Commutative Algebraic Methods in Multivariable Operator Theory and Applications

本文的目标是围绕希尔伯特空间上的交换算子元组的维不变量和塞缪尔多重性，在算子理论中建立一个系统的交换代数不变量理论。这些代数不变量已被应用于算子理论中的纯解析问题，包括解析函数的Hilbert空间和Fredholm理论的研究。与Nevanllina-Pick籽粒的进一步联系将在拟议的研究中进行探讨。希尔伯特空间算子可以看作是经典对象的量子化，如全纯函数。算符理论为有限维空间中的问题提供了一种无限维的解决方法。该学科起源于为量子力学提供数学基础，并在电气工程的质量控制中找到了应用。本研究将交换代数（数学的另一个领域）的新技术引入算子理论，从而不仅丰富了这门学科，而且加强了数学中两门学科之间的联系。

The objective of this proposal is develop a systematic theory of commutative algebraic invariants in operator theory, centering around the dimension invariant and the Samuel multiplicity of a tuple of commuting operators acting on a Hilbert space. These algebraic invariants have found applications to purely analytic problems in operator theory including the study of Hilbert spaces of analytic functions, and Fredholm theory. Further connection with Nevanllina-Pick kernels will be explored in the proposed research.Hilbert space operators can be viewed as quantization of classical objects such as holomorphic functions. Operator theory provides an infinite dimensional approach to problems in a finite dimensional space. The subject originates from providing a mathematical foundation to Quantum Mechanics, and has found applications in quality control in Electrical Engineering. The proposed research will introduce new techniques from commutative algebra, another area of mathematics, into operator theory, hence not only enriching the subject, but also enhancing the ties between two subjects in mathematics.