The Brown-Halmos theorem for a pair of abstract Hardy spaces

TL;DR

This paper extends the Brown-Halmos theorem to abstract Hardy spaces built on Banach function spaces, including variable Lebesgue and Lorentz spaces, under minimal assumptions on the spaces and boundedness of the Riesz projection.

Contribution

It proves an analogue of the Brown-Halmos theorem for Toeplitz operators between abstract Hardy spaces with broad applicability to various function spaces.

Findings

The theorem holds when $X$ is separable and the Riesz projection is bounded on $Y$.

Results are specified for variable Lebesgue spaces $L^{p(ullet)}$ and Lorentz spaces with Muckenhoupt weights.

The paper characterizes Toeplitz operators in these generalized Hardy space settings.

Abstract

Let $H[X]$ and $H[Y]$ be abstract Hardy spaces built upon Banach function spaces $X$ and $Y$ over the unit circle $\mathbb{T}$. We prove an analogue of the Brown-Halmos theorem for Toeplitz operators $T_a$ acting from $H[X]$ to $H[Y]$ under the only assumption that the space $X$ is separable and the Riesz projection $P$ is bounded on the space $Y$. We specify our results to the case of variable Lebesgue spaces $X=L^{p(\cdot)}$ and $Y=L^{q(\cdot)}$ and to the case of Lorentz spaces $X=Y=L^{p,q}(w)$, $1<p<\infty$, $1\le q<\infty$ with Muckenhoupt weights $w\in A_p(\mathbb{T})$.