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.
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