Closed-Range Composition Operators on Weighted Bergman Spaces

For analytic self-maps φ of the unit disk, we show that recently given conditions that are both necessary and sufficient for the composition operator C _φ to be closed-range on the classical Bergman space \mathbbA²{\mathbb{A}^2} are actually necessary and sufficient in the general setting of the weighted Bergman spaces \mathbbA^p_a{\mathbb{A}^p_{\alpha}} , for 1 ≤ p < ∞ and weights of the form (1 - |z|²)^a{(1 - |z|^2)^{\alpha}} ; where α > −1. Along the way, we show how work done in a paper of J. Akeroyd, P. Ghatage and M. Tjani can be used to improve upon the most definitive of these conditions, and we give applications.     

加权Bergman空间到μ-Bloch空间的复合算子

给定α＞-1,p＞0和[0,1)上的正规函数μ,得到了Cn中单位球上加权Bergman空间Apα到μ-Bloch空间βμ的复合算子C(ψ)为有界算子和紧算子的充要条件,同时也给出了几个推论.     

Reproducing Kernel Hilbert Spaces Supporting Nontrivial Hermitian Weighted Composition Operators

We characterize those generating functions ${k(z) = \sum_{j=0}^\infty z^j/\beta(j)^2}$ that produce weighted Hardy spaces H ²(β) of the unit disk ${\mathbb D}$ supporting nontrivial Hermitian weighted composition operators. Our characterization shows that the spaces associated with the “classical reproducing kernels” ${z \mapsto (1 - \bar{w}z)^{-\eta}}$ , where ${w \in \mathbb D}$ and η > 0, as well as certain natural extensions of these spaces, are precisely those that are hospitable to Hermitian weighted composition operators. It also leads to a refinement of a necessary condition for a weighted composition to be Hermitian, obtained recently by Cowen, Gunatillake, and Ko, into one that is both necessary and sufficient.     

Order Bounded Weighted Composition Operators Mapping into the Bergman Space

We will investigate the order boundedness of weighted composition operators ${uC_{\varphi}}$ from weighted Bergman spaces ${L_{a}^p(dA_{\alpha})}$ , weighted-type spaces ${H_{\alpha}^{\infty}}$ or Bloch-type spaces ${\mathcal{B}_{\alpha}}$ into the space ${L_{a}^q(dA_{\beta})}$ .     

Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions

The essential norm of weighted composition operators on weightedBanach spaces of analytic functions is computed in terms ofthe weights and the inducing symbols. As a consequence the boundednessand compactness of these operators is characterized. As anotherconsequence the essential norm of composition operators on weightedBloch spaces is obtained and, consequently, the boundednessand compactness of composition operators on these spaces isalso characterized. Particular instances of weighted Bloch spacesare the Lipschitz spaces. The method used allows a unified treatmentof the problem of boundedness and compactness on these spaces.     

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．     

加权Begman空间的加权复合算子的本性模

利用上极限,给出了单位球上加权Bergman空间的加权复合算子的本性模的表示．     

Cn中有界对称域上不同加权Bergman空间之间的复合算子

本文研究了Cn中有界对称域Ω上不同加权Bergman空间之间的复合算子,引入了η-αCarleson测度,利用它作为工具给出了有界的或紧的复合算子Cφ:Lαp(Ω,dνα)Lαq(Ω,dνβ)的特征.     

Generalized Interpolation in Bergman Spaces and Extremal Functions

In a recent paper A. Schuster and K. Seip [SchS] have characterized interpolating sequences for Bergman spaces in terms of extremal functions (or canonical divisors). As these are natural analogues in Bergman spaces of Blaschke products, this yields a Carleson type condition for interpolation. We intend to generalize this idea to generalized free interpolation in weighted Bergman spaces B^{p, α} as was done by V. Vasyunin [Va1] and N. Nikolski [Ni1] (cf.also [Ha2]) in the case of Hardy spaces. In particular we get a strong necessary condition for free interpolation in B^{p, α} on zero–sets of B^{p, α}–functions that in the special case of finite unions of B^{p, α}–interpolating sequences turns out to be also sufficient.     

