Operator Theory - Involving Toeplitz Operators and Model Spaces.

研究背景分析函数的希尔伯特空间非常重要，无论是就其本身而言，还是就其在系统和控制理论、信号处理和偏微分方程中的应用而言。本文研究Toeplitz算子（最早于100年前提出）与模型空间之间的关系，模型空间是Hardy空间在倒移算子下不变的子空间。随着Hayashi， Hitt和Sarason在20世纪80年代的工作，Toeplitz核（Toeplitz算子的核）的结构得到了很好的理解，并且已经使用模型空间（Toeplitz核的例子）来公式化。大约十年前，Sarason发起了一项关于截断Toeplitz算子的研究计划，尽管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。项目的总体目标是发展截断Toeplitz算子的理论（在某种程度上，截断Hankel算子）与某些块材料Toeplitz算子的理论并行，并回答有关其核，不变子空间和类似特征的各种目前未解决的问题。模型空间上的其他相关算子将酌情介绍和研究。该项目的第一个目标是建立在最近的结果之上，这些结果将截断的Toeplitz算子与块-材料Toeplitz算子（一个称为扩展等效的过程）联系起来，其核已经由Chalendar， Chevrot和Partington在扩展Hayashi/Hitt/Sarason思想的工作中描述。预计最近发展的Toeplitz核的极大向量的概念也将发挥作用。该项目的进一步目标将引导学生寻找截断Toeplitz算子核的近不变性（Toeplitz核拥有的属性）的类似物，并进一步发展模型空间上类似算子的理论。潜在的应用和好处作为一个纯数学项目，虽然模型空间在控制理论（可控性）和信号处理（佩利-维纳空间）中都出现了，但这项工作主要不是由工业应用驱动的。在这一点上，数学以外的好处纯粹是长期的。研究领域：纯数学、数学分析。资格要求：数学博士学位

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