Spaces of analytic functions and their operators

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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.