The essential norms of Toeplitz operators with symbols in $C+H^\infty$ on weighted Hardy spaces are independent of the weights

TL;DR

This paper demonstrates that for symbols in $C+H^ fty$, the essential norms of Toeplitz operators on weighted Hardy spaces are independent of the weights, extending previous results for unweighted spaces.

Contribution

It extends the known invariance of essential norms of Toeplitz operators with symbols in $C$ to symbols in $C+H^ fty$ across all weighted Hardy spaces with $A_p$ weights.

Findings

Essential norms of Toeplitz operators with symbols in $C+H^ Infty$ are the same on $H^p$ and $H^p(w)$ for all $w ext{ in }A_p$.

For $w$ in $A_2$, the essential norm equals the $L^ Infty$ norm of the symbol.

The invariance of essential norms holds across a broad class of weights, generalizing previous results.

Abstract

Let $1<p<\infty$, let $H^p$ be the Hardy space on the unit circle, and let $H^p(w)$ be the Hardy space with a Muckenhoupt weight $w\in A_p$ on the unit circle. In 1988, B\"ottcher, Krupnik and Silbermann proved that the essential norm of the Toeplitz operator $T(a)$ with $a\in C$ on the weighted Hardy space $H^2(\varrho)$ with a power weight $\varrho\in A_2$ is equal to $\|a\|_{L^\infty}$. This implies that the essential norm of $T(a)$ on $H^2(\varrho)$ does not depend on $\varrho$. We extend this result and show that if $a\in C+H^\infty$, then, for $1<p<\infty$, the essential norms of the Toeplitz operator $T(a)$ on $H^p$ and on $H^p(w)$ are the same for all $w\in A_p$. In particular, if $w\in A_2$, then the essential norm of the Toeplitz operator $T(a)$ with $a\in C+H^\infty$ on the weighted Hardy space $H^2(w)$ is equal to $\|a\|_{L^\infty}$.