On Toeplitz operators on $H^1({\mathbb{C}}^{+})$

Abstract

In this paper we consider Toeplitz operators with anti-analytic symbols on $H^1({\mathbb{C}}^{+})$. It is well known that there are no bounded Toeplitz operators $T_{\overline{\hat{I}}˜}:H^1({\mathbb{C}}^{+})\to H^1({\mathbb{C}}^{+})$, where $\hat{I}˜\in H^{\infty }({\mathbb{C}}^{+})$. We consider the subspace $H_{\hat{I}}^1˜=\left\{f\in H^1({\mathbb{C}}^{+}):{\int }_{\mathbb{R}}f\overline{\hat{I}}˜=0\right\}$ and show that it is natural to study the boundedness of $T_{\overline{\hat{I}}˜}:H_{\hat{I}}^1˜\to H^1({\mathbb{C}}^{+})$. We provide several different conditions equivalent to such boundedness. We prove that when $\hat{I}˜=e^{i\ddot{I}„(\cdot )}$, with $\ddot{I}„>0$ $T_{\overline{\hat{I}}˜}:H_{\hat{I}}^1˜\to H^1({\mathbb{C}}^{+})$ is bounded. Finally, we discuss a number of related open questions.

Article accepted in Proceedings of the American Mathematical Society