C. Chen, Y. Wang, Y. X. Liang
For $α>-1$, let $A_α^2(\mathbb{B}_N)$ be the weighted Bergman space on the unit ball $\mathbb{B}_N$ in $\mathbb{C}^N$. In this paper, we prove that the multiplication operator $M_{z^n}$ is quasi-similar to $\oplus_1^{\prod_{i=1}^N n_i}M_z$ on $A_α^2(\mathbb{B}_N)$ for the multi-index $n=(n_1,n_2,\cdots,n_N)$.
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On the norms of the multiplication operators between weighted Bergman spaces
本文研究乘法算子在加权伯格曼空间间的范数，推导了新的范数估计并探讨了伯恩斯坦猜想与单值函数的Schwarzian导数诱导的乘法算子范数之间的联系。
Suppose that $f := (f_1,\ldots,f_d):Ω_1\toΩ_2$ is a proper holomorphic map between two bounded domains in $\mathbb C^d.$ In this paper, we find a non-trivial minimal joint reducing subspace for the multiplication operator (tuple) $M_f=(M_{f_1},\ldots, M_{f_d})$ on the Bergman space $\mathbb A^2(Ω_1)$, say $\mathcal M.$ We further show that the restriction of $(M_{f_1},\ldots,M_{f_d})$ to $\mathcal M$ is unitarily equivalent to Bergman operator on $\mathbb A^2(Ω_2).$
Given a weight function, we define the Bergman type projection with values in the corresponding weighted Bergman space on the unit ball $\mathbb B_n$ of $\mathbb C^n, n>1$. We characterize the radial weights such that this projection is bounded from $L^\infty$ to the Bloch space $\mathcal B$.
In this paper, using the group-like property of local inverses of a finite Blaschke product $ϕ$, we will show that the largest $C^*$-algebra in the commutant of the multiplication operator $M_ϕ$ by $ϕ$ on the Bergman space is finite dimensional, and its dimension equals the number of connected components of the Riemann surface of $ϕ^{-1}\circϕ$ over the unit disk. If the order of the Blaschke product $ϕ$ is less than or equal to eight, then every $C^*$-algebra contained in the commutant of $M_ϕ$ is abelian and hence the number of minimal reducing subspaces of $M_ϕ$ equals the number of connect…
Following upon results of Putinar, Sun, Wang, Zheng and the first author, we provide models for the restrictions of the multiplication by a finite Balschke product on the Bergman space in the unit disc to its reducing subspaces. The models involve a generalization of the notion of bundle shift on the Hardy space introduced by Abrahamse and the first author to the Bergman space. We develop generalized bundle shifts on more general domains. While the characterization of the bundle shift is rather explicit, we have not been able to obtain all the earlier results appeared, in particular, the facts…
Given an n-tuple of multiplication operators on the Bergman space of a bounded pseudoconvex domain in C^n, we study the algebra of their commutants. In particular, we give a geometric description of the maximal C*-subalgebra of this algebra.
Using recent characterizations of the compactness of composition operators on Hardy-Orlicz and Bergman-Orlicz spaces on the ball, we first show that a composition operator which is compact on every Hardy-Orlicz (or Bergman-Orlicz) space has to be compact on H^{\infty}. Then, we prove that, for each Koranyi region Γ, there exists a map ϕtaking the ball into Γsuch that, C_ϕ is not compact on H^ψ\left(\mathbb{B}_{N}\right), when ψgrows fast. Finally, we give another characterization of the compactness of composition operator on weighted Bergman-Orlicz spaces in terms of an Orlicz "Angular derivat…
We study the Carleson measures and the Toeplitz operators on the class of so-called small weighted Bergman spaces, introduced recently by Seip. A characterization of Carleson measures is obtained which extends Seip's results from the unit disc of $\mathbb C$ to the unit ball of $\mathbb C^n$. We use this characterization to give necessary and sufficient conditions for the boundedness and compactness of Toeplitz operators. Finally, we study the Schatten $p$ classes membership of Toeplitz operators for $1<p<\infty$.
