Abstract. We obtain a necessary and sufficient condition for a noninjective Toeplitz operator on $H^2$ of the unit disk to be surjective. The condition involves the extremal function for the kernel of the operator. The canonical right inverse of a surjective Toeplitz operator is shown to be a product of three Toeplitz operators.

AMS Subject Classification (1991): 30D55, 46C07, 46E22, 47B32, 47B35

Keyword(s): {Toeplitz operators, Devinatz--Widom theorem, de Branges--Rovnyak spaces, reproducing kernels, Helson--Szegő weight, extremal function}

Received May 3, 2004. (Registered under 5834/2009.)