Topics in Multivariable Operator Theory and Interpolation

本文的主要研究方向是：（1）Fock空间上的调和分析，（2）熵与多元插值，（3）自由半群代数上Hilbert模的数值不变量。本文结合多重Toeplitz算子和多重解析算子的因子分解、多变量插值和完全正映射的数值不变量，提出了Fock空间上算子熵的一个新概念.在第一个标题下还发现了一些与Fock空间上的调和分析、由左生成算子生成的非交换解析Toeplitz代数的单位球的几何、插值序列和多元费耶尔型不等式有关的问题。这些结果对多复变函数论、预测和多变量随机过程有潜在的应用。近年来，多变量插值方法取得了令人振奋的进展。PI将继续他在这一领域的研究工作，并期望找到几个多变量插值问题（Sarason，Caratheodory-Schur，Nevanlinna-Pick）的最大熵解，包括行压缩的抽象非交换提升定理。这一研究对多变量控制理论和系统理论的发展具有一定的指导意义。本文提出了一个新的算子不变量熵，它似乎是对曲率不变量的补充。其目标是在建立一套完整的数值不变量集方面取得重大进展，这些数值不变量集可以对大类完全正映射进行分类。算子理论起源于量子化的概念，将数学的几个分支联系在一起，与数学物理密切相关。本研究的动机是最近世界范围内对调和分析和多变量算子理论的非对易方面的兴趣。其目的是推进这些相对较新的研究领域的理解，并将结果应用于研究完全正映射及其不变量，函数理论和插值在几个变量，多变量线性系统和控制理论，预测和随机过程。在电子物理学和图像处理等领域也有潜在的应用前景。

The main directions of this proposed research are the following: (1) Harmonic analysis on Fock spaces, (2) Entropy and multivariable interpolation, (3) Numerical in-variants for Hilbert modules over free semigroup algebras. A new notion of entropy for operators on Fock spaces is proposed in connection with factorizations of multi-Toeplitz and multi-analytic operators, multivariable interpolation, and numerical invariants for completely positive maps. Under the first heading are also found a number of problems pertaining to harmonic analysis on Fock spaces, the geometry of the unit ball of non-commutative analytic Toeplitz algebras generated by the left creation operators, interpo-lation sequences, and Fejer type inequalities in several variables. These results have po-tential applications to function theory in several complex variables, prediction and multi-variate stochastic processes. In recent years, there has been exciting progress in multi-variable interpolation. The PI will continue his work in this area of research and expects to find the maximal entropy solutions of several multivariable interpolation problems (Sarason, Caratheodory-Schur, Nevanlinna-Pick) including the abstract noncommutative commu-tant lifting theorem for row contractions. This proposed research is expected to play a role in multivariable control theory and systems theory. A new invariant, entropy, is pro-posed for n-tuples of operators, that seems to complement the curvature invariant. The goal is to make significant progress towards a complete set of numerical invariants that classify large classes of completely positive maps.Originated from the concept of quantization, operator theory links together several branches of mathematics and is closely related to mathematical physics. The motivation of this research is the recent worldwide interest in the noncommutative aspect of har-monic analysis and multivariable operator theory. The objective is to advance the under-standing of these relatively new areas of research and apply the results to the study of completely positive maps and their invariants, function theory and interpolation in several variables, multivariable linear systems and control theory, and prediction and stochastic processes. Potential applications in fields such as geophysics and image processing are also expected.