A Toeplitz-type operator on Hardy spaces in the unit ball

Abstract

We study a Toeplitz-type operator $Q_\mu$ between the holomorphic Hardy spaces $H^p$ and $H^q$ of the unit ball. Here the generating symbol $\mu$ is assumed to be a positive Borel measure. This kind of operator is related to many classical mappings acting on Hardy spaces, such as composition operators, the Volterra-type integration operators, and Carleson embeddings. We completely characterize the boundedness and compactness of $Q_\mu: H^p \rightarrow H^q$ for the full range $1<p, q<\infty$; and also describe the membership in the Schatten classes of $H^2$. In the last section of the paper, we demonstrate the usefulness of $Q_\mu$ through applications.