{"title":"The essential norms of Toeplitz operators with symbols in $C+H^\\infty$ on weighted Hardy spaces are independent of the weights","authors":"Oleksiy Karlovych, Eugene Shargorodsky","doi":"arxiv-2409.03548","DOIUrl":null,"url":null,"abstract":"Let $1<p<\\infty$, let $H^p$ be the Hardy space on the unit circle, and let\n$H^p(w)$ be the Hardy space with a Muckenhoupt weight $w\\in A_p$ on the unit\ncircle. In 1988, B\\\"ottcher, Krupnik and Silbermann proved that the essential\nnorm of the Toeplitz operator $T(a)$ with $a\\in C$ on the weighted Hardy space\n$H^2(\\varrho)$ with a power weight $\\varrho\\in A_2$ is equal to\n$\\|a\\|_{L^\\infty}$. This implies that the essential norm of $T(a)$ on\n$H^2(\\varrho)$ does not depend on $\\varrho$. We extend this result and show\nthat if $a\\in C+H^\\infty$, then, for $1<p<\\infty$, the essential norms of the\nToeplitz operator $T(a)$ on $H^p$ and on $H^p(w)$ are the same for all $w\\in\nA_p$. In particular, if $w\\in A_2$, then the essential norm of the Toeplitz\noperator $T(a)$ with $a\\in C+H^\\infty$ on the weighted Hardy space $H^2(w)$ is\nequal to $\\|a\\|_{L^\\infty}$.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"24 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Functional Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.03548","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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