The function-operator convolution algebra over the Bergman space of the ball and its Gelfand theory
本文研究了球体上伯格曼空间的函数-算子卷积代数结构及其盖尔范德理论，探讨了该代数的性质及相关算子的傅里叶变换。
We provide a characterization of the commutant of analytic Toeplitz operators $T_B$ induced by finite Blachke products $B$ acting on weighted Bergman spaces which, as a particular instance, yields the case $B(z)=z^n$ on the Bergman space solved recently by by Abkar, Cao and Zhu. Moreover, it extends previous results by Cowen and Wahl in this context and applies to other Banach spaces of analytic functions such as Hardy spaces $H^p$ for $1<p<\infty$. Finally, we apply this approach to study the reducing subspaces of $T_{B}$ in weighted Bergman spaces.
We investigate which boundary points in the closed unit ball of the Bergman space of integrable holomorphic functions on the unit disc are strongly exposed. This requires study of the Bergman projection and its kernel, the annihilator of Bergman space. We show that all polynomials in the boundary of the unit ball are strongly exposed.
Let $\mathfrak{T}$ denote the full Toeplitz algebra on the Bergman space of the unit ball $\mathbb{B}_n.$ For each subset $G$ of $L^{\infty},$ let $\mathfrak{CI}(G)$ denote the closed two-sided ideal of $\mathfrak{T}$ generated by all $T_fT_g-T_gT_f$ with $f,g\in G.$ It is known that $\mathfrak{CI}(C(\bar{\mathbb{B}}_n))=\mathcal{K}$ - the ideal of compact operators and $\mathfrak{CI}(C(\mathbb{B}_n))=\mathfrak{T}.$ Despite these ``extremal cases'', $\mathfrak{T}$ does contain other non-trivial ideals. This paper gives a construction of a class of subsets $G$ of $L^{\infty}$ so that $\mathcal{…
We establish weak factorizations for a weighted Bergman space $A^p_{\a}$, with $1<p<\infty$, into two weighted Bergman spaces on the unit ball of $\C^n$. To obtain this result, we characterize bounded Hankel forms on weighted Bergman spaces on the unit ball of $\C^n$.
We give embedding theorems for weighted Bergman-Orlicz spaces on the ball and then apply our results to the study of composition operators in this context. As one of the motivations of this work, we show that there exist some weighted Bergman-Orlicz spaces, different from $H^{\infinity}$, on which every composition operator is bounded.
Recently representation theory has been used to provide atomic decompositions for a large collection of classical Banach spaces. In this paper we extend the techniques to also include projective representations. As our main application we obtain atomic decompositions of Bergman spaces on the unit ball through the holomorphic discrete series for the group of isometries of the ball.
On the density of Toeplitz operators in the Toeplitz algebra over the Bergman space of the unit ball
该研究采用量子调和分析与表示论方法，给出单位球Bergman空间上Toeplitz代数中Toeplitz算子按范数稠密的夏氏定理的新证明。
Absolutely Summing Toeplitz operators on Bergman spaces in the unit ball of $\mathbb{C}^n$
本文研究了复数n维单位球上Bergman空间中Toeplitz算子的绝对可和性，并给出了其范数等价条件，同时扩展了Carleson嵌入的相关结果。
Some new characterizations on Carleson measures for weighted Bergman spaces on the unit ball involving product of functions are obtained. For these we characterize bounded and compact Toeplitz operators between weighted Bergman spaces. The above results are applied to characterize bounded and compact extended Cesàro operators and pointwise multiplication operators. The results are new even in the case of the unit disk.
For $α>-1$ and $0<p<\infty$, we study weighted Bergman spaces $\mathcal B^p_α$ of harmonic functions on the real hyperbolic ball and obtain an atomic decomposition of these spaces in terms of reproducing kernels. We show that an $r$-separated sequence $\{a_m\}$ with sufficiently large $r$ is an interpolating sequence for $\mathcal B^p_α$. Using these we determine precisely when a Bergman space $\mathcal B^p_α$ is included in another Bergman space $\mathcal B^q_β$.
In this paper, we present maximal and area integral characterizations of Bergman spaces in the unit ball of $\mathbb{C}^n.$ The characterizations are in terms of maximal functions and area integral functions on Bergman balls involving the radial derivative, the complex gradient, and the invariant gradient. As an application, we obtain new maximal and area integral characterizations of Besov spaces. Moreover, we give an atomic decomposition of real-variable type with respect to Carleson tubes for Bergman spaces.
