Complex analysis and spectral theory

点击翻译按钮获取中文摘要

The goal of my research is to develop and apply methods in complex analysis and potential theory to solve problems in other areas of mathematics, in particular in operator theory and spectral theory. This proposal outlines three specific projects on which I plan to work in the next few years: (1) analytic capacity, (2) Dirichlet spaces and (3) pseudospectra. Each project contains a component for the training of research students or postdoctoral fellows. Analytic capacity measures the size of compact plane sets from the point of view of certain aspects of complex analysis. It was first introduced by Ahlfors in the 1940's in connection with the so-called Painlevé problem of characterizing removable singularities of bounded holomorphic functions. Later, in the 1960's, it played a key role in the solution of several fundamental problems in rational approximation. The last twenty years have seen significant advances in the understanding of analytic capacity, one of the most striking breakthroughs being Tolsa's proof of the long-standing conjecture that analytic capacity is semiadditive (the capacity of the union of two sets is bounded by C times the sum of their individual capacities, where C is a universal constant). However, whether analytic capacity is subadditive (can we take C=1?) still remains an open problem. I propose to attack this problem, exploiting recently developed expertise in the rigorous computation of analytic capacity. The Dirichlet space (together with its close relatives the Hardy and Bergman spaces) is one of the three fundamental Hilbert spaces of holomorphic functions on the unit disk. Even though it has been intensively studied over the years, a number of fundamental questions remain unresolved, notably the problem of characterizing the closed shift-invariant subspaces and the cyclic functions (the analogues in this setting of closed ideals and invertible elements). The characterization of the cyclic functions is the subject of a celebrated 30-year-old conjecture of Brown and Shields, on which I propose to work. I also plan to study the several-variable analogue of this conjecture in the special case of polynomials. The theory of pseudospectra is a modern outgrowth of classical eigenvalue analysis. The general idea is to analyse a matrix by studying how fast its resolvent blows up as one approaches the eigenvalues. In the case of non-normal matrices, this yields considerably more information than merely identifying the eigenvalues. Efficient computation methods have also made pseudospectra a practical tool in numerical linear algebra. It is therefore of interest to determine to what extent the pseudospectra of a matrix really determine its behavior. I plan to study this problem, both for pseudospectra and for certain higher-order generalizations.