{"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.
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Annals of Functional Analysis is published by Birkhäuser on behalf of the Tusi Mathematical Research Group.
Ann. Funct. Anal. is a peer-reviewed electronic journal publishing papers of high standards with deep results, new ideas, profound impact, and significant implications in all areas of functional analysis and all modern related topics (e.g., operator theory). Ann. Funct. Anal. normally publishes original research papers numbering 18 or fewer pages in the journal’s style. Longer papers may be submitted to the Banach Journal of Mathematical Analysis or Advances in Operator Theory.
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