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 $T_{\overline{\Theta}}\colon H^1(\mathbb{C}^+) \to H^1(\mathbb{C}^+)$, where $\Theta \in H^\infty(\mathbb{C}^+)$. We consider the subspace $H^1_{\Theta}=\left\lbrace f \in H^1(\mathbb{C}^+)\colon \int_{\mathbb{R}}f \overline{\Theta}=0\right\rbrace$ and show that it is natural to study the boundedness of $T_{\overline{\Theta}}\colon H^1_\Theta \to H^1(\mathbb{C}^+)$. We provide several different conditions equivalent to such boundedness. We prove that when $\Theta=e^{i\tau (\cdot)}$, with $\tau>0$ $T_{\overline{\Theta}}\colon H^1_\Theta \to H^1(\mathbb{C}^+)$ is bounded. Finally, we discuss a number of related open questions.