On the essential norms of Toeplitz operators on abstract Hardy spaces built upon Banach function spaces
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917758695899136 |
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| author | Karlovych, Oleksiy Shargorodsky, Eugene |
| author_facet | Karlovych, Oleksiy Shargorodsky, Eugene |
| contents | Let $X$ be a Banach function space over the unit circle such that the Riesz projection $P$ is bounded on $X$ and let $H[X]$ be the abstract Hardy space built upon $X$. We show that the essential norm of the Toeplitz operator $T(a):H[X]\to H[X]$ coincides with $\|a\|_{L^\infty}$ for every $a\in C+H^\infty$ if and only if the essential norm of the backward shift operator $T(\mathbf{e}_{-1}):H[X]\to H[X]$ is equal to one, where $\mathbf{e}_{-1}(z)=z^{-1}$. This result extends an observation by Böttcher, Krupnik, and Silbermann for the case of classical Hardy spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_13907 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the essential norms of Toeplitz operators on abstract Hardy spaces built upon Banach function spaces Karlovych, Oleksiy Shargorodsky, Eugene Functional Analysis Let $X$ be a Banach function space over the unit circle such that the Riesz projection $P$ is bounded on $X$ and let $H[X]$ be the abstract Hardy space built upon $X$. We show that the essential norm of the Toeplitz operator $T(a):H[X]\to H[X]$ coincides with $\|a\|_{L^\infty}$ for every $a\in C+H^\infty$ if and only if the essential norm of the backward shift operator $T(\mathbf{e}_{-1}):H[X]\to H[X]$ is equal to one, where $\mathbf{e}_{-1}(z)=z^{-1}$. This result extends an observation by Böttcher, Krupnik, and Silbermann for the case of classical Hardy spaces. |
| title | On the essential norms of Toeplitz operators on abstract Hardy spaces built upon Banach function spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2408.13907 |