Hankel and Toeplitz transforms on \(H^1\): continuity, compactness and Fredholm properties
The paper is devoted to the study of continuity, compactness and Fredholm properties of Hankel operators \(H_\varphi\) and Toeplitz operators \(T_\varphi\) on the Hardy space \(H^1(\mathbb{T})\), where \(\mathbb{T}\) denotes the unit circle in the complex plane. A new proof for the characterization of the boundedness of \(H_\varphi\) (by means of the symbol \(\varphi\)) is provided. Additionally, the authors obtain a precise estimate for the operator norm of \(H_\varphi : H^1(\mathbb{T}) \rightarrow H^1(\mathbb{T})\) and a characterization of the compactness of \(H_\varphi \) on \(H^1(\mathbb{T})\). The Fredholm property of \(T_\varphi\) on \(H^1(\mathbb{T})\) is characterized in the case of not necessarily continuous functions \(\varphi\), and a formula for the corresponding Fredholm index of \(T_\varphi\) is also derived.
scientific article; zbMATH DE number 5617558