Operator Theory, Complex Analysis and Applications Seminar 

21/03/2013, 16:30 — 17:30 — Room P3.10, Mathematics Building
Cristina Câmara, Instituto Superior Técnico

A Riemann-Hilbert approach to Toeplitz operators and the corona theorem

Together with differential operators, Toeplitz operators (TO) constitute one of the most important classes of non-self adjoint operators , and they appear in connection with various problems in physics and engineering. The main topic of my presentation will be the interplay between TOs and Riemann-Hilbert problems (RHP), and the relations of both with the corona theorem. It has been shown that the existence of a solution to a RHP with $2\times 2$ coefficient $G$, satisfying some corona type condition, implies – and in some cases is equivalent to – Fredholmness of the TO with symbol $G$. Moreover, explicit formulas for an appropriate factorization of $G$ were obtained, allowing to solve different RHPs with coefficient $G$, and to determine the inverse, or a generalized inverse, of the TO with symbol $G$. However, those formulas depend on the solutions to 2 meromorphic corona problems. These solutions being unknown or rather complicated in general, the question whether the factorization of $G$ can be obtained without the corona solutions is a pertinent one. In some cases, it already has a positive answer; how to solve this question in general is open, and all the more so in the case of $n\times n$ matrix functions $G$, for which the results regarding the $2\times 2$ case have recently been generalized.