Luciano Abadias, José E. Galé, Carlos Lizama
We characterize the solutions of the Poisson equation and the domain of its associated one-sided Hilbert transform for Cesàro bounded operators of fractional order. The results obtained fairly generalize the corresponding ones for power-bounded operators. In passing, we give an extension of the mean ergodic theorem. Examples are given to illustrate the theory.
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A more general notion of weight called admissible is introduced and then an investigation is carried out on the a.e. convergence of weighted strong laws of large numbers and their applications to weighted one-sided ergodic Hilbert transforms on vector-valued $L_p$-spaces. Furthermore, the obtained results are applied to the a.e. convergence of random modulated weighted ergodic series, which is also new in the classical setting.
For operators on Hilbert spaces of any dimension, we show that equivalence after extension coincides with equivalence after one-sided extension, thus obtaining a proof of their coincidence with Schur coupling. We also provide a concrete description of this equivalence relation in several cases, in particular for compact operators.
Uniform ergodicity and the one-sided ergodic Hilbert transform
本文证明了Banach空间上有界线性算子的一致遍历性等价于单侧遍历Hilbert变换的收敛性，研究了旋转一致遍历性，并指出无穷维空间上的正等距算子绝不一致遍历。
One-sided Fractional Integrals and Riesz Potentials on a Spherical Cap
本文在球冠上定义分数次积分，研究其加权Lp有界性并给出显式反演公式，进而解决Riesz位势的反演问题，主要工具包括球极投影、Marchaud型超奇异积分等。
The theory of one-sided $M$-ideals and multipliers of operator spaces is simultaneously a generalization of classical $M$-ideals, ideals in operator algebras, and aspects of the theory of Hilbert $C^*$-modules and their maps. Here we give a systematic exposition of this theory; a reference tool for `noncommutative functional analysts' who may encounter a one-sided $M$-ideal or multiplier in their work.
Functional Gaussian approximations on Hilbert-Poisson spaces
本文在 Hilbert-Poisson 空间上建立函数型 Stein-Malliavin 方法，结合无穷维可交换对与 Gamma 演算，给出高斯随机元逼近的定量中心极限定理和四阶矩界，并应用于 Poisson 过程的 Brown 逼近及随机几何图边计数统计量。
Vector-Carleson, Calderón, and Poisson-Atomic Criteria for Generalized Hilbert Operators on $H^p$, $p>2$
该研究针对p>2时广义希尔伯特算子在H^p上的有界性，建立了向量卡尔松等判据，构造了特殊符号并证明泊松原子检验定理，给出原子复杂度公式与检验界。
General results of interpolation (eg. Nevanlinna-Pick) by elements in the noncommutative analytic Toeplitz algebra $F^\infty$ (resp. noncommutative disc algebra $A_n$) with consequences to the interpolation by bounded operator-valued analytic functions in the unit ball of ${\bf C}^n$ are obtained. Non-commutative Poisson transforms are used to provide new von Neumann type inequalities. Completely isometric representations of the quotient algebra $F^\infty/J$ on Hilbert spaces, where $J$ is any $w^*$-closed, 2-sided ideal of $F^\infty$, are obtained and used to construct a $w^*$-continuous, $F^…
Bounded Composition Operators on Hilbert Space of Complex-Valued Harmonic Functions
本文研究复值调和函数希尔伯特空间上的复合算子，刻画了其有界性、等距性与生成自映射类型，并建立其与再生核的关系。
In this paper we establish the semi-Fredholm theory on Hilbert C*-modules as a continuation of Fredholm theory on Hilbert C*-modules established by Mishchenko and Fomenko. We give a definition of a semi-Fredholm operator on Hilbert C*-module and prove that these semi-Fredholm operators are those that are one-sided invertible modulo compact operators, that the set of proper semi-Fredholm operators is open and many other results that generalize their classical counterparts.
