Multiplicative order continuous operators on Riesz algebras

一般黎曼流形上 Riesz 变换类型的乘法不等式

A multiplicative inequality of Riesz transform type on general Riemannian manifolds

本文在任意完备黎曼流形上证明了一个与维度无关的 Riesz 变换类型乘法不等式，并将结果推广至 sub-Markov 半群生成子的抽象设定。

Strongly order continuous operators on Riesz spaces

In this paper we introduce two new classes of operators that we call strongly order continuous and strongly $σ$-order continuous operators. An operator $T:E\rightarrow F$ between two Riesz spaces is said to be strongly order continuous (resp. strongly $σ$-order continuous), if $x _α\xrightarrow{uo}0$ (resp. $x _n \xrightarrow{uo}0$) in $E$ implies $Tx _α\xrightarrow{o}0$ (resp. $Tx _n \xrightarrow{o}0$) in $F$. We give some conditions under which order continuity will be equivalent to strongly order continuity of operators on Riesz spaces. We show that the collection of all $so$-continuous lin…

Continuous Operators with Convergence in Lattice-Normed Locally Solid Riesz Spaces

A linear operator $T$ between two lattice-normed locally solid Riesz spaces is said to be $p_τ$-continuous if, for any $p_τ$-null net $(x_α)$, the net $(Tx_α)$ is $p_τ$-null, and $T$ is also said to be $p_τ$-bounded operator if it sends $p_τ$-bounded subsets to $p_τ$-bounded subsets. They are generalize several known classes of operators such as continuous, order continuous, $p$-continuous, order bounded, $p$-bounded operators, etc. We also study $up_τ$-continuous operators between lattice-normed locally solids Riesz spaces.

Order continuous operators on pre-Riesz spaces and embeddings

We investigate properties of order continuous operators on pre-Riesz spaces with respect to the embedding of the range space into a vector lattice cover or, in particular, into its Dedekind completion. We show that order continuity is preserved under this embedding for positive operators, but not in general. For the vector lattice $\ell_0^\infty$ of eventually constant sequences, we consider the pre-Riesz space of regular operators on $\ell_0^\infty$ and show that making the range space Dedekind complete does not provide a vector lattice cover of the pre-Riesz space. A similar counterexample i…

Full Lattice Convergence on Riesz Spaces

The full lattice convergence on a locally solid Riesz space is an abstraction of the topological, order, and relatively uniform convergences. We investigate four modifications of a full convergence $\mathbb{c}$ on a Riesz space. The first one produces a sequential convergence $\mathbb{sc}$. The second makes an absolute $\mathbb{c}$-convergence and generalizes the absolute weak convergence. The third modification makes an unbounded $\mathbb{c}$-convergence and generalizes various unbounded convergences recently studied in the literature. The last one is applicable whenever $\mathbb{c}$ is a ful…

Stone-Weierstraß Theorems for Riesz Ideals of Continuous Functions

Notions of convergence and continuity specifically adapted to Riesz ideals I of the space of continuous real-valued functions on a Lindelöf locally compact Hausdorff space are given, and used to prove Stone-Weierstraß-type theorems for I. As applications, sufficient conditions are discussed that guarantee that various types of positive linear maps on I are uniquely determined by their restriction to various point-separating subsets of I. A very special case of this is the characterization of the strong determinacy of moment problems, which is rederived here in a rather general setting and with…

Domination properties and extension of positive compact operators on pre-Riesz spaces

This paper concerns positive domination property of compact operators on pre-Riesz spaces. The method is embedding the pre-Riesz space to the Riesz completion. It extends the order continuous norms in pre-Riesz spaces to Riesz completions. The compactness of third power of a positive operator is obtained in a pre-Riesz space which has an order unit.

序连续正交可加算子的序有界性

Order Boundedness of Order Continuous Orthogonally Additive Operators

该研究证明适当假设下序连续正交可加算子空间是Riesz空间，否定了Mykhaylyuk等人的开放问题，建立了序有界性结果并刻画了抽象Uryson算子的判定条件。

希尔伯特空间中的连续双框架

Continuous biframes in Hilbert Spaces

该研究提出希尔伯特空间中连续双框架的概念，探究其以双框架算子为核心的基本性质，并引入新型连续双框架-Riesz基，丰富了希尔伯特空间的框架理论。

On the relationship between three classes of operators on Riesz spaces

Following several papers in the prior literature, we study the relationship between order bounded operators, topologically bounded operators and topologically continuous operators. Our main contribution is two folded: (i) we provide a set of counterexamples to illustrate several extant results in the literature; (ii) we give conditions for the space of order bounded operators to coincide with the space of topologically bounded operators as well as conditions for these two spaces to coincide with the space of topologically continuous operators.

