  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 $1<p,q<\infty$; 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.
