pinecone/core-2020-05-10-deduplication huggingface dataset:
Unitary Colligations, Reproducing Kernel Hilbert Spaces, and Nevanlinna–Pick Interpolation in Several Variables

AbstractRecently J. Agler studied the class Sdof scalar-valued, analytic functions ofdcomplex variablesffor whichf(T1, …, Td) has norm at most 1 for any collection ofdcommuting contractions (T1, …, Td) on a Hilbert space H. Among other results he obtained a characterization of such functions in terms of a positivity property and in terms of a representation as the transfer function of a certain type of d-variable linear system, as well as a Nevanlinna–Pick interpolation theorem for this class of functions. In this note we examine the system theory aspects and uniqueness of the transfer function representation, and give a simpler proof of the Nevanlinna–Pick interpolation theorem for the class Sdand obtain ad-variable version of the Toeplitz corona theorem. By using ideas of Arov and Grossman introduced for 1-variable problems, as a bonus we obtain a collection of linear fractional maps which parametrize the set of all Sdsolutions of an interpolation problem
Principal Bimodules of Nest Algebras

AbstractWe classify thewot-closed bimodules over a pair of nest algebras which are singly generated as algebraic and as norm-closed bimodules. The obstructions relate to the finite rank atoms. In particular, if both nest algebras have infinite multiplicity, then every wot-closed bimodule is (algebraically) principal. Another important special case is the ideal of strictly upper triangular operators, which is always principal; and the generator is a sum of commutators. In general, every countably generated wot-closed bimodule is singly generated, and we obtain explicit bounds on the number and norms of the terms in a factorization through the generator
Module Homomorphisms and Topological Centres Associated with Weakly Sequentially Complete Banach Algebras

AbstractThis paper is a contribution to the theory of weakly sequentially complete Banach algebrasA. We require them to have bounded approximate identities and, for the most part, to be ideals in their second duals, so that examples are the group algebrasL1(G) for compact groupsGor the Fourier algebrasA(G) for discrete amenable groupsG. In Section 2 we present our main result, that the topological centre (or set of weak* bicontinuous elements) ofA** is identifiable withA. As a corollary, we deduce in Section 3 that each leftA-module homomorphism fromA* toA*Acan be realized as right translation by an element ofA. These conclusions generalise recent advances in the subject. In Section 4 we take a special algebra,l1(S) for a commutative discrete semigroupS, and show that if its second dual has an identity then that identity must lie inl1(S)
Diffeomorphism-Invariant Spin Network States

AbstractWe extend the theory of diffeomorphism-invariant spin network states from the real-analytic category to the smooth category. Suppose thatGis a compact connected semisimple Lie group andP→Mis a smooth principalG-bundle. A “cylinder function” on the space of smooth connections onPis a continuous complex function of the holonomies along finitely many piecewise smoothly immersed curves inM. We construct diffeomorphism-invariant functionals on the space of cylinder functions from “spin networks”: graphs inMwith edges labeled by representations ofGand vertices labeled by intertwining operators. Using the “group averaging” technique of Ashtekar, Marolf, Mourão, and Thiemann, we equip the space spanned by these diffeomorphism-invariant spin network states with a natural inner product
The Complex Moment Problem and Subnormality: A Polar Decomposition Approach

AbstractIt has been known that positive definiteness does not guarantee a bisequence to be a complex moment. However, it turns out that positive definite extendibility does (Theorems 1 and 22), and this is the main theme of this paper. The main tool is, generally understood, polar decomposition. To strengthen applicability of our approach we work out a criterion for positive definite extendibility in a fairly wide context (Theorems 9 and 29). All this enables us to prove characterizations of subnormality of unbounded operators having invariant domain (Theorems 37 and 39) and their further applications (Theorems 41 and 43) and a description of the complex moment problem on real algebraic curves (Theorems 52 and 56). The latter question is completed in the Appendix, in which we relate the complex moment problem to the two-dimensional real one, with emphasis on real algebraic sets
Resolvent Estimates for Fleming–Viot Operators with Brownian Drift

