Noncommutative Multivariable Operator Theory and Free Holomorphic Functions

在过去的二十年里，一个免费的类似物Sz。Nagy-Foias关于希尔伯特空间算子的n元组空间中的单位球的理论已经被研究者和其他人所追求。这一理论在n维复插值、多变量预测和熵优化、控制理论、系统理论、散射理论和小波理论等方面都有显著的应用。研究人员将继续进行正在进行的计划，以开发一个免费的类似物的Nagy-Foias理论的收缩更一般的非交换域和品种的空间中的n元组的运营商，在一个统一的框架，其中包括两个方面：非交换和交换。调查员将继续工作的功能理论的非交换球，重点是几何方面的理论自由全纯函数和连接与双曲几何。本研究的主要方向如下：（i）非交换域，通用模型，算子代数，和分类;（ii）酉不变量的非交换域;（iii）非交换双曲几何;（iv）自由全纯函数的非交换球。每个非交换域（分别为品种）进行研究是与一个普遍的模型，包括对运营商的Fock空间，域（分别。哈代，C*-）代数，和Reinhardt（resp. n维复空间中的圆域。提议者将工作的分类某些类别的非交换域代数，分类相应的非交换域自由双全纯映射，和连接的分类莱因哈特域。 研究者将试图发展一个非交换多域模型理论，并在此基础上建立一个曲率不变的理论，试图推广和统一现有的结果（交换和非交换）。对一个非交换域的元素到酉等价的分类的重大进展是预期的，所提出的研究的一个重要部分涉及酉不变量：特征函数，曲率和熵。研究人员将继续研究双曲几何的单位球的n元组的运营商在密切联系的理论自由全纯函数，并将试图扩大理论更一般的noncommutative domains.起源于概念的量化，运营商理论连接在一起的几个分支的数学，是密切相关的数学物理，并在工程中有许多应用.拟议的研究的动机是最近全球范围内的多变量算子理论和函数理论的非交换方面的兴趣，以及它们与多复变量全纯函数理论，算子代数的表示理论和Fock空间上的调和分析的相互作用。其目的是促进对这些相对较新的研究领域的理解，并与双曲复分析和代数几何建立新的联系。所得结果在多复变函数的插值和双全纯分类、系统理论和散射理论等方面具有潜在的应用价值。在控制理论、熵优化、小波理论和图像处理等领域也有潜在的影响。研究结果将在研究生研讨会、会议和讲习班上传播。

In the last two decades, a free analogue of Sz.-Nagy-Foias theory on the unit ball in the space of n-tuples of Hilbert space operators has been pursued by the investigator and others. This theory has already had remarkable applications in n-dimensional complex interpolation, multivariable prediction and entropy optimization, control theory, systems theory, scattering theory, and wavelet theory. The investigator will continue the ongoing program to develop a free analogue of the Nagy-Foias theory of contractions for more general noncommutative domains and varieties in the space of n-tuples of operators, in a uniform framework that includes both aspects: noncommutative and commutative. The investigator will continue to work on function theory on noncommutative balls, with the emphasis on the geometric aspects of the theory of free holomorphic functions and the connection with the hyperbolic geometry. The main directions of this proposed research are the following: (i) Noncommutative domains, universal models, operator algebras, and classification; (ii) Unitary invariants on noncommutative domains; (iii) Noncommutative hyperbolic geometry; (iv) Free holomorphic functions on noncommutative balls. Each noncommutative domain (resp. variety) to be studied is associated with a universal model consisting on operators on Fock spaces, a domain (resp. Hardy, C*-) algebra, and Reinhardt (resp. circular) domains in n-dimensional complex space. The proposer will work on the classification of certain classes of noncommutative domain algebras, the classification of the corresponding noncommutative domains up to free biholomorphic maps, and the connections with the classification of Reinhardt domains. The investigator will try to develop a model theory on noncommutative polydomains and formulate a theory of curvature invariant in this setting, in the attempt to extend and unify the existent results (commutative and noncommutative). Significant progress towards the classification of the elements of a noncommutative domain up to unitary equivalence is expected; an important part of the proposed research concerns unitary invariants: characteristic function, curvature, and entropy. The investigator will continue to study the hyperbolic geometry of the unit ball of n-tuples of operators in close connection with the theory free holomorphic functions, and will try to extend the theory to more general noncommutative domains.Originated from the concept of quantization, operator theory links together several branches of mathematics, is closely related to mathematical physics, and has numerous applications in engineering. The motivation of the proposed research is the recent worldwide interest in the noncommutative aspects of multivariable operator theory and function theory, and their interplay with the theory of holomorphic functions in several complex variables, the representation theory of operator algebras, and the harmonic analysis on Fock spaces. The objective is to advance the understanding of these relatively new areas of research and make new connections with hyperbolic complex analysis and algebraic geometry. The expected results have potential applications in interpolation and biholomorphic classification in several complex variables, systems theory, and scattering theory. Potential impact in fields such as control theory, entropy optimization, wavelet theory, and image processing is also expected. The results will be disseminated in graduate seminars, conferences and workshops.