Representation theorems are proved for Banach ideal spaces with the Fatou property which are built by the Calder{\'o}n--Lozanovski\u\i\ construction. Factorization theorems for operators in spaces more general than the Lebesgue $L^{p}$ 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 Calder{\'o}n--Lozanovski\u\i\ construction are involved in the proofs.

Keywords:

Banach ideal spaces, weighted spaces, weight functions, Calderón--Lozanovski\u\i\ spaces, Orlicz spaces, representation of, spaces, uniqueness problem, positive linear operators, positive sublinear, operators, Schur test, factorization of operators, f Banach ideal spaces, weighted spaces, weight functions, Calderón--Lozanovski\u\i\ spaces, Orlicz spaces, representation of, spaces, uniqueness problem, positive linear operators, positive sublinear, operators, Schur test, factorization of operators, f

MSC Classifications:

46E30, 46B42, 46B70 show english descriptions Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Kothe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Banach lattices [See also 46A40, 46B40]
Interpolation between normed linear spaces [See also 46M35] 46E30 - Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Kothe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B42 - Banach lattices [See also 46A40, 46B40]
46B70 - Interpolation between normed linear spaces [See also 46M35]

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