AbstractThis article is a supplement to the paper of D. A. Dawson and P. March (J. Funct. Anal.132(1995), 417–472). We define a two-parameter scale of Banach spaces of functions defined on M1(Rd), the space of probability measures ond-dimensional euclidean space, using weighted sums of the classical Sobolev norms. We prove that the resolvent of the Fleming–Viot operator with constant diffusion coefficient and Brownian drift acts boundedly between certain members of the scale. These estimates gauge the degree of smoothing performed by the resolvent and separate the contribution due to the diffusion coefficient and that due to the drift coefficient
Large Time Asymptotics of Solutions to the Generalized Korteweg–de Vries Equation

AbstractWe study the asymptotic behavior for large time of solutions to the Cauchy problem for the generalized Korteweg–de Vries (gKdV) equationut+(|u|ρ−1u)x+13uxxx=0, wherex,t∈Rwhen the initial data are small enough. If the powerρof the nonlinearity is greater than 3 then the solution of the Cauchy problem has a quasilinear asymptotic behavior for large time. More precisely, we show that the solutionu(t) satisfies the decay estimate ‖u(t)‖Lβ⩽C(1+t)−(1/3)(1−1/β)forβ∈(4,∞], ‖uux(t)‖L∞⩽Ct−2/3(1+t)−1/3and using these estimates we prove the existence of the scattering stateu+∈L2such that ‖u(t)−U(t)u+‖L2⩽Ct−(ρ−3)/3for any small initial data belonging to the weighted Sobolev spaceH1,1={f∈L2; ‖(1+|x|2)1/2(1−∂2x)1/2f‖L2<∞}, whereU(t) is the Airy free evolution group
Invariant Subspaces for Semigroups of Algebraic Operators

AbstractT. Laffey showed (Linear and Multilinear Algebra6(1978), 269–305) that a semigroup of matrices is triangularizable if the ranks of all the commutators of elements of the semigroup are at most 1. Our main theorem is an extension of Hthis result to semigroups of algebraic operators on a Banach space. We also obtain a related theorem for a pair {A,B} of arbitrary bounded operators satisfying rank (AB−BA)=1 and several related conditions. In addition, it is shown that a semigroup of algebraically unipotent operators of bounded degree is triangularizable
Tensor Algebras overC*-Correspondences: Representations, Dilations, andC*-Envelopes

AbstractTensor algebras overC*-correspondences are noncommutative generalizations of the disk algebra. They contain, as special cases, analytic crossed products, semi-crossed products, and Popescu's noncommutative disk algebras. In this paper, a dilation theorem and a commutant lifting theorem for representations of tensor algebras is proved. These are used to show that for certainC*-correspondences, theC*-envelopes (in the sense of Arveson) of the tensor algebras are the Cuntz–Pimsner algebras of the correspondences
Crossed Products byC0(X)-Actions

AbstractSuppose thatGhas a representation groupH, thatGab≔G/[G,G]is compactly generated, and thatAis aC*-algebra for which the complete regularization of Prim(A) is a locally compact Hausdorff spaceX. In a previous article, we showed that there is a bijectionα↦(Zα,fα) between the collection of exterior equivalence classes of locally inner actionsα:G→Aut(A), and the collection of principalGab-bundlesZαtogether with continuous functionsfα:X→H2(G,T). In this paper, we compute the crossed productsA⋊αGin terms of the dataZα,fα, andC*(H)
Estimates for Hyperbolic Equations of Space Dimension 3

AbstractWe discuss the problem of boundedness fromLp(Rn) toLp′(Rn) (1/p+1/p′=1, 1⩽p⩽2) of operators of the typeM=F−1eiϕ(ξ)a(ξ)F, which is related to the study of hyperbolic equations with constant coefficients. The boundedness is dependent on a geometrical property ofΣ=ϕ−1(1), and its dependence has been exactly determined in the casesn=2, 1⩽p⩽2 andn⩾3,p=1,2 (M. Sugimoto,Math. Z.215(1994), 519–531;222(1996), 521–531). This paper is devoted to the unsolved case 1<p<2, and a strange phenomenon is exhibited in the simplest casen=3
Spectral Analysis on Infinite Sierpiński Gaskets

