A Toeplitz type operator on Hardy spaces in the unit ball

TL;DR

This paper investigates a Toeplitz type operator on Hardy spaces in the unit ball, providing a full characterization of its boundedness, compactness, and Schatten class membership, with applications demonstrating its utility.

Contribution

It introduces a comprehensive analysis of a Toeplitz type operator on Hardy spaces, including criteria for boundedness, compactness, and Schatten class membership, extending classical operator theory.

Findings

Complete characterization of boundedness and compactness of $Q_\mu$

Description of Schatten class membership for $H^2$

Applications illustrating the operator's usefulness

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 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.