Fredholm properties of a class of Toeplitz operators

Cristina Diogo

ISCTE -- Instituto Superior de Ciências do Trabalho e da Empresa
Departamento de Métodos Quantitativos
Lisboa

In this talk the Riemann-Hilbert problem with matrix coefficient $G\in (L^{\infty}(\mathbb{R})^{2\times 2}$ is considered, assuming the existence of a non trivial solution $(\phi_+, \phi_-)$ with $\phi_{\pm}$ belonging to the Hardy spaces $H_{\infty}(\mathbb{C}^{\pm})$ and such that $\phi_+$ or $\phi_{-}$ vanishes in some point of the corresponding half-plane $\mathbb{C}^+$ or $\mathbb{C}^-$, respectively. The results are used to study the Fredholm properties of Toeplitz operators with such a matrix symbol $G$ in the Hardy spaces $H_2(\mathbb{C}^{\pm})$.