AbstractWe study the spectral properties of the Laplacian on infinite Sierpiński gaskets. We prove that the Laplacian with the Neumann boundary condition has pure point spectrum. Moreover, the set of eigenfunctions with compact support is complete. The same is true if the infinite Sierpiński gasket has no boundary, but is false for the Laplacian with the Dirichlet boundary condition. In all these cases we describe the spectrum of the Laplacian and all the eigenfunctions with compact support. To obtain these results, first we prove certain new formulas for eigenprojectors of the Laplacian on finite Sierpiński pre-gaskets. Then we prove that the spectrum of the discrete Laplacian on a Sierpiński lattice is pure point, and the eigenfunctions are localized
Semilinear Parabolic Equations, Diffusions, and Superdiffusions

AbstractThe semilinear equationu+L0u=uα, whereL0is a second-order elliptic differential operator without zero-order term and 1<α⩽2, has been studied by the author in [4] and [5] by using superdiffusions. In the present paper, we apply superdiffusions to a more general equationu+Lu=ψ(u), whereLu=L0u+cu(with a bounded coefficientc) andψbelongs to a convex class which containskuαwith 1<α⩽2 and positive locally bounded coefficientk. We also cover a substantially wider class of functionsψwhich do not correspond to any superdiffusion (for instance,kuαwithα>1). Related problems are treated with the help of diffusion processes. This approach is useful even in the linear theory. For instance, the first boundary value problem for equationu+Lu=−fcan be investigated for a class of domains described in probabilistic terms which is substantially larger than the class considered in the literature on PDEs
On Martingale Inequalities in Non-commutative Stochastic Analysis

AbstractWe develop a non-commutativeLpstochastic calculus for the Clifford stochastic integral, anL2theory of which has been developed by Barnett, Streater, and Wilde. The main results are certain non-commutativeLpinequalities relating Clifford integrals and their integrands. These results are applied to extend the domain of the Clifford integral fromL2toL1integrands, and we give applications to optional stopping of Clifford martingales, proving an analog of a Theorem of Burkholder: The stopped Clifford processFThas zero expectation providedET<∞. In proving these results, we establish a number of results relating the Clifford integral to the differential calculus in the Clifford algebra. In particular, we show that the Clifford integral is given by the divergence operator, and we prove an explicit martingale representation theorem. Both of these results correspond closely to basic results for stochastic analysis on Wiener space, thus furthering the analogy between the Clifford process and Brownian motion
Nodal Sets and Morse Indices of Solutions of Super-linear Elliptic PDEs

AbstractIn this paper, we develop a Sturm–Liouville type theory for the nodal sets and Morse indices of solutions of super-linear elliptic PDEs with Dirichlet boundary condition. It shows that there are some relationships between analytic properties (e.g.,Lp-norm, vanishing order of the nodal point, andN−1 dimensional Hausdorff measure of the nodal set) of the solutions as the functions on the domainΩand Morse indices of the solutions as the critical points of the functionalJ(u)=∫Ω[12|∇u|2−F(x,u)]dxonH10(Ω)
Martin Boundary and Integral Representation for Harmonic Functions of Symmetric Stable Processes

