One-sided invertibility of Toeplitz operators on the space of all holomorphic functions on finitely connected domains

Abstract

We prove that if the symbol of a Toeplitz operator acting on the space of all holomorphic functions on a finitely connected domain is non-degenerate and vanishes then the range of this operator is not complemented. As a result, we obtain that a Toeplitz operator on the space of all holomorphic functions on finitely connected domains is left invertible if and only if it is an injective Fredholm operator. Also, such an operator is right invertible if and only if it is a surjective Fredholm operator.