M. Eshaghi Gordji, F. Habibian, A. Rejali
The aim of this work is to generalize Johnson's techniques in order to apply them to establish a bijective correspondence between $S$-derivations and continuous derivations on $M_a(S,ω),$ where $S$ is a locally compact foundation semigroup with identity $e$, and $ω$ is a weight function on $S$, and apply it to find a necessary condition for amenability of weighted group algebras.
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In this short note, we give some new results on continuous bounded cohomology groups of topological semigroups with values in complex field. We show that the second continuous bounded cohomology group of a compact metrizable semigroup, is a Banach space. Also, we study cohomology groups of amenable topological semigroups, and we show that cohomology groups of rank greater than one of a compact left or right amenable semigroup, are trivial. Also, we give some examples and applications about topological lattices.
The main objective of this paper is to determine the simplicial and cyclic cohomology groups of the Cuntz semigroup algebra $\ell^1(\Cuntz)$. In order to do so, we first establish some general results which can be used when studying simplicial and cyclic cohomology of Banach algebras in general. We then turn our attention to $\ell^1(\Cuntz)$, showing that the cyclic cohomology groups of degree $n$ vanish when $n$ is odd and are one-dimensional when $n$ is even ($n\ge 2$). Using the Connes-Tzygan exact sequence, these results are used to show that the simplicial cohomology groups of degree $n$ …
We establish simplicial triviality of the convolution algebra $\ell^1(S)$, where $S$ is a band semigroup. This generalizes results of the first author [Glasgow Math. J. 2005, Houston J. Math. 2010]. To do so, we show that the cyclic cohomology of this algebra vanishes in all odd degrees, and is isomorphic in even degrees to the space of continuous traces on $\ell^1(S)$. Crucial to our approach is the use of the structure semilattice of $S$, and the associated grading of $S$, together with an inductive normalization procedure in cyclic cohomology; the latter technique appears to be new, and its…
This paper considers discrete and continuous semigroups of (weighted) composition operators on the Fock space. For discrete semigroups consisting of powers of a single operator, the asymptotic behaviour of the semigroups is analysed. For continuous semigroups and groups, a full classification of possible semigroups is given, and the generator is calculated.
We study the relation between module and Hochschild cohomology groups of Banach algebras with a compatible module structure. More precisely, we show that for every commutative Banach $ \mathcal{A} $-$ \mathfrak{A}$-bimodule $ X $ and every $ k \in \mathbb{N}$, the seminormed spaces $ \mathcal{H}^{k}_{\mathfrak{A}} (\mathcal{A},X^*)$ and $ \mathcal{H}^k (\frac{\mathcal{A}}{J}, X^*) $ are isomorphic, where $ J $ is the closed ideal of $ \mathcal{A} $ generated by the elements of the form $ a (α\cdot b)-(a\cdot α)b$ with $ a,b \in \mathcal{A} $ and $α\in \mathfrak{A}. $ As an example, we calculat…
Stability of weighted composition strongly continuous semigroups acting on Lebesgue and Sobolev spaces is studied, without the use of spectral conditions on the generator of the semigroup. Applications to the generalized von Foerster - Lasota semigroup and a comparison with hypercylicity conditions are presented.\
M. Embry and A. Lambert initiated the study of a weighted translation semigroup $\{S_t\}$ in ${\cal B}(L^2({\mathbb R_+})),$ with a view to explore a continuous analogue of a weighted shift operator. We continued the work, characterized some special types of semigroups and developed an analytic model for the left invertible weighted translation semigroup. The present paper deals with the generalization of the weighted translation semigroup in multi-variable set up. We develop the toral analogue of the analytic model and also describe the spectral picture. We provide many examples of weighted t…
It is known that every countable semigroup admits a weight w for which the semigroup algebra l_1(S,w) is Arens regular and no uncountable group admits such a weight; see [4]. In this paper, among other things, we show that for a large class of semigroups, the Arens regularity of the weighted semigroup algebra l_1(S,w) implies the countability of S.