AbstractMartin boundaries and integral representations of positive functions which are harmonic in a bounded domainDwith respect to Brownian motion are well understood. Unlike the Brownian case, there are two different kinds of harmonicity with respect to a discontinuous symmetric stable process. One kind are functions harmonic inDwith respect to the whole processX, and the other are functions harmonic inDwith respect to the processXDkilled upon leavingD. In this paper we show that for bounded Lipschitz domains, the Martin boundary with respect to the killed stable processXDcan be identified with the Euclidean boundary. We further give integral representations for both kinds of positive harmonic functions. Also given is the conditional gauge theorem conditioned according to Martin kernels and the limiting behaviors of theh-conditional stable process, wherehis a positive harmonic function ofXD. In the case whenDis a boundedC1,1domain, sharp estimate on the Martin kernel ofDis obtained
Applications of Connes' Geodesic Flow to Trace Formulas in Noncommutative Geometry

AbstractThe “trace formula” of Chazarain, Duistermaat, and Guillemin expresses that the singularities of the distribution trace of the wave group on a compact Riemannian manifoldXis included in the set of periods of the geodesic flow restricted toS*X. Most of the objects involved in this trace formula have analogues in Connes' Noncommutative Geometry. This paper shows, on several significant examples of Noncommutative Geometry, that Connes' definition of geodesic flow leads to statements analogous to the classical trace formula of Chazarain, Duistermaat, and Guillemin
Ideal Structure and Simplicity of theC*-Algebras Generated by Hilbert Bimodules

AbstractPimsner introduced theC*-algebra OXgenerated by a Hilbert bimoduleXover aC*-algebra A. We look for additional conditions thatXshould satisfy in order to study the simplicity and, more generally, the ideal structure of OXwhenXis finite projective. We introduce two conditions, “(I)-freeness” and “(II)-freeness,” stronger than the former, in analogy with J. Cuntz and W. Krieger (Invent. Math.56, 1980, 251–268) and J. Cuntz (Invent. Math.63, 1981, 25–40), respectively. (I)-freeness comprehends the case of the bimodules associated with an inclusion of simpleC*-algebras with finite index, real or pseudoreal bimodules with finite intrinsic dimension, and the case of “Cuntz–Krieger bimodules.” IfXsatisfies this condition theC*-algebra OXdoes not depend on the choice of the generators when A is faithfully represented. As a consequence, ifXis (I)-free and A isX-simple, then OXis simple. In the case of Cuntz–Krieger algebras OA,X-simplicity corresponds to the irreducibility of the matrixA. If A is simple and p.i. then OXis p.i.; if A is nonnuclear then OXis nonnuclear. Thus we provide many examples of (purely) infinite nonnuclear simpleC*-algebras. Furthermore ifXis (II)-free, we determine the ideal structure of OX
Hopf Algebras and Subfactors Associated to Vertex Models

AbstractIfHis a Hopf algebra whose square of the antipode is the identity,v∈L(V)⊗His a corepresentation, andπ:H→L(W) is a representation, thenu=(id⊗π)vsatisfies the equation (t⊗id)u−1=((t⊗id)u)−1of the vertex models for subfactors. A universal construction shows that any solutionuof this equation arises in this way. A more elaborate construction shows that there exists a “minimal” triple (H,v,π) satisfying (id⊗π)v=u. This paper is devoted to the study of this latter construction of Hopf algebras. Ifuis unitary we construct aC*-norm onHand we find a new description of the standard invariant of the subfactor associated tou. We discuss also the “twisted” (i.e.,S2≠id) case
Modified Mixed Tsirelson Spaces

AbstractWe study the modified and boundedly modified mixed Tsirelson spacesTM[(Fkn,θn)∞n=1] andTM(s)[(Fkn,θn)∞n=1], respectively, defined by a subsequence (Fkn) of the sequence of Schreier families (Fn). These are reflexive asymptotic ℓ1spaces with an unconditional basis (ei)ihaving the property that every sequence {xi}ni=1of normalized disjointly supported vectors contained in ⦠ei⦔∞i=nis equivalent to the basis of ℓn1. We show that if limθ1/nn=1 then the spaceT[(Fn,θn)∞n=1] and its modified variationsTM[(Fn,θn)∞n=1] orTM(s)[(Fn,θn)∞n=1] are totally incomparable by proving thatc0is finitely disjointly representable in every block subspace ofT[(Fn,θn)∞n=1]. Next, we present an example of a boundedly modified mixed Tsirelson spaceXM(1),u=TM(1)[(Fkn,θn)∞n=1] which is arbitrarily distortable. Finally, we construct a variation of the spaceXM(1),uwhich is hereditarily indecomposable
Stability of the Peaceman–Rachford Approximation

