Spaces of analytic functions and their operators

复分析和算子论是数学的两个经典分支。几十年来，许多聪明人给这些领域带来了新的想法，也完善和更好地解释了旧的想法。这就是为什么到目前为止，主要的开放问题非常困难，并抵制了数学家的尝试。近一个世纪前，G.哈代提出了一个以他的名字命名的丰富函数空间的第一个突破：哈代空间。从那时起，分析家们一直在研究这些空间的不同方面，或它们的近亲，如Bergman空间、模型子空间、de Brange-Rovnyak空间和Dirichlet空间。解析函数空间现在有了坚实的基础。然而，在每一种情况下，这都是一个活跃的研究领域，有许多悬而未决的问题一直让我们忙于工作。对函数空间上的算子的研究已被证明是非常有成果的。一方面，它揭示了环境空间结构，并帮助我们更好地了解其元素的属性。例如，元素及其导数在边界上给定点的边界行为与操作符的图像有关。另一方面，我们可以利用关于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函数空间的丰富性和美观性。函数空间在数学的其他分支以及科学技术中也有重要的应用。一个著名的例子是H-无穷控制理论。字母H代表哈代空格。这一理论由已故教授赞姆斯于80年代初在麦吉尔大学的S教授创立，从此改变了控制理论的世界。模型子空间，特别是Paley-Wiener空间，在数字通信中扮演着重要的角色。Blaschke产品用于过滤器设计。哈代空间用于模拟激光光束。在物理学和工程学中还有许多其他的应用。这一建议涉及复函数论及其算子的前沿领域中的一些基本的开放问题。因此，这一研究领域的任何进展都将直接影响到科学和工程领域的许多其他领域。

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.