On Toeplitz operators on $H^1(\mathbb{C}^+)$
Carlo Bellavita, Marco M. Peloso
TL;DR
This work studies Toeplitz operators with anti-analytic symbols on the Hardy space H^1(C^+) and shows that such operators are not bounded on the full space for non-constant symbols. To obtain meaningful boundedness, the authors introduce the subspace H^1_Theta = { f in H^1(C^+): <f, Theta> = 0 } and establish equivalent criteria for the boundedness of T_bar_Theta on H^1_Theta via the associated Hankel operator H_bar_Theta and the multiplier M_Theta on BMOA. For inner symbols, these criteria are further equivalent to the Beurling-type decomposition H^1_Theta = K^1_Theta ⊕ Theta H^1. They prove a positive result for meromorphic inner symbols of the form Theta = e^{i tau(x)} with tau > 0, showing T_bar_Theta is bounded from H^1_Theta to H^1(C^+). The paper concludes with open questions about extending boundedness to other inner thetas and explores real-variable analogues and related operator-theoretic questions.
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.