AbstractLetAanBbe two self-adjoint operators in a Hilbert space; it is assumed thatAandBare bounded from below. The following operator expressionp(t)=(1+tA)−1(1−tB)(1+tB)−1(1−tA) called the Peaceman–Rachford formula, satisfies formally fortsmall the relationp(t)−exp(−2t(A+B))=O(t2). IfAandBdo not commute, it is not true in general thatp(t) is a stable approximation: we give an example ofAandBsuch that |p(t)|B(H), the operator norm ofp(t), is not bounded by 1+O(t); sufficient conditions of stability are given; they involve conditions on the commutators ofAandB. The proof relies on estimates on sandwiches, i.e., products of a finite number of copies ofA,B, (1+tA)−1, (1+tB)−1 and elements of an algebra M of bounded operators inH
Approximation by Ratios of Bounded Analytic Functions

AbstractLet D be the unit disk andEbe a compact subset of theH∞maximal ideal space,M(H∞). We show that any continuous functionfon an open neighborhoodU⊂M(H∞) ofEsuch thatf|U∩D∈H∞(U∩D) can be uniformly approximated onEby ratiosh/g, whereh,g∈H∞andgis zero free onE. We also characterize those setsEfor whichh/gcan be replaced byhfor everyf. Finally, we prove that any inner functionucan be written as a rational function of interpolating Blaschke products. It follows thatucan be uniformly approximated by ratios of interpolating Blaschke products on any compact setE⊂M(H∞) whereuis zero free
Boundary Layers on Sobolev–Besov Spaces and Poisson's Equation for the Laplacian in Lipschitz Domains

AbstractWe study inhomogeneous boundary value problems for the Laplacian in arbitrary Lipschitz domains with data in Sobolev–Besov spaces. As such, this is a natural continuation of work in [Jerison and Kenig,J. Funct. Anal.(1995), 16–219] where the inhomogeneous Dirichlet problem is treated via harmonic measure techniques. The novelty of our approach resides in the systematic use of boundary integral methods. In this regard, the key results are establishing the invertibility of the classical layer potential operators on scales of Sobolev–Besov spaces on Lipschitz boundaries for optimal ranges of indices. Applications toLp-based Helmholtz type decompositions of vector fields in Lipschitz domains are also presented
L2-Concrete Spectral Analysis of the Invariant LaplacianΔαβin the Unit Complex BallBn

AbstractFor every fixed real numberλrelated to the continuous spectrum of the invariant LaplaciansΔαβ=4(1−|z|2)∑i,j=1n(δij−zizj)∂2∂zj∂zj+α∑j=1nzj∂∂zj+β∑j=1nzj∂∂zj−αβinBn, we characterize the eigenfunctions ofΔαβthat are Poisson integrals ofL2-functions on the boundary ofBn. These eigenfunctions occurred in the weighted Plancherel formula of the unit complex ball. (See Zhang,Studia Math.102(2) (1992), for the weighted Plancherel formula in the ball.) The obtained characterization generalizes our announced result in Boussejra and Intissar,C. R. Acad. Sci.315(1992), 1353–1357, for the Bergman LaplacianΔ00of the unit complex ball to the LaplaciansΔαβ, for arbitraryα,βinR
On the Schur Test forL2-Boundedness of Positive Integral Operators with a Wiener–Hopf Example

