested in the paper for the study of essential spectrum. It is proved for symmetric diffusions that, such an inequality is equivalent to empty essential spectrum of the corresponding diffusion operator. This inequality recovers known Sobolev and Nash type ones. It is also equivalent to an isoperimetric inequality provided the curvature of the operator is bounded from below. Some results are also proved for a more general setting including symmetric jump processes. Moreover, estimates of inequality constants are also presented, which lead to a proof of a result on ultracontractivity suggested recently by D. Stroock. Finally, concentration of reference measures for super-Poincaré inequalities is studied, the resulting estimates extend previous ones for Poincaré and log-Sobolev inequalities
Sub-Laplacians of Holomorphic Lp-type on Rank One AN-Groups and Related Solvable Groups

AbstractConsider a right-invariant sub-Laplacian L on an exponential solvable Lie group G, endowed with a left-invariant Haar measure. Depending on the structure of G and possibly also that of L, L may admit differentiable Lp-functional calculi, or may be of holomorphic Lp-type for a given p≠2, as recent studies of specific classes of groups G and sub-Laplacians L have revealed. By “holomorphic Lp-type” we mean that every Lp-spectral multiplier for L is necessarily holomorphic in a complex neighborhood of some point in the L2-spectrum of L. This can only arise if the group algebra L1(G) is non-symmetric. In this article we prove that, for large classes of exponential groups, including all rank one AN-groups, a certain Lie algebraic condition, which characterizes the non-symmetry of L1(G) [37], also suffices for L to be of holomorphic L1-type. Moreover, if this condition, which was first introduced by J. Boidol [6] in a different context, holds for generic points in the dual g* of the Lie algebra of G, then L is of holomorphic Lp-type for every p≠2. Besides the non-symmetry of L1(G), also the closedness of coadjoint orbits plays a crucial role. We also discuss an example of a higher rank AN-group. This example and our results in the rank one case suggest that sub-Laplacians on exponential Lie groups may be of holomorphic L1-type if and only if there exists a closed coadjoint orbit Ω⊂g* such that the points of Ω satisfy Boidol's condition. In the course of the proof of our main results, whose principal strategy is similar as in [8], we develop various tools which may be of independent interest and largely apply to more general Lie groups. Some of them are certainly known as “folklore” results. For instance, we study subelliptic estimates on representation spaces, the relation between spectral multipliers and unitary representations, and develop some “holomorphic” and “continuous” perturbation theory for images of sub-Laplacians under “smoothly varying” families of irreducible unitary representations
Isometries with Isomorphic Invariant Subspace Lattices

AbstractJ. B. Conway and T. A. Gillespie (J. Funct. Anal.64 (1985), 178–189) characterized those reductive normal operators which have isomorphic invariant subspace lattices. In a subsequent paper (J. Operator Theory22 (1989), 31–49) they gave several necessary conditions of isomorphism in the class of nonreductive isometries. In this paper, we provide a new necessary condition when the isometry contains a bilateral shift. Furthermore, we give complete characterization if the nonreductive components of the isometries are cyclic. It turns out that this characterization is of different types in the unitary and in the nonunitary case. We describe also when absolutely continuous unitary operators have spatially isomorphic invariant subspace lattices. Our results provide answers for questions posed in the second Conway and Gillespie paper referenced above
Inclusions of von Neumann Algebras, and Quantum Groupoı&#x0308;ds

AbstractFrom a depth 2 inclusion of von Neumann algebras M0 ⊂M1 , with an operator-valued weight verifying a regularity condition, we construct a pseudo-multiplicative unitary, which leads to two structures of Hopf bimodules, dual to each other. Moreover, we construct an action of one of these structures on the algebra M1 such that M0 is the fixed point subalgebra, the algebra M2 given by the basic construction being then isomorphic to the crossed-product. We construct on M2 an action of the other structure, which can be considered as the dual action. If the inclusion M0 ⊂M1 is irreducible, we recover quantum groups, as proved in former papers. This construction generalizes the situation which occurs for actions (or co-actions) of groupoı&#x0308;ds. Other examples of “quantum groupoı&#x0308;ds” are given
A Rationality Criterion for Unbounded Operators

RésuméLet G be a group, let U(G) denote the set of unbounded operators on L2(G) which are affiliated to the group von Neumann algebra W(G) of G, and let D(G) denote the division closure of CG in U(G). Thus D(G) is the smallest subring of U(G) containing CG which is closed under taking inverses. If G is a free group then D(G) is a division ring, and in this case we shall give a criterion for an element of U(G) to be in D(G). This extends a result of Duchamp and Reutenauer, which was concerned with proving a conjecture of Connes. Copyright 2000 Academic Press.Soient G un groupe, U(G) l'ensemble d'opérateurs non bornés affiliés à l'algèbre de von Neumann de groupe de G, et D(G) la clôture de division de CG dans U(G). Ainsi D(G) est le plus petit anneau qui est fermé sous l'opération d'inverse. Si G est un group libre, nous donnons un critère pour qu'un élément de U(G) soit dans D(G)
Solving Boundary Value Problems on Networks Using Equilibrium Measures

AbstractThe purpose of this paper is to construct solutions of self-adjoint boundary value problems on finite networks. To this end, we obtain explicit expressions of the Green functions for all different boundary value problems. The method consists of reducing each boundary value problem either to a Dirichlet problem or to a Poisson equation on a new network closely related with the former boundary value problem. In this process we also get an explicit expression of the Poisson kernel for the Dirichlet problem. In all cases, we express the Green function in terms of equilibrium measures solely, which can be obtained as the unique solution of linear programming problems. In particular, we get analytic expressions of the Green function for the following problems: the Poisson equation on a distance-regular graph, the Dirichlet problem on an infinite distance-regular graph, and the Neumann problem on a ball of an homogeneous tree