Affiliation:
1. School of Information Technology and Engineering, Guangzhou College of Commerce , Guangzhou, G. D. 511363, P.R. China
Abstract
AbstractLet (X, 𝒜,μ) be a σ−finite measure space. A transformationϕ:X→Xis non-singular ifμ∘ϕ−1is absolutely continuous with respect withμ. For this non-singular transformation, the composition operatorCϕ: 𝒟(Cϕ) →L2(μ) is defined byCϕf=f∘ϕ,f∈ 𝒟(Cϕ).For a fixed positive integern≥ 2, basic properties of productCϕn· · ·Cϕ1inL2(μ) are presented in Section 2, including the boundedness and adjoint. Under the assistance of these properties, normality and quasinormality of specific boundedCϕn· · ·Cϕ1inL2(μ) are characterized in Section 3 and 4 respectively, whereCϕ1,Cϕ2, · · ·,Cϕnare all densely defined.
Subject
Applied Mathematics,Analysis
Reference16 articles.
1. [1] R.B. Ash, C. A. Doléans-Dade, Probability and Measure Theory, Harcourt/Academic Press, Burlington, 2000.
2. [2] P. Budzyński, Z. J. Jabłoński, I. B. Jung and J. Stochel, On unbounded composition operators in L2-spaces, Annali di Matematica, 193 (2014), 663–688.
3. [3] P. Budzyński, Z. Jabłoński, I. B. Jung and J. Stochel, Unbounded weighted composition operators in L2-Spaces, Lecture Notes in Mathematics 2209, Springer, Switzerland, 2018.
4. [4] D. L. Cohn, Measure Theory, Second Edition, Springer Science+Business Media, 2013.
5. [5] C. C. Cowen and B. D. Maccluer, Composition operators on spaces of analytic functions, CRC Press, 1995.