Commuting Toeplitz Operators on Fock-Sobolev Spaces of Negative Orders

In the setting of Fock-Sobolev spaces of positive orders over the complex plane,Choe and Yang showed that if the one of the symbols of two commuting Toeplitz operators with bounded symbols is non-trivially radial,then the other must also be radial.In this paper,we extend this result to the Fock-Sobolev space of negative order using the Fock-type space with a confluent hyper geometric function.     

Products of Toeplitz and Hankel Operators on Fock-Sobolev Spaces

In this paper, the authors investigate the boundedness of Toeplitz product T_f T_g and Hankel product H^*_fH_g on Fock-Sobolev space for f, g ∈ P. As a result, the boundedness of Toeplitz operator T_f and Hankel operator H_g with f ∈ P is characterized.     

Mixed Product of Hankel and Toeplitz Operators on Fock-Sobolev Spaces

Let f and g be functions in Fock-Sobolev space F^2,m. In this paper, we completely characterize the boundedness and compactness of H_f T_g.     

加权Bloch空间上复合算子的线性组合

设λi(i= 1,..,N)是一列非0的数,D是一维复平面C的开单位圆盘,φi (i = 1,...,N)是D的解析自映射,本文研究了定义在加权Bloch空间上复合算子线性组合∑Ni=1λiCψi的紧致性.     

Composition Operators on Small Spaces

We show that if a small holomorphic Sobolev space on the unit disk is not just small but very small, then a trivial necessary condition is also sufficient for a composition operator to be bounded. A similar result for holomorphic Lipschitz spaces is also obtained. These results may be viewed as boundedness analogues of Shapiro’s theorem concerning compact composition operators on small spaces. We also prove the converse of Shapiro’s theorem if the symbol function is already contained in the space under consideration. In the course of the proofs we characterize the bounded composition operators on the Zygmund class. Also, as a by-product of our arguments, we show that small holomorphic Sobolev spaces are algebras.     

Hardy空间上的有界复合算子

本文讨论了由解析映射φ：Ｂｎ→Ｂｍ诱导复合算子的有界性；特别地，研究了由保测内映射诱导复合算子的性质．作为推论，得到了Ｗ．Ｒｕｄｉｎ［１０］中一个公开问题的解答．     

Closed-Range Composition Operators on Dirichlet-Type Spaces

We characterize the bounded composition operators mapping one Dirichlet-type space into another. We also give a geometric characterization for those composition operators having closed range.     

Eigenfunctions of Composition Operators on Bloch-type Spaces

Suppose φ is a holomorphic self map of the unit disk and C_φ is a composition operator with symbol φ that fixes the origin and 0 < |φ'(0)| < 1. This paper explores sufficient conditions that ensure all the holomorphic solutions of Schröder equation for the composition operator C_φ to belong to a Bloch-type space B_α for some α > 0. In the second part of the paper, the results obtained for composition operators are extended to the case of weighted composition operators.     

Bers型空间和复合算子

For α∈（0,∞),let Hα^∞（or Hα^∞,0)denote the collection of all functions f which are analytic on the unit disc D and satisfy |f(z)|(1-|z|^2)^α=O(1)(or|f(z)|(1-|z|^2)^α=o(1) as |z|→1).Hα^∞,0)is called a Bers-type space (or a little Bers-type space).In this paper,we give some basic properties of Hα^∞，Cψ，the composition operator associated with a symbol function ψ which is an analytic self map of D,is difined by Cψf=f o ψ,We characterize the boundedness,and compactness of Cψ which sends one Bers-type space to another function space.     

Composition Operators on Orlicz–Lorentz Spaces

In this paper, we study the boundedness and the compactness of composition operators on Orlicz–Lorentz spaces.      

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.     

加权Bloch型空间上的广义复合算子

研究了加权Bloch型空间上的广义复合算子的有界性和紧性,得到了刻画该算子为有界和紧的一些充分必要条件.     

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.     

广义Bergman空间上的紧复合算子

首先证明广义Bergman空间A_(N,α)~p,(α-n-1,p0)上的复合算子C_φ的有界性和紧性是不依赖于p的,进而证明了若对某个q0和-n-1βα,C_φ在A_(N,β)~α上有界,则C_φ在A_(N,α)~p,α(α-n-1,p0)上是紧的当且仅当lim|z|→1-1-(|z|~2/1-|φ(1)|~2)=0.     

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C(ψ)的有界性和紧性特征,得到了以下结论(1)C(ψ)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)C(ψ)是Bergman空间到q-Bloch空间的有界算子或紧算子之充要条件;(3)C(ψ)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C0的范数估计,此处C0(f)(z)=fo(ψ)(z)-f((ψ)(0)).     

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))．     

Besov空间和Zygmund空间上的复合算子

讨论单位圆盘上Besov空间B（p,q）和Zygmund空间Z及小Zygmund空间Z_0之间的复合算子,得到了B（p,q）到Z（Z_0）的复合算子以及Z（Z0）到B（p,q）的复合算子有界或紧的充要条件。     

Differences of Weighted Composition Operators on the Bloch Spaces

We study the boundedness and the compactness of the differences of weighted composition operators on the Bloch spaces. The results generalize the corresponding results on the single weighted composition operators and on the differences of composition operators.     

Approximation Numbers of Operators on Normed Linear Spaces

In [1], B?ttcher et. al. showed that if T is a bounded linear operator on a separable Hilbert space H, {e_j}_j=1^￥H, \{e_{j}\}_{j=1}^{\infty} is an orthonormal basis of H and P_n is the orthogonal projection onto the span of {e_j}_j=1ⁿ\{e_{j}\}_{j=1}^{n}, then for each k ? \mathbbNk \in {\mathbb{N}}, the sequence {s_k(P_nTP_n)}\{s_{k}(P_{n}TP_{n})\} converges to s_k(T), where for a bounded operator A on H, s_k(A) denotes the kth approximation number of A, that is, s_k(A) is the distance from A to the set of all bounded linear operators of rank at most k − 1. In this paper we extend the above result to more general cases. In particular, we prove that if T is a bounded linear operator from a separable normed linear space X to a reflexive Banach space Y and if {P_n} and {Q_n} are sequences of bounded linear operators on X and Y, respectively, such that ||P_n|| ||Q_n|| ￡ 1\|P_n\| \|Q_n\| \leq 1 for all n ? \mathbbNn \in {\mathbb{N}} and {Q_nTP_n} converges to T under the weak operator topology, then {s_k(Q_nTP_n)}\{s_{k}(Q_{n}TP_{n})\} converges to s_k(T). We also obtain a similar result for the case of any normed linear space Y which is the dual of some separable normed linear space. For compact operators, we give this convergence of s_k(Q_nTP_n)s_{k}(Q_{n}TP_{n}) to s_k(T) with separability assumptions on X and the dual of Y. Counter examples are given to show that the results do not hold if additional assumptions on the space Y are removed. Under separability assumptions on X and Y, we also show that if there exist sequences of bounded linear operators {P_n} and {Q_n} on X and Y respectively such that (i) Q_nTP_nQ_{n}TP_{n} is compact, (ii) ||P_n|| ||Q_n|| ￡ 1\|P_{n}\| \|Q_{n}\| \leq 1 and (iii) {Q_nTP_n}\{Q_{n}TP_{n}\} converges to T in the weak operator topology, then {s_k(Q_nTP_n)}\{s_k(Q_{n}TP_{n})\} converges to s_k(T) if and only if s_k(T) = s_k(T￠)s_{k}(T) = s_{k}(T^\prime). This leads to a generalization of a result of Hutton [3], proved for compact operators between normed linear spaces.