We obtain some estimates for norm and essential norm of the difference of two composition operators between weighted Bergman spaces $A^p_α$ and $A^q_β$ on the unit ball. In particular, we completely characterize the boundedness and compactness of $C_φ-C_ψ: A^p_α\to A^q_β$ for full range $0<p,q<\infty$, $-1<α,β<\infty$.
Modified Bergman spaces on the unit ball of $\mathbb C^n$ and applications
本文在复单位球上引入推广经典 Bergman 空间的新全纯函数空间，并研究相关算子性质及 Bergman 型投影和 Berezin 变换等积分表示。
We completely describe spaces of multipliers of certain harmonic function spaces of Bergman type in R^n.This is the first sharp result of this kind for Bergman type mixed norm spaces of harmonic functions in the unit ball of R^n
In this paper, a family of holomorphic spaces of tent type in the unit ball of $\mathbb{C}^n$ is introduced, which is closely related to maximal and area integral functions in terms of the Bergman metric. It is shown that these spaces coincide with Bergman spaces. Furthermore, Littlewood-Paley type $g$-functions for the Bergman spaces are introduced in terms of the radial derivative, the complex gradient, and the invariant gradient. The corresponding characterizations for Bergman spaces are obtained as well. As an application, we obtain new maximal and area integral characterizations for Hardy…
Difference of weighted composition operators on weighted Bergman spaces over the unit Ball
本文在单位球上刻画了加权Bergman空间到Lebesgue空间的加权复合算子差的有界性与紧致性，推广了单位圆盘结果，并给出了q-Carleson测度的新刻画。
The boundedness of $P_ω:L^\infty(\mathbb{B})\to \mathcal{B}(\mathbb{B})$ and $P_ω(P_ω^+):L^p(\mathbb{B},\upsilon dV)\to L^p(\mathbb{B},\upsilon dV)$ on the unit ball of $\mathbb{C}^n$ with $p>1$ and $ω,\upsilon\in \mathcal{D}$ are investigated in this paper.
The boundedness and compactness of Toeplitz operator from $A_ω^p$ to $A_ω^q$ with doubling weights $ω$ are studied in this paper. The characterizations of Schatten class Toeplitz operators and Volterra operators on $A_ω^2$ are also investigated.
In this paper, we show that every (weighted) Bergman space $\mathcal{A}^p_α (\mathbb{B}_n)$ in the complex ball admits an atomic decomposition of real-variable type for any $0 < p \le 1$ and $α> -1.$ More precisely, for each $f \in \mathcal{A}^p_α (\mathbb{B}_n)$ there exist a sequence of real-variable $(p, \8)_α$-atoms $a_k$ and a scalar sequence $\{λ_k \}$ with $\sum_k | λ_k |^p < \8$ such that $f = \sum_k λ_k P_α (a_k),$ where $P_α$ is the Bergman projection from $L^2_α (\mathbb{B}_n)$ onto $\mathcal{A}^2_α (\mathbb{B}_n).$ The proof is constructive, and our construction is based on some sh…
In our preprint q-alg/9703005 q-analogues of bounded symmetric domains were defined to be homogeneous spaces of the associated quantum groups. The investigation of a simplest among those domains, the quantum matrix ball, was started in math.QA/9803110. This work presents a construction of q-analogues for Hardy-Bergman spaces of 'functions in those balls', together with an explicit form of the Bergman kernel. Besides that, two auxiliary results are also established: a boundedness of matrix balls is proved, and de Rham complexes of differential forms with finite coefficients in those balls are c…
On Generators of the Hardy and the Bergman Spaces
本文通过同时对生成元与 z 的乘法不变的子空间刻画 Hardy 空间和 Bergman 空间的生成元，并研究其游荡性及有界解析函数的稠密性。
In this paper, we investigate the spectra of invertible weighted composition operators with automorphism symbols, on Hardy space $H^2(\mathbb{B}_N)$ and weighted Bergman spaces $A_α^2(\mathbb{B}_N)$, where $\mathbb{B}_N$ is the unit ball of the $N$-dimensional complex space. By taking $N=1$, $\mathbb{B}_N=\mathbb{D}$ the unit disc, we also complete the discussion about the spectrum of a weighted composition operator when it is invertible on $H^2(\mathbb{D})$ or $A_α^2(\mathbb{D})$.
