Toeplitz operators on the space of all holomorphic functions on finitely connected domains

Abstract

We define and study Toeplitz operators on the Fréchet space of all holomorphic functions on finitely connected domains in the Riemann sphere. We completely characterize Fredholm, semi-Fredholm and invertible operators belonging to this class. As a result, we obtain a characterization of these classes of operators in the unit disk case. As a motivation we formulate and analyze the Riemann–Hilbert problem in the space of all holomorphic functions on the domains which we consider.