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，我们研究了分析功能及其运营商的空间，主要是“组成运营商”。[1] B. Maccluer，S.Ohno和R.Zhao表征了均匀操作员拓扑中H^∞的组成算子拓扑空间的组件和孤立点，并且还紧凑了两个组成算子的差异。借助这些结果，我们表明，组成算子空间中的一个组成部分通常不是与紧凑型操作员不同的所有组成算子的集合。[2] （1）S.Ohno，K.Strothoff和R.Zhao认为加权组合算子是组成和乘法操作员的概括。连续地，我们将考虑小蓝色型空间的情况。（2）S.ohno表征了H^∞和Bloch空间之间的加权组合算子。此外，S.Ohno在H^2和Bloch空间之间考虑了它们。我们认为，对这种情况的投资可能与该问题（Sundberg和Shapiro的问题）具有一定的关系，即在H^2上所有组成操作员的空间中孤立的组成操作员（3）S.Ohno和H.Takagi Studiod Studiod加权构图操作员在Disk algebra和H^∞上。对于这些操作员，我们提供了紧凑性，紧凑性和完全连续性的等效性。此外，我们为加权构图操作员提供了必要和充分的条件，使其具有封闭范围或成为弗雷霍尔姆操作员。[3] S.ohno定义了由组成操作员引起的De Branges-Rovnyak空间，并从D. sarason引起的一般情况下降低了基本结果。我们可以考虑乘数和不变子空间的问题。