Riesz空间之间算子连续性的一个一般框架

A General Framework for Operator Continuity between Riesz Spaces

本文提出一个统一框架，研究Riesz空间间基于广义序收敛的$\mathcal{FG}$-序连续算子，证明其有序有界性、模保持性、序连续性与带刻画，并给出正算子的扩张定理。

Riesz basis for strongly continuous groups

Given a Hilbert space and the generator of a strongly continuous group on this Hilbert space. If the eigenvalues of the generator have a uniform gap, and if the span of the corresponding eigenvectors is dense, then these eigenvectors form a Riesz basis (or unconditional basis) of the Hilbert space. Furthermore, we show that none of the conditions can be weakened. However, if the eigenvalues (counted with multiplicity) can be grouped into subsets of at most $K$ elements, and the distance between the groups is (uniformly) bounded away from zero, then the spectral projections associated to the gr…

On the Riesz Transforms for the inverse Gauss measure

Let $γ_{-1}$ be the absolutely continuous measure on $\mathbb{R}^n$ whose density is the reciprocal of a Gaussian function. Let further $\mathscr{A}$ be the natural self-adjoint Laplacian on $L^2(γ_{-1})$. In this paper, we prove that the Riesz transforms associated with $\mathscr{A}$ of order one or two are of weak type $(1,1)$, but that those of higher order are not.

Riesz-like bases in rigged Hilbert spaces

The notions of Bessel sequence, Riesz-Fischer sequence and Riesz basis are generalized to a rigged Hilbert space $\D[t] \subset \H \subset \D^\times[t^\times]$. A Riesz-like basis, in particular, is obtained by considering a sequence $\{ξ_n\}\subset \D$ which is mapped by a one-to-one continuous operator $T:\D[t]\to\H[\|\cdot\|]$ into an orthonormal basis of the central Hilbert space $\H$ of the triplet. The operator $T$ is, in general, an unbounded operator in $\H$. If $T$ has a bounded inverse then the rigged Hilbert space is shown to be equivalent to a triplet of Hilbert spaces.

On the diagonal of Riesz operators on Banach lattices

This paper extends the well-known Ringrose theory for compact operators to polynomially Riesz operators on Banach spaces. The particular case of an ideal-triangularizable Riesz operator on an order continuous Banach lattice yields that the spectrum of such operator lies on its diagonal, which motivates the systematic study of an abstract diagonal of a regular operator on an order complete vector lattice $E$. We prove that the class $\mathscr D$ of regular operators for which the diagonal coincides with the atomic diagonal is always a band in $\mathcal L_r(E)$, which contains the band of abstra…

关于 Hilbert C*-模中连续框架的扰动

On perturbation of continuous frames in Hilbert C*-modules

本文将 Hilbert 空间框架的 Casazza-Christensen 扰动定理推广至 Hilbert C*-模中的连续框架，得到了 Riesz 型框架扰动保持不变的必要条件，并探讨了算子框架接近性对连续 Bessel 映射成为连续框架的影响。

The statistically unbounded $τ$-convergence on locally solid Riesz spaces

A sequence $(x_n)$ in a locally solid Riesz space $(E,τ)$ is said to be statistically unbounded $τ$-convergent to $x\in E$ if, for every zero neighborhood $U$, $\frac{1}{n}\big\lvert\{k\leq n:\lvert x_k-x\rvert\wedge u\notin U\}\big\rvert\to 0$ as $n\to\infty$. In this paper, we introduce this concept and give the notions $st$-$u_τ$-closed subset, $st$-$u_τ$-Cauchy sequence, $st$-$u_τ$-continuous and $st$-$u_τ$-complete locally solid vector lattice. Also, we give some relations between the order convergence and the $st$-$u_τ$-convergence.

Characterization of Riesz spaces with topologically full center

Let $E$ be a Riesz space and let $E^{\sim}$ denote its order dual. The orthomorphisms $Orth(E)$ on $E,$ and the ideal center $Z(E)$ of $E,$ are naturally embedded in $Orth(E^{\sim})$ and $Z(E^{\sim})$ respectively. We construct two unital algebra and order continuous Riesz homomorphisms \[ γ:((Orth(E))^{\sim})_{n}^{\sim}\rightarrow Orth(E^{\sim})\text{ }% \] and \[ m:Z(E)^{\prime\prime}\rightarrow Z(E^{\sim}) \] that extend the above mentioned natural inclusions respectively. Then, the range of $γ$ is an order ideal in $Orth(E^{\sim})$ if and only if $m$ is surjective. Furthermore, $m$ is surj…

Statistical unbounded convergence in Banach lattices

Several recent papers investigated unbounded and statistical versions of order convergence and topology convergence in locally solid Riesz space. In this papers, we study the statistical unbounded order and topology convergence in Riesz spaces and Banach lattices. In particular, we study the relationship of those convergence and characterize order continuous, KB, reflexive Banach lattices in terms of these convergence.

