On Toeplitz Operators on H1(C+)

In this paper we consider Toeplitz operators with anti-analytic symbols on H-1(C+). It is well known that there are no bounded Toeplitz operators T-Theta(sic): H1(C+)-+ H-1(C+), where Theta is an element of H-infinity(C+). We consider the subspace H-Theta(sic)(1) = {f is an element of H-1(C+): fR f Theta = 0} and show that it is natural to study the boundedness of T Theta : H Theta(1)((sic))-+ H-1(C+). We provide several different conditions equivalent to such boundedness. We prove that when Theta = e(i tau()), with tau > 0, T-Theta(sic) : H-Theta(sic)(1)+ H-1(C+) is bounded. Finally, we discuss a number of related open questions.