{"title":"L2空间中密定义复合算子的特定有界积的正态性和拟不规则性","authors":"Hang Zhou","doi":"10.1515/conop-2022-0130","DOIUrl":null,"url":null,"abstract":"Abstract Let (X, 𝒜, μ) be a σ−finite measure space. A transformation ϕ : X → X is non-singular if μ ∘ ϕ−1 is absolutely continuous with respect with μ. For this non-singular transformation, the composition operator Cϕ: 𝒟(Cϕ) → L2(μ) is defined by Cϕf = f ∘ ϕ, f ∈ 𝒟(Cϕ). For a fixed positive integer n ≥ 2, basic properties of product Cϕn · · · Cϕ1 in L2(μ) are presented in Section 2, including the boundedness and adjoint. Under the assistance of these properties, normality and quasinormality of specific bounded Cϕn · · · Cϕ1 in L2(μ) are characterized in Section 3 and 4 respectively, where Cϕ1, Cϕ2, · · ·, Cϕn are all densely defined.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"9 1","pages":"86 - 95"},"PeriodicalIF":0.3000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Normality and Quasinormality of Specific Bounded Product of Densely Defined Composition Operators in L2 Spaces\",\"authors\":\"Hang Zhou\",\"doi\":\"10.1515/conop-2022-0130\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract Let (X, 𝒜, μ) be a σ−finite measure space. A transformation ϕ : X → X is non-singular if μ ∘ ϕ−1 is absolutely continuous with respect with μ. For this non-singular transformation, the composition operator Cϕ: 𝒟(Cϕ) → L2(μ) is defined by Cϕf = f ∘ ϕ, f ∈ 𝒟(Cϕ). For a fixed positive integer n ≥ 2, basic properties of product Cϕn · · · Cϕ1 in L2(μ) are presented in Section 2, including the boundedness and adjoint. Under the assistance of these properties, normality and quasinormality of specific bounded Cϕn · · · Cϕ1 in L2(μ) are characterized in Section 3 and 4 respectively, where Cϕ1, Cϕ2, · · ·, Cϕn are all densely defined.\",\"PeriodicalId\":53800,\"journal\":{\"name\":\"Concrete Operators\",\"volume\":\"9 1\",\"pages\":\"86 - 95\"},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2022-01-01\",\"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-2022-0130\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Concrete Operators","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/conop-2022-0130","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
Normality and Quasinormality of Specific Bounded Product of Densely Defined Composition Operators in L2 Spaces
Abstract Let (X, 𝒜, μ) be a σ−finite measure space. A transformation ϕ : X → X is non-singular if μ ∘ ϕ−1 is absolutely continuous with respect with μ. For this non-singular transformation, the composition operator Cϕ: 𝒟(Cϕ) → L2(μ) is defined by Cϕf = f ∘ ϕ, f ∈ 𝒟(Cϕ). For a fixed positive integer n ≥ 2, basic properties of product Cϕn · · · Cϕ1 in L2(μ) are presented in Section 2, including the boundedness and adjoint. Under the assistance of these properties, normality and quasinormality of specific bounded Cϕn · · · Cϕ1 in L2(μ) are characterized in Section 3 and 4 respectively, where Cϕ1, Cϕ2, · · ·, Cϕn are all densely defined.