Analytic Self Maps of the Disk and Related Operators

皮埃特罗·波吉-科拉迪尼。Pietro Poggi-Corradini教授的研究跨越了复数分析和泛函分析两个领域。研究对象一方面是定义在单位圆盘上的模数为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数分析(调和度量、极值距离、双曲度量等)。结果是双重的，至少在一对一映射和原点为吸引不动点的情况下是这样的。在描述相关复合算子的性质，如谱和本质谱半径的同时，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。映射与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。确定了以前没有考虑过的Koenigs映射的性质，例如它们的Hardy类，这些性质反过来又揭示了主题的内部工作原理。P.Poggi-Corradini打算从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。第二个问题是在Bergman空间而不是Hardy空间上提出类似的问题。最近，Bergman空间理论的许多基本方面受到了极大的关注。第三条研究路线涉及描述复合算符的谱的一般问题。很明显，谱的性质和线性化理论都依赖于Denjoy-Wolff点的位置。上述结果仅适用于该点无模1且乘子不为零的情况。因此，有空间来推广适用于新情况的技术。最后，这一研究可能有助于回答M.Heins提出的关于Denjoy-Wolff定理在任意黎曼曲面上的推广的问题和J.Cima提出的关于Cauchy-Stieltjes积分的一些问题。在更广泛的层面上，由于近年来在几个复变量的背景下对复合算符和复动力学都进行了深入的研究，因此观察复合算符的谱理论和圆盘边界上的解析映射的动力学之间的相互作用是否延续到球或多圆盘的自映射将是有趣的。更广泛地说，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，该领域在工程和应用科学(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单一函数或变换。创新的观点是通过重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。

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.