The essential norms of Toeplitz operators with symbols in $C+H^\infty$ on weighted Hardy spaces are independent of the weights
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916383388860416 |
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| author | Karlovych, Oleksiy Shargorodsky, Eugene |
| author_facet | Karlovych, Oleksiy Shargorodsky, Eugene |
| contents | 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, Böttcher, 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 $\|a\|_{L^\infty}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03548 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The essential norms of Toeplitz operators with symbols in $C+H^\infty$ on weighted Hardy spaces are independent of the weights Karlovych, Oleksiy Shargorodsky, Eugene Functional Analysis 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, Böttcher, 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 $\|a\|_{L^\infty}$. |
| title | The essential norms of Toeplitz operators with symbols in $C+H^\infty$ on weighted Hardy spaces are independent of the weights |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2409.03548 |