单位球上加权Bergman空间上加权复合算子的本性范数

本文给出了单位球上加权Bergman空间上的加权复合算子的本性范数,并刻画了这类加权复合算子的有界性和紧性.     

Topological Structure of the Set of Weighted Composition Operators on Weighted Bergman Spaces of Infinite Order

We consider the topological space of all weighted composition operators on weighted Bergman spaces of infinite order endowed with the operator norm. We show that the set of compact weighted composition operators is path connected. Furthermore, we find conditions to ensure that two weighted composition operators are in the same path connected component if the difference of them is compact. Moreover, we compare the topologies induced by L(H^∞) and L(H^∞_v) on the space of bounded composition operators and give a sufficient condition for a composition operator to be isolated.     

Weighted Composition Operators on the Space of Bounded Harmonic Functions

We determine the norm and the essential norm of the difference of weighted composition operators on the space of bounded harmonic functions on the open unit disk. The argument is done on the boundary.     

Norm Estimates for Weighted Composition Operators on Spaces of Holomorphic Functions

This paper shows that the boundedness of a weighted composition operator on the Hardy–Hilbert space on the disc or half-plane implies its boundedness on a class of related spaces, including weighted Bergman spaces. The methods used involve the study of lower-triangular and causal operators.     

The Essential Norm of Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions

For a wide class of radial weights we calculate the essential norm of a weighted composition operator uC_j{uC_\varphi} on the weighted Banach spaces of analytic functions in terms of the analytic function u \colon \mathbb D ? \mathbb C{u \colon \mathbb D \to \mathbb C} and the nth power of the analytic selfmap j{\varphi} of the open unit disc \mathbb D{\mathbb D} . We also apply our result to calculate the essential norm of composition operators acting on Bloch type spaces with general radial weights.     

A Class of Operators on Weighted Bergman Spaces

This paper studies the connection between an analytic function φ on the unit disk and a certain type of operators induced by φ on weighted Bergman spaces A ^α of analytic functions on . The general theme is the interplay between smoothness conditions on φ and the S _p -norms of the operators induced by φ.      

Hyponormal Toeplitz Operators on the Weighted Bergman Space

Consider j = f +[`(g)]\varphi = f + \overline {g}, where f and g are polynomials, and let T_jT_{\varphi} be the Toeplitz operators with the symbol j\varphi. It is known that if T_jT_{\varphi} is hyponormal then |f￠(z)|² 3 |g￠(z)|²|f'(z)|^{2} \geq |g'(z)|^{2} on the unit circle in the complex plane. In this paper, we show that it is also a necessary and sufficient condition under certain assumptions. Furthermore, we present some necessary conditions for the hyponormality of T_jT_{\varphi} on the weighted Bergman space, which generalize the results of I. S. Hwang and J. Lee.     

Weighted Composition Operators on Weighted Dirichlet Spaces

We characterize the boundedness and compactness of weighted composition operators on weighted Dirichlet spaces in terms of Nevanlinna counting functions and Carleson measure.     

Bergman空间和q-Bloch空间之间的复合算子

本文讨论了Bergman空间和q-Bloch空间(小q-Bloch空间)之间的复合算子Cφ的有界性和紧性特征,得到了以下结论:(1)Cφ是q-Bloch空间(小q-Bloch空间)到Bergman空间的有界算子或紧算子之充要条件; (2)Cφ是Bergman空间到q-Bloch空间的有界算子或紧算子之充要条件; (3)Cφ是Bergman空间到小q-Bloch空间的有界算子或紧算子之充要条件,还给出了算子 Cφ0的范数估计,此处Cφ0(f)(z)=foφ(z)-f(φ(0))．