Multivariable Operator Theory and Several Complex Variables

摘要：作为本项目的主要活动，Salinas计划继续在多变量算子理论的特定领域进行研究。这些包括：n维复空间中（光滑有界）伪凸域上Bergman空间的解析子模（Hilbert和Frechet）和商模；该类域光滑到边界的有界全纯映射在该类模上导出的n元组的谱图和C*-代数性质（如本质正态性和型一性）；相应诱导扩展的平滑性；具有非阿贝尔横向对称的伪凸域（如广义Reinhardt域）上的Toeplitz C*-代数，这类C*-代数的k -理论的计算，以及光滑kk -理论与光滑相对拟对角性的关系。他还打算考虑一种新的几何现象；即，在给定算子T的“数值范围”内的无限曲率点产生T的广义表示谱中的点。Salinas还计划研究在光滑有界伪凸域上某些二阶算子的平滑性，其中Bergman投影不是全局正则的。这个项目的目标是探索和利用数学中两个丰富而历史悠久的领域，算子理论和多复变量函数理论之间的重要相互作用。作为数学跨学科活动的一个新出现的焦点，这个领域充满了兴奋。从纯数学的角度来看，促进交叉授粉自然是有利的，因为它会增加算子理论家和复杂分析人员在各自的子学科中攻击基本问题的工具数量，但这种结合的成功超出了所有人的预期。从更实际和具体的角度来看，这两个领域的结合具有重要意义，因为它为理论和实验物理以及工程提供了更大的应用潜力。例如，作为航天飞机自动着陆软件开发的关键部分，算子理论和多复变量理论已经成功地应用于控制理论中。Salinas的活动旨在为盲人教师创造可供选择的教学方法，他在科学8点盲文的发展中继续发挥作用，为人力资源开发领域的项目增加了价值。

Proposal: DMS-9970518Principal Investigator: Norberto SalinasAbstract: As the main activity of this project, Salinas plans to continue his research in specific areas of multivariable operator theory. These include: analytic (Hilbert and Frechet) submodules and quotient modules of the Bergman space over a (smoothly bounded) pseudoconvex domain in n-dimensional complex space; the spectral picture and C*-algebraic properties (such as essential normality and type oneness) of the n-tuples induced on modules of this type by bounded holomorphic mappings of such domains that are smooth up to the boundary; the smoothness of the corresponding induced extensions; Toeplitz C*-algebras over pseudoconvex domains with nonabelian transverse symmetries (such as generalized Reinhardt domains), the calculation of the K-theory of such C*-algebras, and the relationship between the smooth KK-theory and smooth relative quasidiagonality. He also intends to consider a new geometric phenomenon; namely, points of infinite curvature in the "numerical range" of a given operator T that produce points in the generalized representation spectrum of T. Salinas also plans to investigate the smoothness of certain operators of second order on smoothly bounded pseudoconvex domains where the Bergman projection is not globally regular.The goal of this project is to explore and to exploit the important interplay between two rich and time-honored areas of mathematics, operator theory and the theory of functions of several complex variables. A newly emerging focal point of mathematical interdisciplinary activity, this area is brimming with excitement. From a purely mathematical point of view, it was naturally advantageous to foster the cross-pollination because it would increase the number of tools that operator theorists and complex analysts alike have to attack fundamental problems in their respective subdisciplines, but the success of the marriage has exceeded all expectations. From a more practical and concrete vantage point, the combination of the two fields is significant because of the expanded potential it offers for applications to both theoretical and experimental physics, as well as to engineering. For example, operator theory and the theory of several complex variables have already been used successfully in control theory as a crucial part of the development of the software that automatically lands the space shuttle. The activities of Salinas aimed at creating alternative teaching methods for use by blind instructors and his continuing role in the evolution of scientific 8-dot Braille add value to the project in the area of human resource development.