Analytic Self Maps of the Disk and Related Operators

磁盘及相关操作符的分析自映射

• 批准号: 9706408
• 负责人: Pietro Poggi-Corradini
• 金额: $ 5.13万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-06-01 至 1998-09-01
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706408&HistoricalAwards=false
• 关键词: Analytic; Self; Maps; Disk; Related

ABSTRACT Pietro Poggi-Corradini. The research of Professor Pietro Poggi-Corradini stretches across the fields of complex analysis and functional analysis. The objects of study are on the one hand the analytic maps defined on the unit disk that are bounded by one in modulus, and on the other hand, the composition operators that these maps induce on classical function spaces of the disk. The motivations and the questions come from the theory of operators, but the tools used are drawn from complex analysis (harmonic measure, extremal distance, hyperbolic metric, etc...). The results, at least in the case when the map is one-to-one and has the origin as an attracting fixed point, are twofold. While describing properties of the associated composition operator, such as the spectrum and the essential spectral radius, new properties of the analytic map, such as the dynamical behavior near the boundary of the unit disk, were discovered. The link between the map and the operator is given by the linearization map near the fixed point provided by iteration theory, i.e. the Koenigs map. Properties of the Koenigs maps which had not been considered before, such as their Hardy class, are determined and these in turn shed light on the inner workings of the subject. P. Poggi-Corradini intends to continue his study in three different directions. The first consists in dropping the requirement that the maps be one-to-one. A second topic of inquiry is to ask similar questions on the Bergman spaces instead of the Hardy spaces. Many fundamental aspects of the theory of Bergman spaces have received much attention lately. A third line of research deals with the general problem of describing the spectrum of composition operators. It is clear that the properties of the spectrum and the linearization theory both depend on the location of the Denjoy-Wolff point. The results mentioned above only dealt with the case when this point does not have modulus one and the multiplier there is non-vanishing. So ther e is space for generalizing the techniques used to new situations. Finally, this study might help answer a question of M. Heins about generalizations of the Denjoy-Wolff Theorem to arbitrary Riemann surfaces and some questions of J. Cima about Cauchy-Stieltjes integrals. On a wider level, since, in recent years, both composition operators and complex dynamics have been studied intensively in the context of several complex variables, it would be interesting to see if the interplay between spectral theory of composition operators and the dynamics of analytic maps at the boundary of the disk carries over to self-maps of the ball or the polydisk. More generally, the research of P. Poggi-Corradini tackles fundamental issues in a classical mathematical field known as function theory, which has many applications to engineering and the applied sciences (e.g. iterative methods, aerodynamics, control theory, etc...). Much of the classical theory is devoted to the study of a single function or transformation. The innovative point of view is to extract information about a single transformation by iterating it repeatedly and analyzing the behavior "in the limit" of the ensuing dynamical system.

摘要 皮埃特罗·波吉-科拉迪尼 Pietro Poggi-Corradini教授的研究横跨复分析和泛函分析领域。研究对象一方面是定义在单位圆盘上的模为1的解析映射，另一方面是这些映射在圆盘的经典函数空间上诱导的复合算子。动机和问题来自算子理论，但使用的工具来自复分析（调和测度，极值距离，双曲度量， 等）。至少在地图是一对一的并且原点是吸引不动点的情况下，结果是双重的。在描述伴随复合算子的谱、本质谱半径等性质的同时，发现了解析映射的新性质，如单位圆盘边界附近的动力学行为.映射和算子之间的联系由迭代理论提供的不动点附近的线性化映射，即Koenigs映射给出。以前没有考虑过的Koenigs映射的性质，如它们的哈代 类，是确定的，这些反过来又揭示了主题的内部运作。P. Poggi-Corradini打算在三个不同的方向继续他的研究。第一个是放弃映射是一对一的要求。 第二个主题的调查是问类似的问题伯格曼空间，而不是哈代空间。伯格曼空间理论的许多基本方面最近受到了广泛的关注。 第三条线的研究涉及的一般问题，描述谱的组成运营商。很明显，谱的性质和线性化理论都依赖于Denjoy-Wolff点的位置。上述结果只处理了该点不具有模1且其乘子不为零的情况。因此，有空间来推广用于新情况的技术。最后，本研究可能有助于回答M. Heins关于Denjoy-Wolff定理在任意Riemann曲面上的推广以及J. Cima关于Cauchy-Stieltjes积分的一些问题。在更广泛的层面上，由于近年来，复合算子和复动力学都在多个复变量的背景下得到了深入研究，因此很有趣的是，看看复合算子的谱理论与边界处解析映射的动力学之间的相互作用。圆盘延续到球或多圆盘的自映射。 更一般地说，P. Poggi-Corradini的研究解决了被称为函数论的经典数学领域的基本问题，该理论在工程和应用科学（例如迭代方法，空气动力学，控制理论等）中有许多应用。许多经典理论都致力于研究一个单一的函数或变换。创新的观点是通过反复迭代并分析随后的动力系统的“极限”行为来提取关于单个变换的信息。