Normality and Quasinormality of Specific Bounded Product of Densely Defined Composition Operators in L2 Spaces

Let (X, A, mu) be a sigma-finite measure space. A transformation phi : X -> X is non-singular if mu circle phi(-1) is absolutely continuous with respect with mu. For this non-singular transformation, the composition operator C-phi : D(C-phi) -> L-2 (mu) is defined by C(phi)f = f circle phi, f is an element of D(C-phi). For a fixed positive integer n >= 2, basic properties of product C-phi n ... C-phi 1 in L-2 (mu) are presented in Section 2, including the boundedness and adjoint. Under the assistance of these properties, normality and quasi-normality of specific bounded C-phi n ... C-phi 1 in L-2(mu) are characterized in Section 3 and 4 respectively, where C-phi 1, C-phi 2, ..., C-phi n are all densely defined.