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.
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