{"title":"巴拿赫空间的扩张理论方法","authors":"Swapan Jana, Sourav Pal, Saikat Roy","doi":"arxiv-2407.15112","DOIUrl":null,"url":null,"abstract":"For a complex Banach space $\\mathbb X$, we prove that $\\mathbb X$ is a\nHilbert space if and only if every strict contraction $T$ on $\\mathbb X$\ndilates to an isometry if and only if for every strict contraction $T$ on\n$\\mathbb X$ the function $A_T: \\mathbb X \\rightarrow [0, \\infty]$ defined by\n$A_T(x)=(\\|x\\|^2 -\\|Tx\\|^2)^{\\frac{1}{2}}$ gives a norm on $\\mathbb X$. We also\nfind several other necessary and sufficient conditions in this thread such that\na Banach sapce becomes a Hilbert space. We construct examples of strict\ncontractions on non-Hilbert Banach spaces that do not dilate to isometries.\nThen we characterize all strict contractions on a non-Hilbert Banach space that\ndilate to isometries and find explicit isometric dilation for them. We prove\nseveral other results including characterizations of complemented subspaces in\na Banach space, extension of a Wold isometry to a Banach space unitary and\ndescribing norm attainment sets of Banach space operators in terms of\ndilations.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A dilation theoretic approach to Banach spaces\",\"authors\":\"Swapan Jana, Sourav Pal, Saikat Roy\",\"doi\":\"arxiv-2407.15112\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For a complex Banach space $\\\\mathbb X$, we prove that $\\\\mathbb X$ is a\\nHilbert space if and only if every strict contraction $T$ on $\\\\mathbb X$\\ndilates to an isometry if and only if for every strict contraction $T$ on\\n$\\\\mathbb X$ the function $A_T: \\\\mathbb X \\\\rightarrow [0, \\\\infty]$ defined by\\n$A_T(x)=(\\\\|x\\\\|^2 -\\\\|Tx\\\\|^2)^{\\\\frac{1}{2}}$ gives a norm on $\\\\mathbb X$. We also\\nfind several other necessary and sufficient conditions in this thread such that\\na Banach sapce becomes a Hilbert space. We construct examples of strict\\ncontractions on non-Hilbert Banach spaces that do not dilate to isometries.\\nThen we characterize all strict contractions on a non-Hilbert Banach space that\\ndilate to isometries and find explicit isometric dilation for them. We prove\\nseveral other results including characterizations of complemented subspaces in\\na Banach space, extension of a Wold isometry to a Banach space unitary and\\ndescribing norm attainment sets of Banach space operators in terms of\\ndilations.\",\"PeriodicalId\":501114,\"journal\":{\"name\":\"arXiv - MATH - Operator Algebras\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-07-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Operator Algebras\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2407.15112\",\"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 - Operator Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.15112","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
For a complex Banach space $\mathbb X$, we prove that $\mathbb X$ is a
Hilbert space if and only if every strict contraction $T$ on $\mathbb X$
dilates to an isometry if and only if for every strict contraction $T$ on
$\mathbb X$ the function $A_T: \mathbb X \rightarrow [0, \infty]$ defined by
$A_T(x)=(\|x\|^2 -\|Tx\|^2)^{\frac{1}{2}}$ gives a norm on $\mathbb X$. We also
find several other necessary and sufficient conditions in this thread such that
a Banach sapce becomes a Hilbert space. We construct examples of strict
contractions on non-Hilbert Banach spaces that do not dilate to isometries.
Then we characterize all strict contractions on a non-Hilbert Banach space that
dilate to isometries and find explicit isometric dilation for them. We prove
several other results including characterizations of complemented subspaces in
a Banach space, extension of a Wold isometry to a Banach space unitary and
describing norm attainment sets of Banach space operators in terms of
dilations.