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

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 $T_{\bar{\mathrm{\Theta }}}\text{\hspace{0.17em}⁣}:H^1({\mathbb{C}}^{+})\to H^1({\mathbb{C}}^{+})$, where $\mathrm{\Theta }\in H^{\mathrm{\infty }}({\mathbb{C}}^{+})$. We consider the subspace $H_{\mathrm{\Theta }}^1=\{f\in H^1({\mathbb{C}}^{+})\text{\hspace{0.17em}⁣}:{\int }_{\mathbb{R}}f\bar{\mathrm{\Theta }}=0\}$ and show that it is natural to study the boundedness of $T_{\bar{\mathrm{\Theta }}}\text{\hspace{0.17em}⁣}:H_{\mathrm{\Theta }}^1\to H^1({\mathbb{C}}^{+})$. We provide several different conditions equivalent to such boundedness. We prove that when $\mathrm{\Theta }=e^{i\tau (\cdot )}$, with $\tau >0$ $T_{\bar{\mathrm{\Theta }}}\text{\hspace{0.17em}⁣}:H_{\mathrm{\Theta }}^1\to H^1({\mathbb{C}}^{+})$ is bounded. Finally, we discuss a number of related open questions.