Operator Theory - Involving Toeplitz Operators and Model Spaces.

研究背景解析函数的希尔伯特空间具有重要意义，无论是它们本身，还是它们在系统和控制理论、信号处理和偏微分方程组中的应用。这个项目是关于Toeplitz算子和模型空间之间的关系的，模型空间是Hardy空间在后向移位算子下不变的子空间。在Hayashi，Hitt和Sarason于20世纪80年代的工作之后，Toeplitz核(Toeplitz算子的核)的结构得到了很好的理解，这是用Toeplitz核的例子模型空间来表达的。大约10年前，Sarason发起了一个关于截断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。目的和目标本项目的总体目标是将截断Toeplitz算子(在某种程度上，截断Hankel算子)的理论发展到某些块矩阵Toeplitz算子的理论，并回答目前尚未解决的关于其核、不变子空间和类似特征的各种问题。这个项目的第一个目标是在C？mara和Partington的最新结果的基础上将截断的Toeplitz算子与块-矩阵Toeplitz算子(称为扩张等价)联系起来，其核心已经由Chalendar，Chevrot和Partington在扩展Hayashi/Hitt/Sarason思想的工作中描述了。预计最近开发的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核的近不变性(Toeplitz核所具有的性质)的类似，并进一步发展模型空间上的相似算子理论。潜在的应用和益处作为纯数学中的一个项目，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，数学以外的好处纯粹是长期的。研究领域：纯数学，数学分析需要获得的资格：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