On reducibility of weighted composition operators

In this paper, we study two types of the reducing subspaces for the weighted composition operator W : f →u. f ◦ φ on L²(Σ). A necessary and sufficient condition is given for W to possess the reducing subspaces of the form L²(ΣB) where B Є Σ σ(u). Moreover, we pose some necessary and some sufficient conditions under which the subspaces of the form L²(A) reduce W. All of these are basically discussed using the con- ditional expectation properties. To explain the results, some examples are then presented.