Multivariable operator theory and analytic operator spaces

摘要：本文主要研究高阶Hankel形式、乘积域上的相关解析函数理论、解析乘子代数上的算子空间结构以及相关的相似/扩张问题。高阶hankelform研究的主要目标之一是提供一个统一的框架来理解结构理论，同时充分利用其与上同调、代数几何、解析泛函理论和多变量算子理论的接口。所提出的算子空间理论的一个主要目标是确定乘数代数上与底层再现核和与该代数相关的“规范模型”的最小算子空间结构项完全同构的充分必要条件。汉克尔形式（或算子）的研究出现在预测理论、系统理论、插值和控制理论中。这些形式的高阶类似物出现在一个纯粹的数学框架中，即群表示。然而，这些形式的非交换类似物存在于加权的Fock空间上，并且与粒子物理学中左右创造算子的模型理论密切相关。算子代数理论是量子力学的数学框架，而对算子代数上的算子空间结构的研究是这一理论的“量子化”。

ABSTRACT:The proposed research involves the study of higher-order Hankel forms,related analytic function theory on product domains, operator spacestructures on algebras of analytic multipliers and relatedsimilarity/dilation problems. One of the main goals in the study of higher-order Hankelforms is to provide a unified framework in which to understand thestructure theory and, at the same time, fully exploitthe interface with cohomology, algebraic geometry, analytic functiontheory and multivariable operator theory. A main goal of the proposed operatorspace theory is to determine necessary and sufficient conditions on a multiplieralgebra to be completely isomorphic to the minimal operator space structure interms of the underlying reproducing kernel and the "canonical model"associated to the algebra.The study of Hankel forms (or operators) arises in prediction theory,systems theory, interpolation and control theory. Higher-order analogues of these forms arose in a purely mathematical framework, namely, group representations. However, non-commutative analogues of these forms live on weighted Fock space and are closely connected to the model theory of theleft and right creation operators from particle physics. The theory of operator algebras is the mathematical framework of quantum mechanics while the study of operator space structures on operator algebras is the "quantization" of this theory.