{"title":"Composition operators with closed range on the Dirichlet space","authors":"Guangfu Cao, Li He","doi":"10.1007/s43037-024-00334-0","DOIUrl":null,"url":null,"abstract":"<p>It is well known that the composition operator on Hardy or Bergman space has a closed range if and only if its Nevanlinna counting function induces a reverse Carleson measure. Similar conclusion is not available on the Dirichlet space. Specifically, the reverse Carleson measure is not enough to ensure that the range of the corresponding composition operator is closed. However, under certain assumptions, we in this paper set the necessary and sufficient condition for a composition operator on the Dirichlet space to have closed range.</p>","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":1.1000,"publicationDate":"2024-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Banach Journal of Mathematical Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s43037-024-00334-0","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
It is well known that the composition operator on Hardy or Bergman space has a closed range if and only if its Nevanlinna counting function induces a reverse Carleson measure. Similar conclusion is not available on the Dirichlet space. Specifically, the reverse Carleson measure is not enough to ensure that the range of the corresponding composition operator is closed. However, under certain assumptions, we in this paper set the necessary and sufficient condition for a composition operator on the Dirichlet space to have closed range.
期刊介绍：
The Banach Journal of Mathematical Analysis (Banach J. Math. Anal.) is published by Birkhäuser on behalf of the Tusi Mathematical Research Group.
Banach J. Math. 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 operator theory and all modern related topics. Banach J. Math. Anal. normally publishes survey articles and original research papers numbering 15 pages or more in the journal’s style. Shorter papers may be submitted to the Annals of Functional Analysis or Advances in Operator Theory.
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