The weighted shift operators turn out to be extremely useful in supplying interesting examples of operators on Hilbert spaces. With a view to study a continuous analogue of weighted shifts, M. Embry and A. Lambert initiated the study of a semigroup of operators $\{S_t\}$ indexed by a non-negative real number $t$ and termed it as weighted translation semigroup. The operators $S_t$ are defined on $L^2(\mathbb R_+)$ by using a weight function. In this paper, we continue the work carried out there and obtain characterizations of hyperexpansive weighted translation semigroups in terms of their symb…
Frequent hypercyclicity for translation $C_0$-semigroups on weighted spaces of continuous functions is investigated. The results are achieved by establishing an analogy between frequent hypercyclicity for the translation semigroup and for weighted pseudo-shifts and by characterizing frequent hypercyclic weighted pseudo-shifts in spaces of vanishing sequences. Frequent hypercylic translation semigroups in weighted $L^p$-spaces are also characterized.
We generalize a result for the translation $C_0$-semigroup on $L^p(\R_+,μ)$ about the equivalence of being chaotic and satisfying the Frequent Hypercyclicity criterion due to Mangino and Peris to certain weighted composition $C_0$-semigroups. Such $C_0$-semigroups appear in a natural way when dealing with initial value problems for linear first order partial differential operators. We apply our result to the linear von Foerster-Lasota equation arising in mathematical biology. Weighted composition $C_0$-semigroups on Sobolev spaces are also considered.
Spectra of generators of hyperbolic composition and weighted composition semigroups
本文研究了由超几何半流在贝格曼空间上诱导的加权复合$C_0$-半群生成元的谱、本质谱和点谱的完整刻画，并通过一个例子展示了半群在非固定自接触点附近的行为如何影响谱性质。
M. Embry and A. Lambert initiated the study of a semigroup of operators $\{S_t\}$ indexed by a non-negative real number $t$ and termed it as weighted translation semigroup. The operators $S_t$ are defined on $L^2(\mathbb R_+)$ by using a weight function. The operator $S_t$ can be thought of as a continuous analogue of a weighted shift operator. In this paper, we show that every left invertible operator $S_t$ can be modeled as a multiplication by $z$ on a reproducing kernel Hilbert space $\cal H$ of vector-valued analytic functions on a certain disc centered at the origin and the reproducing ke…
In this paper the controllabillity and admissibility of perturbed semigroup systems are studied, using tools from the theory of interpolation and Carleson measures. In addition, there are new results on the perturbation of Carleson measures and on the weighted interpolation of functions and their derivatives in Hardy spaces, which are of interest in their own right.
We give a simple characterization of chaos for weighted composition $C_0$-semigroups on $L^p_ρ(Ω)$ for an open interval $Ω\subseteq\mathbb{R}$. Moreover, we characterize chaos for these classes of $C_0$-semigroups on the closed subspace $W^{1,p}_*(Ω)$ of the Sobolev space $W^{1,p}(Ω)$ for a bounded interval $Ω\subset\mathbb{R}$. These characterizations simplify previously obtained characterization of chaos for these classes of $C_0$-semigroups.
In this paper we study the chaotic behaviour of the heat semigroup generated by the Dunkl-Laplacian on weighted $ L^p$ spaces. In the case of the heat semigroup associated to the standard Laplacian we obtain a complete picture on the spaces $ L^p(\R^n, (φ_{iρ}(x))^2 dx) $ where $ φ_{iρ} $ is the Euclidean spherical function. The behaviour is very similar to the case of the Laplace-Beltrami operator on non-compact Riemannian symmetric spaces studied by Pramanik and Sarkar.
