Representation of Banach ideal spaces and factorization of operators

Representation theorems are proved for Banach ideal spaces with the Fatou property which are built by the Calderon-Lozanovskii construction. Factorization theorems for operators in spaces more general than the Lebesgue LP spaces are investigated. It is natural to extend the Gagliardo theorem on the Schur test and the Rubio de Francia theorem on factorization of the Muckenhoupt A(P) weights to reflexive Orlicz spaces. However, it turns out that for the scales far from L-P-spaces this is impossible. For the concrete integral operators it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces are not valid. Representation theorems for the Calderon-Lozanovskii construction are involved in the proofs.