{"title":"Weighted holomorphic mappings attaining their norms","authors":"A. Jiménez-Vargas","doi":"10.1007/s43034-023-00297-7","DOIUrl":null,"url":null,"abstract":"<div><p>Given an open subset <i>U</i> of <span>\\({\\mathbb {C}}^n,\\)</span> a weight <i>v</i> on <i>U</i> and a complex Banach space <i>F</i>, let <span>\\(\\mathcal {H}_v(U,F)\\)</span> denote the Banach space of all weighted holomorphic mappings <span>\\(f:U\\rightarrow F,\\)</span> under the weighted supremum norm <span>\\(\\left\\| f\\right\\| _v:=\\sup \\left\\{ v(z)\\left\\| f(z)\\right\\| :z\\in U\\right\\} .\\)</span> We prove that the set of all mappings <span>\\(f\\in \\mathcal {H}_v(U,F)\\)</span> that attain their weighted supremum norms is norm dense in <span>\\(\\mathcal {H}_v(U,F),\\)</span> provided that the closed unit ball of the little weighted holomorphic space <span>\\(\\mathcal {H}_{v_0}(U,F)\\)</span> is compact-open dense in the closed unit ball of <span>\\(\\mathcal {H}_v(U,F).\\)</span> We also prove a similar result for mappings <span>\\(f\\in \\mathcal {H}_v(U,F)\\)</span> such that <i>vf</i> has a relatively compact range.</p></div>","PeriodicalId":48858,"journal":{"name":"Annals of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2023-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s43034-023-00297-7.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s43034-023-00297-7","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Given an open subset U of \({\mathbb {C}}^n,\) a weight v on U and a complex Banach space F, let \(\mathcal {H}_v(U,F)\) denote the Banach space of all weighted holomorphic mappings \(f:U\rightarrow F,\) under the weighted supremum norm \(\left\| f\right\| _v:=\sup \left\{ v(z)\left\| f(z)\right\| :z\in U\right\} .\) We prove that the set of all mappings \(f\in \mathcal {H}_v(U,F)\) that attain their weighted supremum norms is norm dense in \(\mathcal {H}_v(U,F),\) provided that the closed unit ball of the little weighted holomorphic space \(\mathcal {H}_{v_0}(U,F)\) is compact-open dense in the closed unit ball of \(\mathcal {H}_v(U,F).\) We also prove a similar result for mappings \(f\in \mathcal {H}_v(U,F)\) such that vf has a relatively compact range.
期刊介绍:
Annals of Functional Analysis is published by Birkhäuser on behalf of the Tusi Mathematical Research Group.
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