Let $ω$ be a weight on a right cancellative semigroup $S$. Let $\|\cdot\|_ω$ be the weighted norm on the weighted discrete semigroup algebra $\ell^1(S, ω)$. In this paper, we prove that the weight $ω$ satisfies F-property if and only if the operator norm $\| \cdot \|_{ωop}$ of $\| \cdot \|_ω$ is exactly equal to another weighted norm $\| \cdot \|_{\widetildeω_1}$ [Theorem 2.5 ($iii$)]. Though its proof is elementary, the result is unexpectedly surprising. In particular, $\| \cdot \|_{1 op}$ is same as $\| \cdot \|_1$ on $\ell^1(S)$. Moreover, various examples are discussed to understand the re…
Hochschild cohomology for free semigroup algebras
本文研究自由半群代数的上同调，证明导子自动连续，并利用新方法及 Cesàro 算子等工具，得出第一 Hochschild 上同调群为零、正规上同调平凡及有限生成时高阶上同调消失的结论，且方法可推广至 Thompson 半群等。
We prove that in a large class of Banach spaces of analytic functions in the unit disc $\mathbb{D}$ an (unbounded) operator $Af=G\cdot f'+g\cdot f$ with $G,\, g$ analytic in $\mathbb{D}$ generates a $C_0$-semigroup of weighted composition operators if and only if it generates a $C_0$-semigroup. Particular instances of such spaces are the classical Hardy spaces. Our result generalizes previous results in this context and it is related to cocycles of flows of analytic functions on Banach spaces. Likewise, for a large class of non-separable Banach spaces $X $ of analytic functions in $\mathbb{D}$…
Let $S$ be a Rees semigroup, and let $\ell^1(S)$ be its convolution semigroup algebra. Using Morita equivalence we show that bounded Hochschild homology and cohomology of $\ell^1(S)$ is isomorphic to those of the underlying discrete group algebra.
In this paper we introduce the notion of weighted (weakly) almost periodic compactifcation of a semitopological semigroup and generalize this notion to corresponding notion for transformation semigroup.The inclusion relation and equality of some well known function spaces on a weighted transformation semigroup is also investigated.
Banach algebra A for which the natural embedding x into x^ of A into WAP(A)* is bounded below; that is, for some m in R with m > 0 we have ||x^|| > m ||x||, is called a WAP-algebra. Through we mainly concern with weighted measure algebra M_b(S;w); where w is a weight on a semi-topological semigroup S. We study those con- ditions under which M_b(S;w) is a WAP-algebra (respectively dual Banach algebra). In particular, M_b(S) is a WAP-algebra (respectively dual Banach algebra) if and only if wap(S) separates the points of S (respectively S is compactly cancellative semigroup). We apply our result…
In this paper, we apply the theory of algebraic cohomology to study the amenability of Thompson's group $\mathcal{F}$. We introduce the notion of unique factorization semigroup which contains Thompson's semigroup $\mathcal{S}$ and the free semigroup $\mathcal{F}_n$ on $n$ generators ($\geq2$). Let $\mathfrak{B}(\mathcal{S})$ and $\mathfrak{B}(\mathcal{F}_n)$ be the Banach algebras generated by the left regular representations of $\mathcal{S}$ and $\mathcal{F}_n$, respectively. It is proved that all derivations on $\mathfrak{B}(\mathcal{S})$ and $\mathfrak{B}(\mathcal{F}_n)$ are automatically c…
We characterize strong continuity of general operator semigroups on some Lebesgue spaces. In particular, a characterization of strong continuity of weighted composition semigroups on classical Hardy spaces and weighted Bergman spaces with regular weights is given. As applications, our result improves the results of Siskakis, A. G. \cite{AG1} and König, W. \cite{K} and answers a question of Siskakis, A. G. proposed in \cite{AG4}. We also characterize strongly continuous semigroups of weighted composition operators on weighted Bergman spaces in terms of abelian intertwiners of multiplication ope…
We study the strong continuity of weighted composition semigroups of the form $T_tf=φ_t'\left(f\circφ_t\right)$ in several spaces of analytic functions. First we give a general result on separable spaces and use it to prove that these semigroups are always strongly continuous in the Hardy and Bergman spaces. Then we focus on two non-separable family of spaces, the mixed norm and the weighted Banach spaces. We characterize the maximal subspace in which a semigroup of analytic functions induces a strongly continuous semigroup of weighted composition operators depending on its Denjoy-Wolff point,…
In this paper, among other things, we study those conditions under which the weighted semigroup algebra $\ell^1(S,ω)$ is a dual Banach algebra with respect to predual $c_0(S)$. Some useful examples, illustrating the results, are also included.
