More on the density of analytic polynomials in abstract Hardy spaces

Alexei Karlovich*, Eugene Shargorodsky

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference paperpeer-review

2 Citations (Scopus)

Abstract

Let $\{F_n\}$ be the sequence of the Fej\'er kernels on the unit circle $\mathbb{T}$. The first author recently proved that if $X$ is a separable Banach function space on $\mathbb{T}$ such that the Hardy-Littlewood maximal
operator $M$ is bounded on its associate space $X'$, then $\|f*F_n-f\|_X\to 0$
for every $f\in X$ as $n\to\infty$. This implies that the set of analytic
polynomials $\mathcal{P}_A$ is dense in the abstract Hardy space $H[X]$ built upon a separable Banach function space $X$ such that $M$ is bounded on $X'$. In this note we show that there exists a separable weighted $L^1$ space $X$ such that the sequence $f*F_n$ does not always converge to $f\in X$ in the norm of $X$. On the other hand, we prove that the set $\mathcal{P}_A$ is dense in $H[X]$ under the assumption that $X$ is merely separable.
Original languageEnglish
Title of host publicationOperator Theory
Subtitle of host publicationAdvances and Applications
EditorsAlbrecht Böttcher, Daniel Potts, Peter Stollmann, David Wenzel
PublisherSpringer International Publishing
Pages319-329
Number of pages11
Volume268
DOIs
Publication statusPublished - 2018

Publication series

NameOperator Theory: Advances and Applications
Volume268
ISSN (Print)0255-0156
ISSN (Electronic)2296-4878

Keywords

  • Banach function space, abstract Hardy space, analytic polynomial, Fej\'er kernel

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