A Toeplitz type operator on Hardy spaces in the unit ball

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 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 $Q_{\mu }:H^p\to H^q$ for the full range $1<p,q<\mathrm{\infty }$; 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.