In this paper we show that the realization in $L^p(X,ν_\infty)$ of the nonsymmetric Ornstein-Uhlenbeck operator $L$ is sectorial for any $p\in(1,+\infty)$ and we provide an explicit sector of analyticity. Here $(X,μ_\infty,H_\infty)$ is an abstract Wiener space, i.e., $X$ is a separable Banach space, $μ_\infty$ is a centred non degenerate Gaussian measure on $X$ and $H_\infty$ is the associated Cameron-Martin space. Further, $ν_\infty$ is a weighted Gaussian measure, that is, $ν_\infty=e^{-U}μ_\infty$ where $U$ is a convex function which satisfies some minimal conditions. Our results strongly …
We study, to certain Banach spaces $X$, families of weighted composition operators. Notably, we show that if this family form a strongly continuous semigroup, then its infinitesimal generator ($Γ, D(Γ)$) is given by $Γf = gf+Gf^\prime$ with $D(Γ) = \{ f\in X \ | \ gf+Gf^\prime\in X \}$ where $g, G$ are holomorphic functions. Moreover, our second maim result is to study the reciprocal implication. That is if $(Γ, D(Γ))$, define like above, generate a strongly continuous semigroup, then this one is a family of weighted composition operators.
A weighted composition semigroup related to three open problems
本文研究 Hardy-Hilbert 空间上的加权复合算子半群，揭示其循环向量、谱及子空间性质，并建立相关基本问题与 RH 的等价性及 Báez-Duarte 判据的推广。
In this text we study a generalized conservation property for the heat semigroup generated by a Schrödinger operator with nonnegative potential on a weighted manifold. We establish Khasminskii's criterion for the generalized conservation property and discuss several applications.
This paper is based on three hours of lectures given by the first author in the "Focus Program on Analytic Function Spaces and their Applications" July 1 -- December 31, 2021, organized by the Fields Institute for Research in Mathematical Sciences. The goal of this paper is to give an introduction to the properties of discrete and continuous $C_0$-semigroups of (weighted) composition operators on various spaces of analytic functions.
Phase-Space Analysis of generalised Fractional Anharmonic and Ornstein-Uhlenbeck Semigroups on Weighted Modulation Spaces
本文研究了广义分数阶非谐振子及其热半群在加权调和空间中的相空间分析，推导了分数阶算子的符号估计，并证明了热半群的有界性和光滑性，同时建立了非线性热方程的全局解的存在性。
We investigate the spectral properties of a class of weighted shift semigroups associated to abstract transport equations with a Lipschitz--continuous vector field with no--reentry boundary conditions. We illustrate our results with various examples taken from collisionless kinetic theory.
