HYPONORMALITY OF SPECIFIC UNBOUNDED 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 nonsingular if mu o phi-1 is absolutely continuous with respect with mu . For this non-singular transformation, the composition operator C phi : D(C phi) -> L2( mu) is defined by C phi f= f o phi , f E D(C phi ). For a fixed positive integer n 2, basic properties of product C phi n center dot center dot center dot C phi 1 in L2( mu) are conveyed in Section 3-5, including the dense definiteness, kernel, adjoint of (not necessarily bounded) C phi n center dot center dot center dot C phi 1 . Under the assistance of these properties, when C phi 1 ,C phi 2, center dot center dot center dot ,C phi n are densely defined, hyponormality of specific (not necessarily bounded) C phi n center dot center dot center dot C phi 1 in L2( mu) is characterized in Section 6.