Multivariable Operator Theory on Noncommutative Domains

AbstractPopescuThe建议的研究认为在非交换调和分析，算子代数，插值在几个变量的问题。 这个建议的框架主要是完整的Fock空间，某些非交换的（分别）。交换）解析Toeplitz代数，以及Hilbert空间上所有有界线性算子的代数。 非交换膨胀理论，泊松变换的$C^*$-代数生成的等距，和交换提升定理被认为是为了找到非交换（分别）。交换的）多变量类似的一些经典的结果。本研究的主要方向如下：Fock空间上的调和分析;幂有界序列的算子、结构和数值不变量;中心缠绕提升、次优化和解析插值在几个变量;膨胀理论的元组的运营商（noncontractions）和非解析插值在几个变量。这项研究的动机是最近世界范围内的兴趣，在非交换方面的谐波分析起源于量子化的概念，它连接在一起的几个分支的数学，并密切相关的数学物理。本研究的目的是推进了解相对较新的领域的多变量算子理论和应用这些结果的研究完全正的地图和他们的不变量，函数理论和插值在几个变量，多变量线性系统，散射，控制理论和模型理论的元组的运营商。

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.