Regularization for Lozanovskii's type factorization with applications

We say that a function space Z is factorable by X when there exists a third function space Y such that each f from Z admits factorization f = gh, where g, h belong to X, Y , respectively, and f Z g X h Y . We consider a problem of regularization of such a factorization; namely, suppose that f like above satisfies some additional regularity condition (i.e., is holomorphic, smooth or is a simple function). May g, h be chosen to have the same property? Answer to such a question when f is holomorphic leads us to factorization of Hardy type spaces. We also apply these considerations to get factorization for Toeplitz operators on Hardy spaces.