二进 Riesz 向量 II

The dyadic Riesz vector II

本文为 Riesz 向量构造二进模型算子，证明其有界性蕴含 Riesz 向量的有界性，且对 Banach 空间值函数成立，常数最优为 1。

范数下的加法和乘法映射在连续函数代数正锥中的研究

Additive and multiplicative maps in norm on the positive cone of continuous function algebras

本文研究了在连续函数代数正锥中满足范数加法和乘法条件的映射，证明此类映射为由同胚诱导的复合算子。

关于向量格和Banach格中正规算子的Riesz表示定理

Riesz representation theorems for vector lattices and Banach lattices of regular operators

本文研究了从C_c(X)或C_0(X)到向量格E的正规有界算子格与E值正规Borel测度格的同构关系，探讨了Banach格中的等距同构性质及正规算子的有界性。

齐次树上流拉普拉斯算子的 Riesz 变换

Riesz transform for a flow Laplacian on homogeneous trees

本文证明了齐次树上带典范流测度的流拉普拉斯算子所关联的一阶 Riesz 变换在 $p\in(1,\infty)$ 时的 $L^p$ 有界性，并研究了水平 Riesz 变换及其伴随算子的弱 $(1,1)$ 型有界性。

Multiplicative order convergence in $f$-algebras

A net $(x_α)$ in an $f$-algebra $E$ is said to be multiplicative order convergent to $x\in E$ if $\x_α-x\u\oc 0$ for all $u\in E_+$. In this paper, we introduce the notions $mo$-convergence, $mo$-Cauchy, $mo$-complete, $mo$-continuous and $mo$-KB-space. Moreover, we study the basic properties of these notions.

Multiplicative bijections of semigroups of interval-valued continuous functions

We characterize all compact and Hausdorff spaces $X$ which satisfy that for every multiplicative bijection $ϕ$ on $C(X, I)$, there exist a homeomorphism $μ: X \to X$ and a continuous map $p: X \to (0, +\infty)$ such that $$ϕ(f) (x) = f(μ(x))^{p(x)}$$ for every $f \in C(X,I)$ and $x \in X$. This allows us to disprove a conjecture of Marovt (Proc. Amer. Math. Soc. {\bf 134} (2006), 1065-1075). Some related results on other semigroups of functions are also given.

A general Fubini theorem for the Riesz paradigm

We prove an abstract Fubini-type theorem in the context of monoidal and enriched category theory, and as a corollary we establish a Fubini theorem for integrals on arbitrary convergence spaces that generalizes (and entails) the classical Fubini theorem for Radon measures on compact Hausdorff spaces. Given a symmetric monoidal closed adjunction satisfying certain hypotheses, we show that an associated monad of natural distributions D is commutative. Applying this result to the monoidal adjunction between convergence spaces and convergence vector spaces, the commutativity of D amounts to a Fubin…

正线性算子的Riesz表示定理

Riesz representation theorems for positive linear operators

本文将经典Riesz表示定理从正线性泛函推广到取值于偏序向量空间的正线性算子，构造了取值于扩展正锥的Borel表示测度并给出显式公式，且不要求目标空间为向量格或赋范空间。

On the spectrum of the multiplicative Hilbert matrix

We study the multiplicative Hilbert matrix, i.e. the infinite matrix with entries $(\sqrt{mn}\log(mn))^{-1}$ for $m,n\geq2$. This matrix was recently introduced within the context of the theory of Dirichlet series, and it was shown that the multiplicative Hilbert matrix has no eigenvalues and that its continuous spectrum coincides with $[0,π]$. Here we prove that the multiplicative Hilbert matrix has no singular continuous spectrum and that its absolutely continuous spectrum has multiplicity one. Our argument relies on the tools of spectral perturbation theory and scattering theory. Finding an…

Some multiplicative preservers on $BH)$

In this paper we describe the form of those continuous multiplicative maps on B(H) (H being a separable complex Hilbert space of dimension not less than 3) which preserve the rank, or the corank. Furthermore, we characterize those continuous *-semigroup endomorphisms of B(H) which are spectrum non-increasing.

Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of local boundary conditions

On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that the Atiyah-Singer Dirac operator $\mathrm{D}_{\mathcal B}$ in $\mathrm{L}^{2}$ depends Riesz continuously on $\mathrm{L}^{\infty}$ perturbations of local boundary conditions ${\mathcal B}$. The Lipschitz bound for the map ${\mathcal B} \to {\mathrm{D}}_{\mathcal B}(1 + {\mathrm{D}}_{\mathcal B}^2)^{-\frac{1}{2}}$ depends on Lipschitz smoothness and ellipticity of ${\mathcal B}$ and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. More generally, we …

里斯空间中强单调算子方程的解

Solutions for Strongly Monotone Operator Equations in Riesz Spaces

研究实里斯空间中强单调算子方程的解，利用巴拿赫格内在格结构建立非线性项增长假设，结合下降流不变集技术与强单调算子理论得抽象存在定理，并应用于\((p,q)\)-拉普拉斯边值问题。

Algebraic structures spanned by differential-like operators on toy Fock spaces

We study multiplicative systems of linear mappings acting on the toy Fock space, a.k.a.\ Rademacher chaos or Walsh-Fourier series, related to the creation, annihilation, and conservation operators in quantum probability. Like differential operators they entail analogs of the Leibnitz Formula and the Chain Rule, derived with the help of Riesz products (or discrete coherent vectors). Two symmetries among these operators entail groups and algebras of varying signatures and mixed commutativity, generalizing Pauli spin matrices and quaternions. In particular, anticommutative Rademacher systems are …

Riesz summability on boundary lines of holomorphic functions generated by Dirichlet series

A particular consequence of the famous Carleson-Hunt theorem is that the Taylor series expansions of bounded holomorphic functions on the open unit disk converge almost everywhere on the boundary, whereas on single points the convergence may fail. In contrast, Bayart, Konyagin, and Queffélec constructed an example of an ordinary Dirichlet series $\sum a_n n^{-s}$, which on the open right half-plane $[Re >0]$ converges pointwise to a bounded, holomorphic function -- but diverges at each point of the imaginary line, although its limit function extends continuously to the closed right half plane.…

里斯空间中的抽象不定问题及其应用

Abstract Indefinite Problems in Riesz Spaces with Its Applications

研究希尔伯特空间\(X\)（连续嵌入巴拿赫格\(E\)）上泛函临界点存在性，通过构造格分解获空间分解，结合下降流不变集法与莫尔斯理论得抽象不定问题多临界点存在性，最后应用于椭圆边值问题。

The multiplicative Hilbert matrix

It is observed that the infinite matrix with entries $(\sqrt{mn}\log (mn))^{-1}$ for $m, n\ge 2$ appears as the matrix of the integral operator $\mathbf{H}f(s):=\int_{1/2}^{+\infty}f(w)(ζ(w+s)-1)dw$ with respect to the basis $(n^{-s})_{n\ge 2}$; here $ζ(s)$ is the Riemann zeta function and $H$ is defined on the Hilbert space ${\mathcal H}^2_0$ of Dirichlet series vanishing at $+\infty$ and with square-summable coefficients. This infinite matrix defines a multiplicative Hankel operator according to Helson's terminology or, alternatively, it can be viewed as a bona fide (small) Hankel operator o…

An application of the multiplicative Sewing Lemma to the high order weak approximation of stochastic differential equations

We introduce a variant of the multiplicative Sewing Lemma in [Gerasimovičs, Hocquet, Nilssen; J. Funct. Anal. 281 (2021)] which yields arbitrary high order weak approximations to stochastic differential equations, extending the cubature approximation on Wiener space introduced by Lyons and Victoir. Our analysis allows to derive stability estimates and explicit weak convergence rates. As a particular example, a cubature approximation for stochastic differential equations driven by continuous Gaussian martingales is given.

A Characterization of the Vector Lattice of Measurable Functions

Given a probability measure space $(X,Σ,μ)$, it is well known that the Riesz space $L^0(μ)$ of equivalence classes of measurable functions $f: X \to \mathbf{R}$ is universally complete and the constant function $\mathbf{1}$ is a weak order unit. Moreover, the linear functional $L^\infty(μ)\to \mathbf{R}$ defined by $f \mapsto \int f\,\mathrm{d}μ$ is strictly positive and order continuous. Here we show, in particular, that the converse holds true, i.e., any universally complete Riesz space $E$ with a weak order unit $e>0$ which admits a strictly positive order continuous linear functional on th…

On semibounded Wiener-Hopf operators

We show that a semibounded Wiener-Hopf quadratic form is closable in the space $L^2({\Bbb R}_{+})$ if and only if its integral kernel is the Fourier transform of an absolutely continuous measure. This allows us to define semibounded Wiener-Hopf operators and their symbols under minimal assumptions on their integral kernels. Our proof relies on a continuous analogue of the Riesz Brothers theorem obtained in the paper.