21/03/2013, 16:30 — 17:30 — Sala P3.10, Pavilhão de Matemática
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
coefficient , satisfying some corona type condition, implies –
and in some cases is equivalent to – Fredholmness of the TO with
symbol . Moreover, explicit formulas for an appropriate
factorization of were obtained, allowing to solve different
RHPs with coefficient , and to determine the inverse, or a
generalized inverse, of the TO with symbol . 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 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 matrix
functions , for which the results regarding the case
have recently been generalized.