Mathematical Sciences: Singly Generated Dual Algebras

这个项目的目的是积累更多的知识，从而更好地理解作用在无限维复Hilbert空间上的有界线性算子的结构。在过去的十年里，通过使用由单个算子生成的对偶代数的概念，在这个方向上取得了很大的进展。Pearcy教授等人对这种对偶代数的前偶进行了广泛的分析，这使得压缩算子的伸缩理论和不变子空间问题取得了长足的进步。皮尔西教授计划继续这个数学研究项目，现在专注于一类谱半径为1的压缩算子。这里要追求的数学是算子理论。运算符可以非常粗略地认为是浓缩的(实数或复数)数字。人们可以用运算符进行算术运算，就像处理数字一样，除了运算符的乘法通常取决于因子的取值顺序，并且并不是每个非零运算符都有逆运算。此外，数字本质上是一维的，而运算符通常作用于无限维空间。用算符而不是用数字来表示物理量，这一概念是量子力学的核心，该学科的发展为算符理论提供了很大的原始推动力。

The purpose of this project is to accumulate more knowledge leading to a better understanding of the structure of bounded linear operators acting on a complex Hilbert space of infinite dimension. In the past ten years much progress has been made in this direction by employing the concept of the dual algebra generated by a single operator. Professor Pearcy and others have made an extensive analysis of the predual of such a dual algebra, and this has led to considerable progress on the dilation theory of contraction operators, and on the invariant subspace problem. Professor Pearcy plans to continue this program of mathematical research, focusing now on the the class of contraction operators having spectral radius one. The kind of mathematics to be pursued here is operator theory. Operators may be thought of, very roughly, as enriched (real or complex) numbers. One can do arithmetic with operators just as with numbers, except that multiplication of operators depends in general on the order in which the factors are taken, and not every non-zero operator has an inverse. Furthermore, numbers are inherently one-dimensional, whereas operators generally act on an infinite-dimensional space. The notion that physical quantities are more realistically represented by operators than by numbers lies at the heart of quantum mechanics, a subject whose development provided much of the original impetus to operator theory.