加权哈代空间上符号为 $C+H^\infty$ 的托普利兹算子的基本规范与权重无关

arXiv - MATH - Functional Analysis Pub Date : 2024-09-05 DOI:arxiv-2409.03548

Oleksiy Karlovych, Eugene Shargorodsky

{"title":"加权哈代空间上符号为 $C+H^\\infty$ 的托普利兹算子的基本规范与权重无关","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":"{\"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}","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}