Spaces of analytic functions and their operators

复分析和算子理论是数学的两个经典分支。几十年来，许多聪明的人给这三个领域带来了新的思想，也对旧的思想进行了完善和更好的解释。这就是为什么主要的开放问题是非常困难的，并抵制数学家的尝试到现在。近一个世纪以前，G. Hardy第一次打破了一个以他的名字命名的富函数空间：Hardy空间。从那时起，分析人员开始研究这些空间的不同方面，或者它们的近亲，如Bergman空间、模型子空间、de Branges-Rovnyak空间和Dirichlet空间。解析函数的空间已经有了坚实的基础。然而，在每种情况下，它都是一个活跃的研究领域，有许多悬而未决的问题使我们忙碌。对函数空间上算子的研究已被证明是卓有成效的。一方面，它揭示了周围空间的结构，帮助我们更好地理解其元素的性质。例如，一个元素的边界行为及其导数在边界上的给定点与一个算子的像有关。另一方面，我们可以利用函数空间的已知事实来回答算子理论中的一些问题。两个经典学科之间的相互作用是函数空间及其算子的主要特征，在某种意义上解释了它的丰富性和美感。函数空间在数学的其他分支以及科学和技术中也有重要的应用。一个著名的例子是h∞控制理论。字母H代表哈代空间。这一理论是由已故的Zames教授于80年代初在麦吉尔大学创立的，从那时起就改变了控制理论的世界。模型子空间，特别是paly - wiener空间，在数字通信中起着重要的作用。Blaschke产品用于过滤器设计。Hardy空间用于对激光束进行建模。在物理和工程中还有许多其他应用。本文讨论了复变函数理论及其算子前沿的一些重要开放性问题。因此，这方面的任何进展都将直接影响到科学和工程的许多其他领域。

Complex analysis and operator theory are two classical branches of mathematics. For several decades, many intelligent people have brought new ideas into thee areas and also polished and better explained the old ones. That is why the main open questions are very difficult and resisted the attempts of mathematicians by now. Almost a century ago, G. Hardy put the first break of a rich function space which bears his name: Hardy spaces. Since then, analysts have work on different aspects of these spaces, or their close relatives like Bergman spaces, model subspaces, de Branges-Rovnyak spaces and Dirichlet spaces. Spaces of analytic functions have now a solid foundation. Nevertheless, in each case, it is an active domain of research and there are numerous open questions which have kept us busy. Studying the operators on function spaces have proved to be very fruitful. On one hand, it sheds light to the structure of ambient space and helps us to better understand the properties of its elements. For example, the boundary behavior of an element and its derivative at a given point of the frontier is related to the image of an operator. On the other hand, we can exploit the known facts about function spaces to answer some questions in operator theory. The interplay between two classical disciplines is the main feature of function spaces and their operators and, in a sense, explains its richness and beauty. Function spaces have also found essential applications in other branches of mathematics as well as in science and technology. A celebrated example is the H-infintity control theory. The letter H stands for the Hardy space. This theory was founded by the late professor Zames in early 80's at McGill University, and since then has changed the world of control theory. Model subspaces, in particular the Paley-Wiener space, play an important role in digital communication. Blaschke products are used in filter design. Hardy spaces are used in modeling laser beams. There are numerous other applications in Physics and engineering. This proposal deals with some essential open questions in the frontiers of complex function theory and their operators. Hence, any progress on this line of research will directly effect many other fields in science and engineering.