The exact range of the positive real parameter $p$ is determined so that the multiplications by polynomial functions acting on Bergman spaces over complex ellipsoids are Schatten $p$-summable.
$\mathcal H$-Harmonic Bergman Projection on the Hyperbolic Ball
本文刻画了双曲球上 Bergman 投影在 Lebesgue 空间与 $\mathcal H$-调和加权 Bergman 空间之间有界的精确条件，验证了 Stoll 猜想，并给出再生核估计及到 Bloch 空间的投影结果。
While dealing with a class of generalized Bergman spaces on the unit ball, we construct for each of these spaces a set of coherent states to apply a coherent states quantization method. This provides us with another way to recover the Berezin transforms attached to these spaces. Finally, a new formula representing these transforms a functions of the Laplace-Beltrami operator is established in terms ofWilson polynomials by using the Fourier-Helgason transform.
We proceed with studying the q-analogues of Cartan domains introduced in q-alg/9703005 and turn to the case of a ball in the space of complex matrices. An explicit expression for a positive $U_q\frak{su}_{nm}$-invariant integral (see math.QA/9803110) is used to define q-analogues for weighted Bergman spaces. The orthogonal projections onto these spaces are integral operators; this work presents an explicit formula for their kernels.
The weak-type (1,1) estimate of the $\mathcal{H}$-Harmonic Bergman projection
本文通过回顾Calderon-Zygmund理论，推导了$\mathcal{H}$-harmonic Bergman空间投影的弱型(1,1)有界性。
Doubly commutativity of invariant subspaces of the Bergman space and the Dirichlet space over the unit polydisc $\mathbb{D}^n$ (with $ n \geq 2$) is investigated. We show that for any non-empty subset $α=\{α_1,\dots,α_k\}$ of $\{1,\dots,n\}$ and doubly commuting invariant subspace $\s$ of the Bergman space or the Dirichlet space over $\D^n$, the tuple consists of restrictions of co-ordinate multiplication operators $M_α|_\s:=(M_{z_{α_1}}|_\s,\dots, M_{z_{α_k}}|_\s)$ always possesses wandering subspace of the form \[\bigcap_{i=1}^k(\s\ominus z_{α_i}\s). \]
In this work, we extend the theory of Békollè-Bonami $B_p$ weights. Here we replace the constant $p$ by a non-negative measurable function $p(\cdot),$ which is log-Hölder continuous function with lower bound $1$. We show that the Bergman projector on the unit ball of $\mathbb C^n$ is continuous on the weighted variable Lebesgue spaces $L^{p(\cdot)}(w)$ if and only if $w$ belongs to the generalised Békollè-Bonami class $B_{p(\cdot)}$. To achieve this, we define a maximal function and show that it is bounded on $L^{p(\cdot)}(w)$ if $w\in B_{p(\cdot)}$. We next state and prove a weighted extrapol…
The manuscript is devoted to the construction of $Ψ^\star$- algebras containing the Bergman projection on the unit ball $B_n$ of $\mathbb C^n$. We consider the $C^\star$-algebra $\mathcal{L}(L^2(B_n))$ of bounded operator acting on the Hilbert space of square integrable functions on $B_n$ with respect to the standard probability measure. We search for spectral invariant Fréchet subalgebras of $\mathcal{L}(L^2(B_n))$ containing the Bergman projection $P$ which is defined on $L^2(B_n)$ with values in the space of holomorphic functions. We use the commutator method as introduced by Gramsch. This …
A Szegö limit theorem for a class of Toeplitz operators on the Bergman space of the unit ball with singular symbols
本文针对单位球加权 Bergman 空间上符号支撑于迷向或余迷向子流形且关于 Lebesgue 测度奇异的一类 Toeplitz 算子，建立了 Szegö 极限定理，刻画了参数趋于无穷时的渐近行为。
We show that for $f$ a continuous function on the closed polydisc $\bar{\mathbb{D}^n}$ with $n\geq 2$, the Hankel operator $H_{f}$ is compact on the Bergman space of $\mathbb{D}^n$ if and only if there is a decomposition $f=h+g$, where $h$ is in the ball algebra and $g$ vanishes on the boundary of the polydisc.