AbstractThe Schur sufficiency condition for boundedness of any integral operator with non-negative kernel betweenL2-spaces is deduced from an observation, Proposition 1.2, about the central role played byL2-spaces in the general theory of these operators. Suppose (Ω,M,μ) is a measure space and thatK:Ω×Ω→[0,∞) is an M×M-measurable kernel. The special case of Proposition 1.2 for symmetrical kernels says that such a linear integral operator is bounded onanyreasonable normed linear spaceXof M-measurable functions only if it is bounded onL2(Ω,M,μ) where its norm is no larger. The general form of Schur's condition (Halmos and Sunder “Bounded Integral Operators onL2-Spaces,” Springer-Verlag, Berlin/New York, 1978) is a simple corollary which, in the symmetrical case, says that the existence of an M-measurable (not necessarily square-integrable) functionh>0μ-almost-everywhere onΩwithKh(x)=∫ΩK(x,y)h(y)μ(dy)⩽Λh(x)(x∈Ω) (*)implies thatKis a bounded (self-adjoint) operator onL2(Ω,M,μ) of norm at mostΛ. When (Ω,M,μ) isσ-finite, we show that Schur's condition is sharp: in the symmetrical case the boundedness of K onL2(Ω,M,μ) implies, for anyΛ>‖K‖2, the existence of a functionh∈L2(Ω,M,μ) which is positiveμ-almost-everywhere and satisfies (*). Such functionshsatisfying (*), whether inL2(Ω,M,μ) or not, will be calledSchur test functions. They can be found explicitly in significant examples to yield best-possible estimates of the norms for classes of integral operators with non-negative kernels. In the general theory the operators are not required to be symmetrical (a theorem of Chisholm and Everitt (Proc. Roy. Soc. Edinburgh Sect. A69(14) (1970/1971), 199–204) on non-self-adjoint operators is derived in this way). They may even act between differentL2-spaces. Section 2 is a rather substantial study of how this method yields the exact value of the norm of a particular operator between differentL2-spaces which arises naturally in Wiener–Hopf theory and which has several puzzling features
On Inverse Scattering for theN-Body Schrödinger Equation

AbstractWe consider the generalizedN-body Schrödinger operatorH=−Δ+q(x),q(x)=∑a∈Ava(xa),x∈Rd,xa∈Xa,Rd=Xa⊕Xa,with short-range regular interactions. We obtain, in particular, the following results. (1)We give new formulas and proofs relating high energy asymptotics (in a weak sense) of the scattering operatorsŜβα(with the same cluster decompositionaforαandβ) and theX-ray transformPIβα(defined on the set of all lines inXawith nonsingular directions) of the effective potentialsIβα(xa),xa∈Xa. These results significantly clarify some of those given in the literature. (2)We describe completely KerPand give a method for reconstruction ofIβα(mod KerP) fromPIβα. (3)We prove pointwise high energy asymptotics for the two-cluster-two-cluster scattering amplitudesfβαwith the Fourier transformÎβαin the leading term (for the case when the cluster decomposition forαandβis the same). (4)We give several additional results for the case (of perturbed stratified medium) whenq(x)=va(xa)+vb(xb),xa=x1,xb=(x1,…,xd)=xand eachvc(xc) rapidly decreases as |xc|→∞,c=a,b. Our results (including proofs) in some cases significantly simplify methods of high energy inverse scattering for theN-body Schrödinger operator given earlier by Wang and by Enss and Wede
Strongly Harmonic Forms for Representations in the Discrete Series

AbstractLetGbe a semisimple connected Lie group and letKbe a maximal compact subgroup. Assume that rankG=rankK, and letT⊂Kbe a Cartan subgroup ofG. The quotientG/Tcarries an indefiniteG-invariant hermitian form. The standard ∂ Dolbeault operator has a formal adjoint differential operator ∂*invwith respect to the invariant hermitian form. Letsdenote the complex dimension ofK/T. We form the indefinite harmonic space Hs(G/T,Lχ+2ρ)={(0,s)+Lχ+2ρ×valued forms in Ker∂∩Ker∂*inv}. In this paper we show that under some positivity conditions onχthe cohomology space Hs(G/T,Lχ) contains a copy of the representation in the discrete series ofGwith parameterχ