Mathematical Sciences: Dual Algebras of Operators and H-Infinity Control Theory

Li将利用对偶代数的概念来研究作用在无限维Hilbert空间上的有界线性算子的结构。特别是，对于更一般的一类算子和由多个算子生成的对偶代数，将考虑压缩算子的膨胀理论、不变子空间问题和自反性问题。李还将通过交换提升定理继续她的控制理论研究。这个项目的一般数学领域以希尔伯特空间算子的代数理论为基础。运算符可以被认为是复数的有限或无限矩阵。特殊类型的运算符通常放在一个代数中，自然称为运算符代数。这些抽象对象具有令人惊讶的各种应用。例如，它们在纽结理论中扮演着关键的角色，而纽结理论目前正被用于研究DNA的结构，并且它们在非对易几何中具有基本的重要性，这在物理学中变得越来越重要。

Li will use the concept of dual algebra to study the structure of bounded linear operators acting on infinite dimensional Hilbert space. In particular, the dilation theory of contraction operators, the invariant subspace problem, and the reflexivity problem will be considered for more general classes of operators and dual algebras generated by more than one operator. Li will also continue her study of control theory via the commutant lifting theorem. The general area of mathematics of this project has its basis in the theory of algebras of Hilbert space operators. Operators can be thought of as finite or infinite matrices of complex numbers. Special types of operators are often put together in an algebra, naturally called an operator algebra. These abstract objects have a surprising variety of applications. For example, they play a key role in knot theory, which in turn is currently being used to study the structure of DNA, and they are of fundamental importance in noncommutative geometry, which is becoming increasingly important in physics.