Collaborative Research in Operator Theory on Holomorphic Function Spaces

夏皮罗教授将继续与保罗·波尔登就现代线性算子理论和经典解析函数理论之间的相互作用所产生的问题进行长期合作。夏皮罗和波登将研究解析函数空间上自然出现的算子的范数、数值值域、可分解性和混沌行为等问题，最著名的是复合算子和后移交换的算子。线性算子理论起源于数学物理，特别是量子力学，最近在控制系统理论中得到了重要的应用。数值值域是平面的凸子集，已被证明对工程师确定与线性算子有关的控制系统的稳定性很有用。最近，动力系统的混沌行为已经成为跨越数学、物理、工程和生物学的一个重要课题。直到最近才发现，重要的一类线性算子可以产生混沌系统，这导致了算子理论和动力系统之间意想不到的联系。可分解性的概念-研究如何将复杂的线性系统分解成更简单的系统-最近被证明与这种混乱行为有惊人的联系。夏皮罗和波登的研究源于并促进了这些发展：预计本提案中概述的项目将进一步加深我们对动力系统和算符理论之间联系的理解。

Professor Shapiro will continue his long-term collaboration with Paul Bourdon on problems arising from the interaction between the modern theory of linear operators and the classical theory of analytic functions. Shapiro and Bourdon will study problems involving norms, numerical ranges, decomposability, and chaotic behavior for operators that occur naturally on spaces of analytic functions, most notably composition operators and operators that commute with backward shifts.The theory of linear operators originates from mathematical physics, especially from quantum mechanics, and has more recently found significant application in control system theory. The numerical range is a convex subset of the plane that has proven useful to engineers in determining the stability of control systems associated with linear operators. The chaotic behaviour of dynamical systems has recently become an important subject spanning mathematics, physics, engineering, and biology. It has only recently been discovered that significant classes of linear operators can give rise to chaotic systems, and this has led to unexpected connections between operator theory and dynamical systems. The idea of decomposability---the study of how to break a complicated linear system up into simpler ones---has recently been shown to have surprising connections with such chaotic behaviour. The research of Shapiro and Bourdon arises from, and has contributed to, these developments: it is anticipated that the project outlined in this proposal will further enhance our understanding of the connection between dynamical systems and operator theory.