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

Research output: Contribution to journalArticle

Alexei Karlovich, Eugene Shargorodsky

Original languageEnglish
Pages (from-to)246-265
Number of pages20
JournalJournal of Mathematical Analysis and Applications
Volume472
Issue number1
Early online date13 Nov 2018
DOIs
Accepted/In press11 Nov 2018
E-pub ahead of print13 Nov 2018
PublishedApr 2019

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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 $w\in A_p(\T)$.

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