Multivariable Operator Theory on Noncommutative Domains

摘要本文研究了非交换调和分析、算子代数和多变量插值等问题。该方案的框架主要是全Fock空间，某些非交换空间。交换)解析Toeplitz代数，以及Hilbert空间上所有有界线性算子的代数。讨论了非交换扩张理论，等距生成的C^*$代数上的泊松变换，以及交换提升定理。交换)多变量类似于一些经典结果。本文的主要研究方向是：Fock空间的谐波分析；算子、结构和数值不变量的幂有界序列中心交织提升、次优化和多变量解析插值；算子元组（非压缩）的扩张理论和多变量的非解析插值。这项研究的动机是最近世界范围内对谐波分析的非交换方面的兴趣，起源于量子化的概念，它将数学的几个分支联系在一起，并与数学物理密切相关。本研究的目的是促进对多变量算子理论这一相对较新的领域的理解，并将其中一些结果应用于研究完全正映射及其不变量、函数理论和多变量插值、多变量线性系统、散射、控制理论和元算子模型理论。

AbstractPopescuThe proposed research considers problems in noncommutative harmonic analysis, operator algebras, and interpolation in several variables. The framework of this proposal is mainly the full Fock space, certain noncommutative (resp. commutative) analytic Toeplitz algebras, and the algebra of all bounded linear operators on a Hilbert space. Noncommutative dilation theory, Poisson transforms on $C^*$-algebras generated by isometries, and commutant lifting theorems are considered in order to find noncommutative (resp. commutative) multivariable analogues to some classical results. The main directions of this proposed research are the following: harmonic analysis on Fock spaces; power bounded sequences of operators, structure, and numerical invariants; central intertwining lifting, suboptimization, and analytic interpolation in several variables; dilation theory for tuples of operators (noncontractions) and non-analytic interpolation in several variables. The motivation of this research is the recent worldwide interest in the noncommutative aspect of harmonic analysis originated from the concept of quantization which links together several branches of mathematics and is closely related to mathematical physics. The objective of this research is to advance the understanding of the relatively new area of multivariable operator theory and apply some of these results to the study of completely positive maps and their invariants, function theory and interpolation in several variables, multivariable linear systems, scattering, control theory, and model theory for tuples of operators.