Complex Analysis and Spectral Theory

点击翻译按钮获取中文摘要

The long-term objective of my research is to gain a better understanding of the interactions between complex analysis and operator theory. In recent years my work has focused on holomorphic function spaces. These are closely connected to operators. Indeed, the functions that act on a given operator typically emerge from a suitably chosen holomorphic function space. In the other direction, holomorphic function spaces are often best understood in terms of concrete operators that act on them. The proposed research program pursues both lines of enquiry. The first part of the proposal is to study Hilbert-space operators via their numerical range. It comprises two interlinked projects, one concerning mapping theorems for numerical range, and the other to prove the so-called Crouzeix conjecture. A proof of this conjecture would very like lead to an improved structure theorem for operators on a Hilbert space. The second part of the proposal consists of three projects concerning function spaces. The first project is to establish an automatic continuity theorem, and thereby extend the classic Gleason-Kahane-Zelazko theorem to a wide variety of function spaces. The second is to find constructive methods for polynomial approximation in certain function spaces where these methods are currently lacking. And the third project is to investigate a curious, recently-discovered phenomenon arising in the Nyman-Beurling approach to the Riemann hypothesis. For no apparent reason, at least the first billion entries of a certain infinite triangular matrix associated to the Riemann zeta function are all positive. Is this is true of all the entries? If so, what are the implications, if any, for the Riemann hypothesis?