加权复合算子与加权哈代空间的分类
Weighted composition operators and classification of weighted Hardy spaces
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- Technion Israel Institute of Technology（以色列理工学院）
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中文总结 AI 辅助
本文构建自反函数巴拿赫空间定义域概念，将其框架用于分类多变量全纯函数的酉不变空间，推广Hartz结论并改进Ofek等人工作，还研究了希尔伯特函数空间的Banach-Mazur距离。
中文摘要 AI 辅助
本文阐明了如下基础问题：函数空间所依托的“合适”底层集合是什么，该集合是否唯一确定？对函数空间进行分类的恰当态射是什么？加权复合算子何时诱导同构，何时诱导等距同构？我们首先为自反函数巴拿赫空间的定义域构建合适的概念，研究希尔伯特函数空间间对应的同构概念。随后将该框架应用于分类某些多变量全纯函数的酉不变空间。主要结果将Hartz的结论从完全Pick情形推广至加权哈代空间的一般情形，建立了同构空间核间的简单关系，在多个方向上改进了Ofek与Sofer的早期工作。此外，还将所得结果应用于研究和估计希尔伯特函数空间的Banach-Mazur距离。
英文摘要
In this paper, we shed light on basic questions such as: What is the "right" underlying set on which a function space lives, and is this set uniquely determined? What are the appropriate morphisms to classify function spaces? When does a weighted composition operator induce an isomorphism, and when does it induce an isometric isomorphism? We first formulate suitable notions for domains of reflexive functional Banach spaces and study the corresponding notions of isomorphism between Hilbert function spaces. We then apply this framework to classify certain unitarily invariant spaces of holomorphic functions in several variables. Our main results extend Hartz's results from the complete Pick setting to the general setting of weighted Hardy spaces and establish a simple relation between the kernels of isomorphic spaces, improving also on earlier work of Ofek and Sofer in several directions. Additionally, we apply our results to investigate and estimate the Banach-Mazur distance of Hilbert function spaces.
