{"title":"Cyclic Composition operators on Segal-Bargmann space","authors":"G. Ramesh, B. S. Ranjan, D. Naidu","doi":"10.1515/conop-2022-0133","DOIUrl":null,"url":null,"abstract":"Abstract We study the cyclic, supercyclic and hypercyclic properties of a composition operator Cϕ on the Segal-Bargmann space ℋ(ℰ), where ϕ(z) = Az + b, A is a bounded linear operator on ℰ, b ∈ ℰ with ||A|| ⩽ 1 and A*b belongs to the range of (I – A*A)½. Specifically, under some conditions on the symbol ϕ we show that if Cϕ is cyclic then A* is cyclic but the converse need not be true. We also show that if Cϕ* is cyclic then A is cyclic. Further we show that there is no supercyclic composition operator on the space ℋ(ℰ) for certain class of symbols ϕ.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"9 1","pages":"127 - 138"},"PeriodicalIF":0.3000,"publicationDate":"2020-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Concrete Operators","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/conop-2022-0133","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
Abstract We study the cyclic, supercyclic and hypercyclic properties of a composition operator Cϕ on the Segal-Bargmann space ℋ(ℰ), where ϕ(z) = Az + b, A is a bounded linear operator on ℰ, b ∈ ℰ with ||A|| ⩽ 1 and A*b belongs to the range of (I – A*A)½. Specifically, under some conditions on the symbol ϕ we show that if Cϕ is cyclic then A* is cyclic but the converse need not be true. We also show that if Cϕ* is cyclic then A is cyclic. Further we show that there is no supercyclic composition operator on the space ℋ(ℰ) for certain class of symbols ϕ.
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