Construction of Reproducing Kernel Hilbert Spaces Associated with the Laplace--Bessel Operator
该研究受贝塞尔算子偏微分方程问题启发，引入新的再生核希尔伯特空间，此空间是分数加权索伯列夫空间且有正定核，还获得奇异泊松方程的简单实用近似解。
A stochastic differential equation with coefficients defined in a scale of Hilbert spaces is considered. The existence, uniqueness and path-continuity of infinite-time solutions is proved by an extension of the Ovsyannikov method. This result is applied to a system of equations describing non-equilibrium stochastic dynamics of (real-valued) spins of an infinite particle system on a typical realization of a Poisson or Gibbs point process in ${\mathbb{R}}^{n}$. The paper improves the results of the work by the second named author "Stochastic differential equations in a scale of Hilbert spaces", …
We define Schatten classes of adjointable operators on Hilbert modules over abelian $C^*$-algebras. Many key features carry over from the Hilbert space case. In particular, the Schatten classes form two-sided ideals of compact operators and are equipped with a Banach norm and a $C^*$-valued trace with the expected properties. For trivial Hilbert bundles, we show that our Schatten-class operators can be identified bijectively with Schatten-norm-continuous maps from the base space into the Schatten classes on the Hilbert space fiber, with the fiberwise trace. As applications, we introduce the $C…
The generalized spectral theory is an effective approach to analyze a linear operator on a Hilbert space $\mathcal{H}$ with a continuous spectrum. The generalized spectrum is computed via analytic continuations of the resolvent operators using a dense locally convex subspace $X$ of $\mathcal{H}$ and its dual space $X'$. The three topological spaces $X \subset \mathcal{H} \subset X'$ is called the rigged Hilbert space or the Gelfand triplet. In this paper, the generalized spectra of the Perron-Frobenius operators of the one-sided and two-sided shifts of finite types (symbolic dynamical systems)…
Operator orbit frames and frame-like Fourier expansions
本文研究了由算子轨道生成的 Hilbert 空间框架，提出了不满射算子下框架与算子的显式构造理论，证明了此类框架均源于 Kaczmarz 算法，并对具有类框架 Fourier 展开的测度进行了分类。
Convergence of layer potentials and Riemann-Hilbert problem on extension domains
本文证明扩展域上层面势算子收敛，进而推广Cauchy积分与Hilbert变换，方法基于二进逼近与多种收敛概念。
Parameter--elliptic pseudodifferential operators given on a closed smooth manifold are investigated on the extended Sobolev scale. This scale consists of all Hilbert spaces that are interpolation spaces with respect to the Hilbert Sobolev scale. We prove that these operators set isomorphisms between appropriate spaces of the scale provided the parameter is modulo large enough. For solutions to the corresponding parameter--elliptic equations, we establish two-sided a priori estimates, in which the constants are independent of the parameter.
A stochastic differential equation with coefficients defined in a scale of Hilbert spaces is considered. The existence and uniqueness of finite time solutions is proved by an extension of the Ovsyannikov method. This result is applied to a system of equations describing non-equilibrium stochastic dynamics of (real-valued) spins of an infinite particle system on a typical realization of a Poisson or Gibbs point process in a Euclidean space.
A result of Gilbert shows that every completely bounded multiplier $f$ of the Fourier algebra $A(G)$ arises from a pair of bounded continuous maps $α,β:G \rightarrow K$, where $K$ is a Hilbert space, and $f(s^{-1}t) = (β(t)|α(s))$ for all $s,t\in G$. We recast this in terms of adjointable operators acting between certain Hilbert C$^*$-modules, and show that an analogous construction works for completely bounded left multipliers of a locally compact quantum group. We find various ways to deal with right multipliers: one of these involves looking at the opposite quantum group, and this leads to …
We show that a combination of well-known operators, namely $\Iτ\circ{H}\circ\Ps$ is self-adjoint and {\em ad-hoc} related to the $ζ$ function. Here $τ$ is an involution appearing in Weil's positivity criteria needed for hermitrization, $H$ a regularization operator introduced by Connes \cite{Co2} and $\Ps$ essentially Poisson summation. We elaborate on the Hilbert-Pólya conjecture, discuss why the Hermite-Biehler theorem, uncertainty relations and cohomologies are interesting in our scenario.
If the Vlasov-Poisson or Einstein-Vlasov system is linearized about an isotropic steady state, a linear operator arises the properties of which are relevant in the linear as well as nonlinear stability analysis of the given steady state. We prove that when defined on a suitable Hilbert space and equipped with the proper domain of definition this transport operator $\mathcal{T}$ is skew-adjoint, i.e., $\mathcal{T}^{\ast}=-\mathcal{T}$. In the Vlasov-Poisson case we also determine the kernel of this operator.
Peripheral Poisson Boundary on Full Fock space
本文针对可分 Hilbert 空间上的全 Fock 空间，研究左生成算子诱导的完全正映射的外围 Poisson 边界，并考察其与 Poisson 边界的关系。
We associate to an operator valued completely positive linear map $φ$ on a $C^{\ast }$-algebra $A$ and a Hilbert $C^{\ast }$-module $X$ over $A$ a subset $X_{φ}$ of $X,$ called '\textit{ternary domain}' of $φ$ on $X,$ which is a Hilbert $C^{\ast }$-module over the multiplicative domain of $φ$ and every $φ$-map (i.e., associated quaternary map with $φ$) acts on it as a ternary map. We also provide several characterizations for this set. The ternary domain \ of $φ$ on $A\ $ is a closed two-sided $\ast $-ideal $T_{φ}$ of the multiplicative domain of $φ$. We show that $XT_{φ}=X_{φ}$ and give sever…
The two-weight inequality for the Hilbert transform is characterized for an arbitrary pair of positive Radon measures $σ$ and $w$ on $\mathbb R$. In particular, the possibility of common point masses is allowed, lifting a restriction from the recent solution of the two-weight problem by Lacey, Sawyer, Shen and Uriarte-Tuero. Our characterization is in terms of Sawyer-type testing conditions and a variant of the two-weight $A_2$ condition, where $σ$ and $w$ are integrated over complementary intervals only. A key novely of the proof is a two-weight inequality for the Poisson integral with `holes…
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