This paper is devoted to the study of the weighted Bergman space $A_ω^p $ in the unit ball $\mathbb{B}$ of $\mathbb{C}^n$ with doubling weight $ω$ satisfying $$\int_r^1ω(t)dt <C \int_{\frac{1+r}{2}}^1ω(t)dt ,\,\, 0\leq r<1.$$ The $q-$Carleson measures for $A_ω^p$ are characterized in terms of a neat geometric condition involving Carleson block. Some equivalent characterizations for $A_ω^p$ are obtained by using the radial derivative and admissible approach regions. The boundedness and compactness of Volterra integral operator $T_g:A_ω^p\to A_ω^q$ are also investigated in this paper with $0<p\l…
Let $H_m(\mathbb B)$ be the analytic functional Hilbert space on the unit ball $\mathbb B \subset \mathbb C^n$ with reproducing kernel $K_m(z,w) = (1 - \langle z,w \rangle)^{-m}$. Using algebraic operator identities we characterize those commuting row contractions $T \in L(H)^n$ on a Hilbert space $H$ that decompose into the direct sum of a spherical coisometry and copies of the multiplication tuple $M_z \in L(H_m(\mathbb B))^n$. For $m=1$, this leads to a Wold decomposition for partially isometric commuting row contractions that are regular at $z = 0$. For $m = 1 = n$, the results reduce to t…
Quantum Harmonic Analysis on the Unweighted Bergman Space of the Unit Ball
本文在单位球 Bergman 空间上建立量子调和分析框架，给出 Wiener Tauberian 定理与径向 Toeplitz 代数刻画，并探讨 $\alpha$-Berezin 变换及 Toeplitz 算子逼近。
Contraction properties for holomorphic functions via isoperimetric stability on the Bergman ball
本文研究全纯函数在近常数情况下的局部收缩性质，通过伯格曼球上的等周稳定性结果，将加权伯格曼空间的缺陷不等式与全纯函数的收缩性质联系起来。
The results in the paper are related to the classification problem for invariant subspaces of multiplication operators in several variables. The main results consist of characterizations, in the two dimensional case, of ideals of polynomials and analytic functions which are closed in the relative topology induced by Bergman spaces on certain domains. This provides proofs of the Bergman space analogues of a conjecture of R. G. Douglas and V. Paulsen.
In this paper, we study the boundedness and the compactness of the little Hankel operators $h_b$ with operator-valued symbols $b$ between different weighted vector-valued Bergman spaces on the open unit ball $\mathbb{B}_n$ in $\mathbb{C}^n.$ More precisely, given two complex Banach spaces $X,Y,$ and $0 < p,q \leq 1,$ we characterize those operator-valued symbols $b: \mathbb{B}_{n}\rightarrow \mathcal{L}(\overline{X},Y)$ for which the little Hankel operator $h_{b}: A^p_α(\mathbb{B}_{n},X) \longrightarrow A^q_α(\mathbb{B}_{n},Y),$ is a bounded operator. Also, given two reflexive complex Banach s…
Toeplitz operators and weighted composition operators on variable exponent Bergman spaces
本文将Choe等人关于加权复合算子之差有界性与紧性的Carleson测度刻画推广到单位球上的变指数Bergman空间，并给出更简单的新证明，同时刻画了Toeplitz算子的有界性与紧性。
Analytic continuation of weighted $H$-harmonic Bergman spaces
本文部分解决了 Blaschke 等人提出的关于加权 $H$-调和 Bergman 空间解析延拓的问题，识别了离散 Wallach 集并揭示了结构依赖于维数奇偶性。
We study Toeplitz operators with uniformly continuous symbols on generalized harmonic Bergman spaces of the unit ball in $\mathbb{R}^n$. We describe their essential spectra and establish a short exact sequence associated with the $C^{*}$-algebra generated by these operators.
This note completely describes the bounded or compact Riemann-Stieltjes integral operators $T_g$ acting between the weighted Bergman space pairs $(A^p_α,A^q_β)$ in terms of particular regularities of the holomorphic symbols $g$ on the open unit ball of $\Bbb C^n$.
