Multivariable operator theory and analytic operator spaces

摘要：本研究涉及高阶Hankel形式的研究、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性/伸缩问题。 高阶Hankel型研究的主要目标之一是提供一个统一的框架，在这个框架中理解结构理论，同时，充分利用与上同调、代数几何、解析函数理论和多元算子理论的接口。 算子空间理论的一个主要目标是确定乘子代数与极小算子空间结构完全同构的充分必要条件，其中包含与乘子代数相关的再生核和“正则模型“，Hankel形式（或算子）的研究起源于预测理论、系统理论、插值和控制理论。 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，这些形式的非对易类似物存在于加权Fock空间中，并且与粒子物理学中的左和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.