Abstract
Let $X$ be a translation-invariant Banach function space on the unit circle
and let $H[X]$ be the abstract Hardy space built upon $X$.
We suppose the Riesz projection $P$ is bounded on $X$ and estimate the
essential norms $\|T(a)\|_{\mathcal{B}(H[X]),\mathrm{e}}$ of Toeplitz
operators $T(a)f:=P(af)$ with $a\in C+H^\infty$. We prove that in this case
\[
\|a\|_{L^\infty}
\le
\|T(a)\|_{\mathcal{B}(H[X]),\mathrm{e}}
\le
\min\left\{2,\|P\|_{\mathcal{B}(X)}\right\} \|a\|_{L^\infty},
\]
extending the results by the second author \cite{S21} for classical Hardy
spaces $H^p=H[L^p]$, $1<p<\infty$. In contrast to our previous works
\cite{S21} and \cite{KS24}, we do not assume that $X$ is reflexive or
separable, which complicates the matters, but allows us to include the
Hardy-Lorentz spaces $H^{p,q}=H[L^{p,q}]$ with $1<p<\infty$ and
$q=1,\infty$ into consideration.
and let $H[X]$ be the abstract Hardy space built upon $X$.
We suppose the Riesz projection $P$ is bounded on $X$ and estimate the
essential norms $\|T(a)\|_{\mathcal{B}(H[X]),\mathrm{e}}$ of Toeplitz
operators $T(a)f:=P(af)$ with $a\in C+H^\infty$. We prove that in this case
\[
\|a\|_{L^\infty}
\le
\|T(a)\|_{\mathcal{B}(H[X]),\mathrm{e}}
\le
\min\left\{2,\|P\|_{\mathcal{B}(X)}\right\} \|a\|_{L^\infty},
\]
extending the results by the second author \cite{S21} for classical Hardy
spaces $H^p=H[L^p]$, $1<p<\infty$. In contrast to our previous works
\cite{S21} and \cite{KS24}, we do not assume that $X$ is reflexive or
separable, which complicates the matters, but allows us to include the
Hardy-Lorentz spaces $H^{p,q}=H[L^{p,q}]$ with $1<p<\infty$ and
$q=1,\infty$ into consideration.
| Original language | English |
|---|---|
| Article number | 103599 |
| Number of pages | 17 |
| Journal | BULLETIN DES SCIENCES MATHEMATIQUES |
| Volume | May 2025 |
| Early online date | 28 Feb 2025 |
| Publication status | Published - 6 Mar 2025 |
Keywords
- Toeplitz operator
- Essential norm
- abstract Hardy space
- translation-invariant Banach function space
- restricted dual compact approximation property