Hilbert-Schmidt composition-differentiation operators on the unit ball
该研究在单位球与多圆盘的Hardy、Bergman空间上引入基于径向导数的复合微分算子，推导使该算子为Hilbert-Schmidt算子的诱导函数充要条件。
For $α>-1$, let $A^2_α$ be the corresponding weighted Bergman space of the unit ball in $\mathbb{C}^n$. For a bounded measurable function $f$, let $T_f$ be the Toeplitz operator with symbol $f$ on $A^2_α$. This paper describes all the functions $f$ for which $T_f$ commutes with a given $T_g$, where $g(z)=z_{1}^{L_1}... z_{n}^{L_n}$ for strictly positive integers $L_1,..., L_n$, or $g(z)=|z_1|^{s_1}... |z_n|^{s_n}h(|z|)$ for non-negative real numbers $s_1,..., s_n$ and a bounded measurable function $h$ on $[0,1)$.
This is our third work on Bergman-type operator over bounded domains. In the previous two articles, we systematically study the boundedness, compactness and Schatten membership of Bergman-type on the Hilbert unit ball. In the present paper, we investigate singular integral operators induced by the Bergman kernel and Szegö kernel on the irreducible bounded symmetric domain in its standard Harish-Chandra realization. We completely characterize when Bergman-type operators and Szegö-type operators belong to Schatten class operator ideals by several analytic numerical invariants of the bounded symm…
Carleson measures, tent embeddings, and Volterra-type integral operators on the unit ball
本文通过比较Carleson立方体和Bergman度量球条件，结合Berezin型刻画，证明了Besov空间和Bergman空间到对数帐篷空间的嵌入定理，并应用于刻画Volterra型积分算子的有界性、紧性和本质范数。
Bergman-type integral operators are classical operators in complex analysis and operator theory. Recently, the first author and his collaborator \cite{DiW} completely characterized the $L^p$-$L^q$ boundedness of Bergman-type integral operators $K_α,K_α^+$ and the $L^p$-$L^q$ compactness of $K_α$ on the unit ball. In this paper, we will use a substantially new method to completely characterize the $L^p$-$L^q$ compactness of $K_α^+,$ but also prove that the $L^p$-$L^q$ compactness of operators $K_α,K_α^+$ is in fact equivalent. Moreover, we completely characterize Schatten class and Macaev class…
This paper is devoted to the study of reducing subspaces for multiplication operator $M_ϕ$ on the Dirichlet space with symbol of finite Blaschke product. The reducing subspaces of $M_ϕ$ on the Dirichlet space and Bergman space are related. Our strategy is to use local inverses and Riemann surface to study the reducing subspaces of $M_ϕ$ on the Bergman space, and we discover a new way to study the Riemann surface for $ϕ^{-1}\circϕ$. By this means, we determine the reducing subspaces of $M_ϕ$ on the Dirichlet space when the order of $ϕ$ is $5$; $6$; $7$ and answer some questions of Douglas-Putin…
We consider commutative C* -algebras of Toeplitz operators in the weighted Bergman space on the unit ball in $\mathbb{C}^{\mathbf{n}}$. For the algebras of elliptic type we find a new representation, namely as the algebra of operators which are functions of certain collections of commuting unbounded self-adjoint operators in the Bergman space.
We study the boundedness of Toeplitz operators $T_ψ$ with locally integrable symbols on weighted harmonic Bergman spaces over the unit ball of $\mathbb{R}^n$. Generalizing earlier results for analytic function spaces, we derive a general sufficient condition for the boundedness of $T_ψ$ in terms of suitable averages of its symbol. We also obtain a similar "vanishing" condition for compactness. Finally, we show how these results can be transferred to the setting of the standard weighted Bergman spaces of analytic functions.
In this paper, we investigate the $\partial$-complex on weighted Bergman spaces on Hermitian manifolds satisfying a certain holomorphicity/duality condition. This generalizes the situation of the Segal-Bargmann space in $\mathbb{C}^n$, studied earlier by the first-named author, in which the adjoint of the differentiation is the multiplication by $z$. The results are applied to two important examples in the unit ball, namely, the complex hyperbolic metric and a conformally Kähler metric which are related to Bergman spaces with so-called "exponential" and "standard" weights, respectively. In par…
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