# King's College London

## The Brown-Halmos Theorem for a Pair of Abstract Hardy Spaces

Research output: Contribution to journalArticle

Alexei Karlovich, Eugene Shargorodsky

Original language English 246-265 20 Journal of Mathematical Analysis and Applications 472 1 13 Nov 2018 https://doi.org/10.1016/j.jmaa.2018.11.022 11 Nov 2018 13 Nov 2018 Apr 2019 Link to publication in Scopus

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• The_Brown_Halmos_Theorem_for_KARLOVICH_Firstonline13November2018_GREEN_AAM_CC_BY_NC_ND_.pdf, 423 KB, application/pdf

Uploaded date:13 Nov 2018

Version:Accepted author manuscript

Licence:CC BY-NC-ND

## Abstract

Let $H[X]$ and $H[Y]$ be abstract Hardy spaces built upon Banach function spaces $X$ and $Y$ over the unit circle $\T$. 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 $w\in A_p(\T)$.