We investigate the notion of Connes-amenability for dual Banach algebras, as introduced by Runde, for bidual algebras and weighted semigroup algebras. We provide some simplifications to the notion of a $σWC$-virtual diagonal, as introduced by Runde, especially in the case of the bidual of an Arens regular Banach algebra. We apply these results to discrete, weighted, weakly cancellative semigroup algebras, showing that these behave in the same way as C$^*$-algebras with regards Connes-amenability of the bidual algebra. We also show that for each one of these cancellative semigroup algebras $l^1…
Let $G$ be a locally compact group. We prove that the augmentation ideal in $L^1(G)$ is (algebraically) finitely-generated as a left ideal if and only if $G$ is finite. We then investigate weighted versions of this result, as well as a version for semigroup algebras. Weighted measure algebras are also considered. We are motivated by a recent conjecture of Dales and Żelazko, which states that a unital Banach algebra in which every maximal left ideal is finitely-generated is necessarily finite-dimensional. We prove that this conjecture holds for many of the algebras considered. Finally, we use t…
Free semigroupoid algebras and the first cohomology groups
本文研究有向图对应的自由半群胚代数与张量代数的导子，证明强连通情形的弱Dixmier逼近定理，刻画有限图第一上同调群平凡的充要条件，引入交错数探讨无限图情形，并给出非平凡上同调的例子。
In this paper we prove weighted $\ell^p$-inequalities for variation and oscillation operators defined by semigroups of operators associated with discrete Jacobi operators. Also, we establish that certain maximal operators involving sums of differences of discrete Jacobi semigroups are bounded on weighted $\ell^p$-spaces. $\ell^p$-boundedness properties for the considered operators provide information about the convergence of the semigroup of operators defining them.
Let $(T_t)_{t \geq 0}$ be a markovian (resp. submarkovian) semigroup on some $σ$-finite measure space $(Ω,μ)$. We prove that its negative generator $A$ has a bounded $H^\infty(Σ_θ)$ calculus on the weighted space $L^2(Ω,wdμ)$ as long as the weight $w : Ω\to (0,\infty)$ has finite characteristic defined by $Q^A_2(w) = \sup_{t > 0} \left\| T_t(w) T_t \left(w^{-1} \right) \right\|_{L^\infty(Ω)}$ (resp. by a variant for submarkovian semigroups). Some additional technical conditions on the semigroup have to be imposed and their validity in examples is discussed. Any angle $θ> \fracπ{2}$ is admissib…
Weighted composition semigroups on spaces of continuous functions and their subspaces
本文研究连续函数 Banach 空间及其子空间上由半流和半余循环诱导的加权复合半群，证明其在混合拓扑下可非平凡强连续，并给出强连续性、生成元及相关空间的刻画。
Cesàro-Hardy operators on $L^p[0,1]$: fine spectrum, weighted Koopman semigroups and invariant subspaces
本文研究Cesàro-Hardy算子及其泛化在$L^p[0,1]$上的有界性和精细谱性质，利用$C_0$半群理论，通过子ordination表达Cesàro-Hardy算子及其对偶算子，探讨其谱性质及不变子空间问题。
We investigate an infinite, linear system of ordinary differential equations that models the evolution of fragmenting clusters. We assume that each cluster is composed of identical units (monomers) and we allow mass to be lost, gained or conserved during each fragmentation event. By formulating the initial-value problem for the system as an abstract Cauchy problem (ACP), posed in an appropriate weighted $\ell^1$ space, and then applying perturbation results from the theory of operator semigroups, we prove the existence and uniqueness of physically relevant, classical solutions for a wide class…
Quantum Bernoulli noises (QBN, for short) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation (CAR) in equal-time. In this paper, by using QBN, we first introduce a class of self-adjoint operators acting on Bernoulli functionals, which we call the weighted number operators. We then make clear spectral decompositions of these operators, and establish their commutation relations with the annihilation as well as the creation operators. We also obtain a necessary and sufficient condition for a weighted number op…
Spectra of infinitesimal generators of composition semigroups on weighted Bergman spaces induced by doubling weights
本文研究椭圆型复合半群在加倍权加权Bergman空间上诱导的复合半群的无穷小生成元的谱，利用谱映射定理和复合算子谱刻画，并探讨最终范数连续性及紧积分算子。
Optimal semigroup estimates and functional calculus for the Laplacian on weighted Sobolev spaces
该研究推广了Lindemulder等人的成果，证明半空间上带Dirichlet和Neumann边界条件的Laplace算子在幂权Sobolev空间上的最优半群估计，及边界满足相容性条件时具有有界$H^\infty$-泛函演算，且该条件不可省略。
Complex symmetry in the Fock space of several variables
该研究借助加权复合算子刻画多变量 Fock 空间中的无界复对称算子，并据此分析复对称半群及其一阶微分生成元。
A Semigroup Approach to Weighted Weyl-Sonine Operators: Bypassing Bernstein Functions for Well-Posedness
本文建立加权Weyl-Sonine分数阶算子的泛函分析框架，通过引入遗忘权重绕过Bernstein函数限制，统一处理扩散弛豫和振荡非局部动力学，并证明相关抽象分数阶Cauchy问题的适定性。
We prove that for every group $G$ and any two sets $I,J$, the Brandt semigroup algebras $\ell(B(I,G))$ and $\ell(B(J,G))$ are Morita equivalent with respect to the Morita theory of self-induced Banach algebras introduced by Gronbaek. As applications, we show that if $G$ is an amenable group, then for a wide class of Banach $\ell(B(I,G))$-bimodules $E$, and every $n>0$, the bounded Hochschild cohomology groups $H^n(\ell(B(I,G)),E^*)$ are trivial, and also, the notion of approximate amenability is not Morita invariant.
