Researches on Properties of Banach Spaces of Analytic Functions and Their Operators

解析函数及其算子的Banach空间性质研究

基本信息

批准号：
11640179
负责人：
OHNO Shuichi
金额：
$ 1.34万
依托单位：
Nippon Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640179/
关键词：
Hardy Space Bloch space Small Bloch-type spaces Composition Operators Multiplication Operators Boundedness Compactness de Branges-Rovnyak spaces de Branges-Rovnyak空間 disk環荷重合成作用素 compact作用素 Fredholm作用素閉値域作用素

项目摘要

According to our researches' plan, we have investigated the spaces of analytic functions and their operators, mainly, " composition operators".[1] B.MacCluer, S.Ohno and R.Zhao characterized components and isolated points of the topological space of composition operators on H^∞ in the uniform operator topology and also compact differences of two composition operators. With the aid of these results, we showed that a component in the space of composition operators is not in general the set of all composition operators that differ from the given one by a compact operator.[2] We have considered weighted composition operators as the generalization of composition and multiplication operators.(1) S.Ohno, K.Stroethoff and R.Zhao characterized the necessary and sufficient conditions for weighted composition operators to be bounded or compact between Bloch-type spaces of analytic functions on the unit disk including Lipschitz spaces and Bloch spaces. Continuously, we would consider the case of small Bloch-type spaces.(2) S.Ohno characterized weighted composition operators between H^∞ and Bloch space. Moreover, S.Ohno considered them between H^2 and Bloch space. The investigation in this situation, we suppose, might have some relationship to the problem (Sundberg and Shapiro's problem) of characterizing composition operators that are isolated in the space of all composition operator on H^2.(3) S.Ohno and H.Takagi studied weighted composition operators on the disk algebra and H^∞. For these operators, we proved the equivalence of the compactness, the weak compactness and the complete continuity. Moreover, we gave the necessary and sufficient conditions for weighted composition operators to have closed range or to be Fredholm operators.[3] S.Ohno defined de Branges-Rovnyak spaces induced by composition operator and reduced elementary results from the general situation due to D.Sarason. We could consider the problems of multipliers and invariant subspaces.

根据我们的研究计划，我们研究了解析函数的空间及其算子，主要是“复合算子”。 [1] B.MacCluer、S.Ohno 和 R.Zhao 表征了统一算子拓扑中 H^∞ 上的复合以及两个复合算子的紧凑差异借助这些结果，我们表明复合算子空间中的分量通常不是所有与复合算子不同的复合算子的集合。给定的一个[2] 我们将加权复合算子视为复合算子和乘法算子的推广。(1) S.Ohno、K.Stroethoff 和 R.Zhao 描述了加权复合算子有界或的充要条件单位圆盘上解析函数的Bloch型空间之间的紧致，包括Lipschitz空间和Bloch空间。接下来，我们将考虑小Bloch型空间的情况。(2)S.Ohno表征了之间的加权复合算子。此外，S.Ohno 在 H^2 和 Bloch 空间之间考虑了它们，我们认为这种情况的研究可能与表征组合算子的问题（Sundberg 和 Shapiro 问题）有关。在H^2上的所有复合算子的空间中是孤立的。(3)S.Ohno和H.Takagi研究了盘代数上的加权复合算子和H^∞，对于这些算子，我们证明了紧性、弱紧性和完全连续性，给出了加权合成算子具有闭域或成为 Fredholm 算子的充分必要条件。[3] S.Ohno 定义了由合成算子导出的 de Branges-Rovnyak 空间。由于 D.Sarason，我们可以考虑乘数和不变子空间的问题。