Collaborative Research in Operator Theory on Holomorphic Function Spaces

夏皮罗教授将继续他的长期合作与保罗·波登之间的线性算子的现代理论和解析函数的经典理论的相互作用所产生的问题。Shapiro和Bourdon将研究的问题涉及规范，数值范围，可分解性，以及自然发生在解析函数空间上的算子的混沌行为，最著名的是复合算子和与后移交换的算子。线性算子的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发现了重要的应用。 数值范围是一个凸子集的平面，已被证明是有用的工程师在确定控制系统的稳定性与线性算子。近年来，动力系统的混沌行为已成为一个跨越数学、物理、工程和生物学的重要课题。直到最近才发现，重要的线性算子类可以产生混沌系统，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简单的系统--最近被证明与这种混沌行为有着惊人的联系。Shapiro和Bourdon的研究产生于这些发展，并为之做出了贡献：预计本提案中概述的项目将进一步增强我们对动力系统和算子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.