Weighted Composition Operators on Orlicz Spaces

In this paper we study weighted composition operators on Orlicz spaces. Introduction : Let X and Y be two non empty sets and let F(X) and F(Y) be denoted the topological vector spaces of complex valued functions on X and Y respectively. If T : Y → X is a mapping such that f oT ∈ F (Y) whenever f ∈ F (X), then we can define a composition transformation C T : F (X) → F (Y) by C T f = f oT for every f ∈ F (X). If C T is continuous, we call it a composition operator induced by T. Further, if u : X → C is a mapping such that f ∈ F (X) implies that u.f ∈ F (X), then a multiplication transformation M u : F (X) → F (Y) is defined by M u f = u.f for every f ∈ F (X). A continuous multiplication linear transformation is called a multiplication operator. A weighted composition operator is a bounded linear operator M u,T : F (X) → F (Y) defined by M u,T (f) = (uoT)(x)(foT)(x) for every f ∈ F (X) and x ∈ X.These operators received considerable attention over past several decades on L p-spaces, C(X) spaces and Bergman spaces and play an important role in the