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Operator Theory - Involving Toeplitz Operators and Model Spaces.

算子理论 - 涉及 Toeplitz 算子和模型空间。

基本信息

批准号：
1972662
负责人：
金额：
--
依托单位：
University of Leeds
依托单位国家：
英国
项目类别：
Studentship
财政年份：
2017
资助国家：
英国
起止时间：
2017 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=studentship-1972662
关键词：
Operator Theory Involving Toeplitz Operators

项目摘要

Context of researchHilbert spaces of analytic functions are of importance, both in their own right, and for their applications in systems and control theory, signal processing, and PDEs. This project is concerned with the relations between Toeplitz operators (first developed about 100 years ago) and model spaces, which are subspaces of Hardy spaces invariant under the backward shift operator.Following the work of Hayashi, Hitt, and Sarason in the 1980s, the structure of Toeplitz kernels (kernels of Toeplitz operators) is well understood, and this has been formulated using model spaces, which are examples of Toeplitz kernels.About ten years ago, Sarason initiated a programme of research into truncated Toeplitz operators, although examples of these had been studied for over 50 years. These are operators on model spaces, analogous to Toeplitz operators, and here a complete theory is still lacking.Aims and objectivesThe overall aim of the project is to develop the theory of truncated Toeplitz operators (and, to some extent, truncated Hankel operators) in parallel to the theory of certain block-matricial Toeplitz operators, and to answer various currently unsolved questions concerning their kernels, invariant subspaces, and similar features. Other related operators on model spaces will be introduced and studied, as appropriate.The objectives are The first objective of this project is to build on recent results of Câmara and Partington linking truncated Toeplitz operators with block-matricial Toeplitz operators (a process called equivalence by extension), whose kernels have been described by Chalendar, Chevrot and Partington in work extending the Hayashi/Hitt/Sarason ideas. It is expected that the recently-developed notion of maximal vectors for Toeplitz kernels will also play a part.The project's further objectives will lead the student into seeking analogues of near-invariance (a property possessed by Toeplitz kernels) for kernels of truncated Toeplitz operators, and to develop further the theory of similar operators on model spaces. Potential applications and benefitsAs a project in pure mathematics, this work is not primarily driven by industrial applications, although model spaces have appeared in both control theory (controllability) and signal processing (Paley-Wiener spaces). At this point the benefits outside mathematics are purely long-term.Research Areas: Pure Mathematics, Mathematical AnalysisQualification to be attained: Ph.D degree in Mathematics

解析函数的希尔伯特空间是重要的，无论是在他们自己的权利，并为他们在系统和控制理论，信号处理和偏微分方程的应用。本课题主要研究Toeplitz算子之间的关系（大约100年前首次发展）和模型空间，模型空间是在向后移位算子下不变的哈代空间的子空间。（Toeplitz算子的核）是很好理解的，并且这已经使用模型空间来公式化，模型空间是Toeplitz核的示例。大约十年前，Sarason发起了一项研究计划截断Toeplitz运营商，虽然这些例子已经研究了50多年。这些是模型空间上的算子，类似于Toeplitz算子，这里仍然缺乏一个完整的理论。目的和目标该项目的总体目标是发展截断Toeplitz算子的理论（和，在某种程度上，截断汉克尔算子）平行于某些块矩阵Toeplitz算子的理论，并回答各种目前尚未解决的问题，关于他们的内核，不变子空间，和相似的特征。本项目的第一个目标是建立在Câmara和Partington最近的结果上，将截断Toeplitz算子与块矩阵Toeplitz算子（一个被称为扩展等价的过程）联系起来，其内核已经由Chalanet，Chevrot和Partington在扩展Hayashi/Hitt/Sarason思想的工作中描述。预计最近发展的最大向量的概念Toeplitz内核也将发挥作用。该项目的进一步目标将引导学生到寻找近似不变性（一个属性所拥有的Toeplitz内核）的截断Toeplitz算子的内核的类似物，并进一步发展模型空间上的类似算子的理论。作为纯数学的一个项目，这项工作主要不是由工业应用驱动的，尽管模型空间已经出现在控制理论（可控性）和信号处理（Paley-Wiener空间）中。在这一点上，数学以外的好处是纯粹长期的。研究领域：纯数学，数学分析要达到的资格：数学博士学位