Operator Theory, Free Harmonic Analysis, and Related Problems

建议的研究涉及算子及其不变子空间的研究，以及自由随机变量理论中出现的一些问题。关于不变子空间的问题将集中在结构和分类结果上，这些结果在特殊情况下可以从Young表的组合学中推断出来。自由概率中的问题将涉及任意阵列的弱律，以及乘法稳定性的研究。我们还将研究自由卷积的更精细的调和分析。这里提出的工作主要是理论性质的，它旨在澄清不同领域(如算符理论、复分析和组合分析)中研究的对象之间的一些(有时是意想不到的)联系。这类工作往往与更多的应用领域有关。与算子理论密切相关的一个具体应用领域是控制理论和优化，在这些领域，算子确实发现了相当惊人的应用。

ABSTRACTDMS-0070459The proposed research involves the study of operators and their invariantsubspaces, and some problems arising in the theory of free random variables. The problems on invariant subspaces will focus on the structure andclassification results which can in special cases be inferred from the combinatorics ofYoung tableaux. Problems in free probability will concern the weak lawsfor arbitrary arrays, and the study of multiplicative stability. Thefiner harmonic analysis of free convolutions will also be studied.The work proposed here is mostly of a theoretical nature, and it isintended to clarify some (sometimes unexpected) connections between objects studiedin different areas (such as operator theory, complex analysis, andcombinatorial analysis). This kind of work does often have connectionswith more applied areas. The one specific applied area closely related to the operator theoryis control theory and optimization where operators have indeed found rather strikingapplications.