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