Cohyponormality of Unbounded Product of Composition Operators in L2 Spaces
Let X,A,μ be a σ-finite measure space. The composition operators Cϕ:DCϕ⟶L2μ are defined by Cϕf=f∘ϕ,f∈DCϕ, where ϕ are nonsingular transformations and DCϕ=f∈L2μ: f∘ϕ∈L2μ. Suppose that ϕ1,ϕ2,⋯,ϕn are nonsingular transformations of X, where n∈ℕ and n≥2. It is natural to define the product operator of c...
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2025-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/jofs/7403237 |
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| author | Hang Zhou Qing Su |
| author_facet | Hang Zhou Qing Su |
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| description | Let X,A,μ be a σ-finite measure space. The composition operators Cϕ:DCϕ⟶L2μ are defined by Cϕf=f∘ϕ,f∈DCϕ, where ϕ are nonsingular transformations and DCϕ=f∈L2μ: f∘ϕ∈L2μ. Suppose that ϕ1,ϕ2,⋯,ϕn are nonsingular transformations of X, where n∈ℕ and n≥2. It is natural to define the product operator of composition operators Cϕ1,Cϕ2,⋯,Cϕn, denoted by Cϕn⋯Cϕ1 in L2μ. Equivalent conditions of cohyponormality of (not necessarily bounded) Cϕn⋯Cϕ1 in L2μ are given in this paper, where Cϕ1,Cϕ2,⋯,Cϕn are densely defined. In fact, the following statements are equivalent: (i) Cϕk⋯Cϕ1 is cohyponormal; (ii) ∫Xθk2f2dμ≤∫XEΦ~kf2dμ, where f∈L2μ; and (iii) EΦkθk2·f2≤EΦ~kf2 a.e. μΦk−1A∩Φ~k−1A, where f∈L2μ. Moreover, basic properties of product Cϕn⋯Cϕ1 in L2μ are conveyed, including the dense definiteness, boundedness, and adjointness. |
| format | Article |
| id | doaj-art-33ccc8e1563c41b683a465d4911afb36 |
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| language | English |
| publishDate | 2025-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces |
| spelling | doaj-art-33ccc8e1563c41b683a465d4911afb362025-08-20T02:16:22ZengWileyJournal of Function Spaces2314-88882025-01-01202510.1155/jofs/7403237Cohyponormality of Unbounded Product of Composition Operators in L2 SpacesHang Zhou0Qing Su1Department of Information Technology and EngineeringDepartment of Information Technology and EngineeringLet X,A,μ be a σ-finite measure space. The composition operators Cϕ:DCϕ⟶L2μ are defined by Cϕf=f∘ϕ,f∈DCϕ, where ϕ are nonsingular transformations and DCϕ=f∈L2μ: f∘ϕ∈L2μ. Suppose that ϕ1,ϕ2,⋯,ϕn are nonsingular transformations of X, where n∈ℕ and n≥2. It is natural to define the product operator of composition operators Cϕ1,Cϕ2,⋯,Cϕn, denoted by Cϕn⋯Cϕ1 in L2μ. Equivalent conditions of cohyponormality of (not necessarily bounded) Cϕn⋯Cϕ1 in L2μ are given in this paper, where Cϕ1,Cϕ2,⋯,Cϕn are densely defined. In fact, the following statements are equivalent: (i) Cϕk⋯Cϕ1 is cohyponormal; (ii) ∫Xθk2f2dμ≤∫XEΦ~kf2dμ, where f∈L2μ; and (iii) EΦkθk2·f2≤EΦ~kf2 a.e. μΦk−1A∩Φ~k−1A, where f∈L2μ. Moreover, basic properties of product Cϕn⋯Cϕ1 in L2μ are conveyed, including the dense definiteness, boundedness, and adjointness.http://dx.doi.org/10.1155/jofs/7403237 |
| spellingShingle | Hang Zhou Qing Su Cohyponormality of Unbounded Product of Composition Operators in L2 Spaces Journal of Function Spaces |
| title | Cohyponormality of Unbounded Product of Composition Operators in L2 Spaces |
| title_full | Cohyponormality of Unbounded Product of Composition Operators in L2 Spaces |
| title_fullStr | Cohyponormality of Unbounded Product of Composition Operators in L2 Spaces |
| title_full_unstemmed | Cohyponormality of Unbounded Product of Composition Operators in L2 Spaces |
| title_short | Cohyponormality of Unbounded Product of Composition Operators in L2 Spaces |
| title_sort | cohyponormality of unbounded product of composition operators in l2 spaces |
| url | http://dx.doi.org/10.1155/jofs/7403237 |
| work_keys_str_mv | AT hangzhou cohyponormalityofunboundedproductofcompositionoperatorsinl2spaces AT qingsu cohyponormalityofunboundedproductofcompositionoperatorsinl2spaces |