Noncommutative Multivariable Operator Theory and Free Holomorphic Functions

在过去的二十年里，Sz的自由模拟。-Nagy-Foias理论在Hilbert空间算子的n元空间中的单位球已经被研究者和其他人所追求。该理论已经在n维复杂插值、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得到了显著的应用。研究人员将继续进行正在进行的计划，在一个统一的框架中，为更一般的非交换域和n元算子空间中的变异开发一个自由的Nagy-Foias压缩理论，该框架包括两个方面：非交换和交换。研究者将继续研究非交换球的函数理论，重点是自由全纯函数理论的几何方面以及与双曲几何的联系。本研究的主要方向是：(i)非交换域、通用模型、算子代数和分类；（ii）非交换域上的酉不变量；非交换双曲几何；（iv）非交换球上的自由全纯函数。每个非交换域(如：所要研究的是一个由Fock空间上的算子组成的通用模型，Fock空间是一个域（见图1）。Hardy， C*-代数，和Reinhardt （resp.）。n维复空间中的圆域。本文将研究一类非交换域代数的分类，相应的非交换域到自由全纯映射的分类，以及与Reinhardt域分类的联系。研究者将尝试发展非交换多域的模型理论，并在此背景下制定曲率不变理论，试图扩展和统一现有的结果（交换和非交换）。期望在非交换域元的分类方面取得重大进展，直至酉等价；提出的研究的一个重要部分是关于酉不变量：特征函数，曲率和熵。研究者将继续研究n元算子的单位球的双曲几何与理论自由全纯函数的密切联系，并尝试将理论推广到更一般的非交换域。算子理论起源于量子化的概念，它把数学的几个分支联系在一起，与数学物理密切相关，在工程上有许多应用。本研究的动机是近年来世界范围内对多变量算子理论和函数理论的非交换方面的兴趣，以及它们与复变量全纯函数理论、算子代数的表示理论和Fock空间的调和分析的相互作用。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。研究结果在复杂变量插值和生物全纯分类、系统理论和散射理论等方面具有潜在的应用价值。对控制理论、熵优化、小波理论和图像处理等领域的潜在影响也有望实现。研究结果将在研究生讨论会、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.