Arens regularity of weighted convolution algebras that arise from totally ordered semilattices
本文将Dales与Strauss关于全序半格卷积代数Arens正则性的无权结果推广到加权情形，并进一步研究其Banach代数前对偶的存在性。
Let $\mathcal{E}$ be a Hermitian vector bundle over a Riemannian manifold $M$ with metric $g$, let $\nabla$ be a metric covariant derivative on $\mathcal{E}$. We study the generalized Ornstein-Uhlenbeck differential expression $P^{\nabla}=\nabla^{\dagger}\nabla u+\nabla_{(dϕ)^{\sharp}}u-\nabla_{X}u+Vu$, where $\nabla^{\dagger}$ is the formal adjoint of $\nabla$, $(dϕ)^{\sharp}$ is the vector field corresponding to $dϕ$ via $g$, $X$ is a smooth real vector field on $M$, and $V$ is a self-adjoint locally integrable section of the bundle $\textrm{End }\mathcal{E}$. We show that (the negative of) …
With appropriate notions of Hermitian vector bundles and connections over weighted graphs which we allow to be locally infinite, we prove Feynman-Kac-type representations for the corresponding semigroups and derive several applications thereof.
Composition operators on function spaces on the halfplane: spectra and semigroups
本文研究了右半平面 Zen 空间上的复合算子，推广了关于范数、本质范数、（本质）谱及半群的已有工作，并通过 Hardy--Bergman 空间实例说明结果。
Dynamics of the translation semigroup on directed metric trees
研究有向度量树上的加权L^p空间中左翻译半群的动力学，推导出强连续性的必要充分条件，并通过树结构中权重的渐近衰减特性刻画超循环性和弱混属性。
We propose a new approach to construct the eigenvalue expansion in a weighted Hilbert space of the solution to the Cauchy problem associated to Gauss-Laguerre invariant Markov semigroups that we introduce. Their generators turn out to be natural non-self-adjoint and non-local generalizations of the Laguerre differential operator. Our methods rely on intertwining relations that we establish between these semigroups and the classical Laguerre semigroup and combine with techniques based on non-harmonic analysis. As a by-product we also provide regularity properties for the semigroups as well as f…
We provide the spectral expansion in a weighted Hilbert space of a substantial class of invariant non-self-adjoint and non-local Markov operators which appear in limit theorems for positive-valued Markov processes. We show that this class is in bijection with a subset of negative definite functions and we name it the class of generalized Laguerre semigroups. Our approach, which goes beyond the framework of perturbation theory, is based on an in-depth and original analysis of an intertwining relation that we establish between this class and a self-adjoint Markov semigroup, whose spectral expans…
Weighted variational inequalities for the fractional Dunkl heat semigroup
研究由 Dunkl 拉普拉斯生成的分数阶热半群差分变换算子在加权 L^p 和 Dunkl BMO 空间上的有界性，并建立相应极大算子的点态收敛性质。
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