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.