Project: PTDC/MAT/81385/2006 Toeplitz
Operators, Factorization and Corona Problems
Reasearch team: Maria
Cristina Câmara (coordinator)
António Francisco Ferreira dos Santos (IST)
Frank-Ohlme Speck (IST)
Maria Teresa Malheiro (U. Minho)
Maria do Carmo Martins (U. Açores)
Cristina Diogo (ISCTE)
Abstract: The present
project appears in the context of fundamental research in
Mathematics, taking however its motivation from applications in Physics
and Engineering. Its objective is of an interdisciplinary nature. It
aims at increasing the knowledge on some classes of operators,
Riemann-Hilbert problems and factorization of functions in such a way
that progress in one topic leads to advances in the others. It also
intends to give better insights into some problems in the theory of
operator algebras, as it happens when the corona theorem, which
originally appeared as a purely function-theoretic result, is
formulated as a characterization of a dense subset of the maximal ideal
space of the algebra of bounded analytic functions in the unit disk.
The basic mathematical tools come from several areas, such as
Algebra, Real and Complex Analysis, Operator Theory, Riemann Surfaces.
The main aim of this project is to study the invertibility
and Fredholm properties of new classes of Toeplitz operators, in
particular those suggested by applications in Physics and Engineering.
It intends, with this purpose, to develop innovative methods to
simplify and solve associated Riemann-Hilbert problems, namely by
defining appropriate new types of factorization of functions. It also
intends to generalize the corona theorem and establish a relation
between this generalization and the properties of the operators under
study. These results are expected to shed new light on the
interpretation and understanding of some non-standard corona problems
in an operator-algebraic form.