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

李将使用对偶代数的概念来研究结构 无穷维Hilbert上有界线性算子的 空间 特别是收缩的膨胀理论 算子、不变子空间问题和自反性 问题将被认为是更一般类的运营商 以及由多个算子生成的对偶代数。 李将 她还继续通过交换提升控制理论的研究 定理 这个项目的一般数学领域有其基础 希尔伯特空间算子的代数理论。 运营商 可以被认为是复数的有限或无限矩阵 号码 特殊类型的运算符通常放在 代数，自然称为算子代数。 这些抽象 物体的应用范围之广令人惊讶。 比如说， 它们在纽结理论中起着关键作用，而纽结理论目前 被用来研究DNA的结构， 在非对易几何中的重要性， 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.