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函数空间上若干线性算子的特性

更新时间 2009-12-16 18:17:20 点击数:

函数空间上若干线性算子的特性
Some Linear Operators on Certainfunction Spaces
【摘要】 本文所研究的内容分为两部分.一、加权Bergman空间与Zygmund空间之间广义Ces(?)ro算子和复合算子的乘积算子的有界性和紧性特征;二、Schatten-p类Hankel算子在调和Bergman空间上的特征.研究工作集中在以下内容.记D为复平面C上的单位圆盘,以H(D)表示D上全纯函数的全体.给定0<p<∞,α>-1,定义D上的加权Bergman空间为其中dA(z)是D上规范化的Lebesgue面积测度.定义D上的Zygmund空间(?)为其中众所周知在范数||f||=|f(0)|+|f'(0)|+sup(1-|z|2)|f"(z)|下,(?)成为一个Banach空间.定义D上的小Zygmund空间(?)为给定D上的全纯自映射φ和g∈H(D),定义广义Ces(?)ro算子和复合算子的乘积算子T_gC_φ为它是广义Ces(?)ro算子的一种拓广.当φ(z)=z时,T_gC_φ就是广义Ces(?)ro算子广义Ces(?)ro算子是算子理论研究领域中的一个重要内容,以它为工具可以解决某些函数空间上的Gleason问题,而且它与复合算子及算子半群有着密切的关系,可望被用来研究某些偏微分方程.因此对广义Ces(?)ro算子和复合算子的乘积算子的研究也很有必要.我们研究了算子T_gC_φ在加权Bergman空间和Zygmund(小Zygmund)空间之间的特性,得到了T_gC_φ在相应空间上为有界算子和紧算子的充要条件,在算子类型上扩展了研究范围,丰富了人们对该算子的认识.设Ω是Rn(n≥2)中的有界光滑区域,V是Ω上的Lebesgue测度.L2(Ω)是Ω上满足的可测函数f的集合.定义调和Bergman空间L_h2(Ω)为L2(Ω)中所有调和函数的全体.给定f∈L2(Ω),定义乘法算子M_f为M_f(g)=fg.设Q是L2(Ω)到L_h2(Ω)上的正交投影,以f为符号的L2(Ω)上的Hankel算子J_f定义为我们讨论了Schatten-p类Hankel算子在L_h2(Ω)上的特性,得到了当2≤p<∞时,H_f属于S_p的充要条件,推广了这方面已有的结果.

【Abstract】 In this thesis, first, we discuss the boundedness and compactness of the product of extended Ces(?)ro and composition operators between the weighted Bergman space and the Zygmund space. Second, we study the characterizations of Schatten p-class Hankel operator on the harmonic Bergman spaces.Let D be the unit disk in the complex plane C, and let H(D) be the class of all holomorphic functions on D. For 0<p<∞,α>-1, the weighted Bergman space A_αp is defined byHere dA denotes the normalized Lebesgue area measure on D.The Zygmund space£is defined bywhereIt is well known that It is easy to see that (?) is a Banach space under the norm (?), whereThe little Zygmund space of ED, denoted by (?), is the closed subspace of (?) consisting of functions f withGivenφan holomorphic self-map of D and g∈H(B), the product of extended Ces(?)ro and composition operator T_gC_φis defined byThis is a generalization of the extended Ces(?)ro operator. Ifφ(z) = z, then T_gC_φis just the extended Ces(?)ro operatorThe extended Ces(?)ro operator is significant in the operator theory of holomorphic functionspaces. Therefore, it is necessary to study this operator T_gC_φon the holomorphic function spaces. We characterize the boundedness and compactness of the operator T_gC_φbetween the weighted Bergman space and the Zygmund space (little Zygmund space). According to this, we enlarge the research field of operators, and give a deeper clarification of this operator.LetΩbe a bounded smooth domain in Rn(n≥2) and let V be the Lebesgue measure onΩ. Denote by L2(Ω) the set of all measurable functions f onΩsuch that The harmonic Bergman space L_h2(Ω) is the set of all harmonic functions in L2(Ω).For f∈L2(Ω), the multiplication operator M_f is defined by M_f(g)=fg. Let Q be the orthogonal projection from L2(Ω) onto L_h2(Ω), the Hankel operator H_f is defined on L2(Ω) byWe discuss the characterizations of the Schatten p-class Hankel operator on the harmonicBergman spaces L_h2(Ω), and obtain the necessary and sufficient conditions that H_f to belong to S_p(2≤p<∞). Our work extends the results in references. 

【关键词】 函数空间; 线性算子; 有界性; 紧性; Schatten-p类
  函数空间上若干线性算子的特性

摘要 3-6
ABSTRACT 6-8
目录 9-11
1 绪论 11-17
    1.1 引言 11-12
    1.2 研究背景与主要结果 12-17
2 广义Ces(?)ro算子和复合算子的乘积算子 17-34
    2.1 引言 17-19
    2.2 A_α~p到(?)的T_gC_φ算子 19-26
    2.3 (?)到A_α~p的T_gC_φ算子 26-28
    2.4 A_α~p到(?)的T_gC_φ算子 28-31
    2.5 A~p(φ)与(?)之间的T_gC_φ算子 31-33
    2.6 注记 33-34
3 调和Bergman空间上的S_p类Hankel算子 34-48
    3.1 引言 34-36
    3.2 若干引理 36-41
    3.3 调和Bergman空间上Hankel的特性 41-47
    3.4 注记 47-48
参考文献 48-53
攻读学位期间取得的研究成果 53-54
致谢 54-56

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