Compact Operators on Bergman Spaces

We prove that a bounded operator S on L _a ^p for p > 1 is compact if and only if the Berezin transform of S vanishes on the boundary of the unit disk if S satisfies some integrable conditions. Some estimates about the norm and essential norm of Toeplitz operators with symbols in BT are obtained.     

Toeplitz Operators on Generalized Bergman Spaces

We consider the weighted Bergman spaces HL²(\mathbb B^d, m_l){\mathcal {H}L^{2}(\mathbb {B}^{d}, \mu_{\lambda})}, where we set dm_l(z) = c_l(1-|z|²)^l dt(z){d\mu_{\lambda}(z) = c_{\lambda}(1-|z|^2)^{\lambda} d\tau(z)}, with τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ > d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which the corresponding Toeplitz operators can be defined as bounded operators or as a Hilbert–Schmidt operators on the generalized Bergman spaces.     

Hankel Operators on Fock Spaces and Related Bergman Kernel Estimates

Hankel operators with anti-holomorphic symbols are studied for a large class of weighted Fock spaces on ? ⁿ . The weights defining these Hilbert spaces are radial and subject to a mild smoothness condition. In addition, it is assumed that the weights decay at least as fast as the classical Gaussian weight. The main result of the paper says that a Hankel operator on such a Fock space is bounded if and only if the symbol belongs to a certain BMOA space, defined via the Berezin transform. The latter space coincides with a corresponding Bloch space which is defined by means of the Bergman metric. This characterization of boundedness relies on certain precise estimates for the Bergman kernel and the Bergman metric. Characterizations of compact Hankel operators and Schatten class Hankel operators are also given. In the latter case, results on Carleson measures and Toeplitz operators along with Hörmander’s L ² estimates for the $\bar{\partial}$ operator are key ingredients in the proof.     

加权Bergman空间上小Hankel算子的Schatten类

讨论了 C\+n 中有界对称域的加权Bergman空间上符号属于 L\+2\-a(Ω, dV\-λ) 的小Hankel算子, 利用符号 Φ 的某种积分变换,给出了小Hankel算子 h\-Φ 属于Schatten理想 S\-p 的特征.     

Hankel Operators on Bergman Spaces of Tube Domains over Symmetric Cones

We present here some criteria for Schatten-Von Neumann class membership for the small Hankel operator on Bergman space A ²(T _Ω), when T _Ω is the tube over the symmetric cone Ω. The author would like to thank professor Aline Bonami for helpful advices.     

Hankel Operators on Bergman Spaces of Annulus Induced by Regular Weights

This paper is devoted to studying Bergman spaces A_(ω_(1,2)) ~p(M)(1 p ∞) induced by regular-weight ω_(1,2) on annulus M.We characterize the function f in L_(ω_(1,2)) ~1(M) for which the induced Hankel operator H_f is bounded(or compact) from A_(ω_(1,2))~p(M) to L_(ω_(1,2)) ~q(M) with 1 p,q ∞.     

Bergman空间上Hankel算子的本性范数的刻划

在文中,对于Ｃ＾ｍ中有界强拟凸域Ω,得到了Ｂｅｒｇｍａｎ空间Ａ＾２（Ω）上的Ｈａｎｋｅｌ算子Ｈｆ（ｆ∈Ｌ＾２（Ω,ｄｖ）的本性范数∥Ｈｆ∥ｅｓｓ（在Ｌ＾２（Ω,ｄｖ上）的等价刻划。     

加权Bergman空间到Zygmund空间的广义复合算子

术文讨论了加权Bergman空间到Zygmund空间（小Zygmund空间）的广义复合算子Cφ^h的有界性和紧性特征,得到了以下约结果：（1）Cφ^h是加权Rergman空间到Zygmund空间的有界算子和紧算子的充要条件;（2）Cφ^h是加权Bergman空间到小Zygmund空间的有界算子和紧算子的充要条件．     

Compact Differences of Composition Operators on the Bergman Spaces Over the Ball

The compact differences of composition operators acting on the weighted L ²-Bergman space over the unit disk is characterized by the angular derivative cancellation property and due to Moorhouse. In this paper we extend Moorhouse’s characterization, as well as some related results, to the ball and, at the same time, to the weighted L ^p -Bergman space for the full range of p.     

Compact Toeplitz Operators for Weighted Bergman Spaces on Bounded Symmetric Domains

Let W ì \mathbb C^d{\Omega \subset{\mathbb C}^{d}} be an irreducible bounded symmetric domain of type (r, a, b) in its Harish–Chandra realization. We study Toeplitz operators Tⁿ_g{T^{\nu}_{g}} with symbol g acting on the standard weighted Bergman space H_n²{H_\nu^2} over Ω with weight ν. Under some conditions on the weights ν and ν ₀ we show that there exists C(ν, ν ₀) > 0, such that the Berezin transform [(g)\tilde]_n0{\tilde{g}_{\nu_{0}}} of g with respect to H²_n0{H^2_{\nu_0}} satisfies:

Compact Hankel operators on harmonic Bergman spaces

We study Hankel operators on the harmonic Bergman spaceb ²(B), whereB is the open unit ball inR ⁿ,n2. We show that iff is in then the Hankel operator with symbolf is compact. For the proof we have to extend the definition of Hankel operators to the spacesb ^p(B), 1<p<, and use an interpolation theorem. We also use the explicit formula for the orthogonal projection ofL ²(B, dV) ontob ²(B). This result implies that the commutator and semi-commutator of Toeplitz operators with symbols in are compact.     

The Essential Norm of Hankel Operators on the Weighted Bergman Spaces with Exponential Type Weights

Let denote the closed subspace of consisting of analytic functions in the unit disc . For certain class of subharmonic functions and , it is shown that the essential norm of Hankel operator is comparable to the distance norm from H_f to compact Hankel operators.     

Essential Norm Estimates for Little Hankel Operators on Weighted Bergman Spaces of the Unit Ball

We give estimates for the essential norm of a bounded little Hankel operator with $L^2$ symbol on weighted Bergman spaces of the unit ball in terms of a certain integral transform of the symbol. As an application of these estimates, we also give a necessary and sufficient condition for the little Hankel operators to be compact.     

Hankel Operators on Weighted Fock Spaces

We describe a general method to construct completely bounded idempotent mappings on operator spaces, starting from amenable semigroups of completely bounded mappings. We then explore several applications of that method to injective operator spaces, fixed points of completely contractive mappings, Toeplitz operators, dynamical systems and similarity orbits of group representations.     

Compact Operators on the Bergman Spaces of the Unit Ball and the Polydisk in C^n

Let Ω be the unit ban or the polydisk of C^n and La^2 (Ω) the Bergman space. In this paper we prove that if S is a finite sum of finite products of Toeplitz operators on La^2 (Ω), then S is compact if and only if the Berezin transform S↑-(z) of S tends to zero as z→эΩ.     

关于广义Bergman空间的乘子

In this paper,we characterize the multipliers of generalized Bergman spaces Ap,q,αwith 00) in almost every case considered. The corollaries on multipliers of the spaces Ap,q,αextend some related results.     

Hilbert-Schmidt Hankel Operators on the Bergman Space of Planar Domains

In this paper we study the problem of the membership of H _ϕ in the Hilbert-Schmidt class, when and Ω is a planar domain. We find a necessary and sufficient condition.We apply this result to the problem of joint membership of H _φ and in the Hilbert-Schmidt class. Using the notion of Berezin Transform and a result of K. Zhu we are able to give a necessary and sufficient condition. Finally, we recover a result of Arazy, Fisher and Peetre on the case with φ holomorphic.