Toeplitz and Hankel operators between distinct Hardy spaces

TL;DR

This paper characterizes Toeplitz and Hankel operators between different Hardy spaces, providing symbol characterizations, general theorems, and bounds on noncompactness, extending classical results to more general Hardy space settings.

Contribution

It introduces new characterizations and generalizations of Toeplitz and Hankel operators between distinct Hardy spaces, including arbitrary rearrangement invariant spaces.

Findings

Characterization of symbols for Toeplitz and Hankel operators

General versions of Brown-Halmos and Nehari theorems

Lower bounds for measure of noncompactness of Toeplitz operators

Abstract

The paper gives the background for Toeplitz $T_a$ and Hankel $H_a$ operators acting between distinct Hardy type spaces over the unit circle $\mathbb{T}$. We characterize possible symbols of such operators and prove general versions of Brown-Halmos and Nehari theorems. The lower bound for measure of noncomactness of Toeplitz operator is also found. Our approach allows Hardy spaces associated with arbitrary rearrangement invariant spaces, but part of the results is new even for the classical case of $H^p$ spaces.