S-decomposable Banach lattices, optimal sequence spaces and interpolation

We investigate connections between upper/lower estimates for Banach lattices and the notion of relative s-decomposability, which has roots in interpolation theory. To get a characterization of relatively s-decomposable Banach lattices in terms of the above estimates, we assign to each Banach lattice X two sequence spaces $$X_{U}$$ and $$X_{L}$$ that are largely determined by the set of p, for which $$l_{p}$$ is finitely lattice representable in X. As an application, we obtain an orbital factorization of relative K-functional estimates for Banach couples $$\vec {X} =(X_{0},X_{1})$$ and $$\vec {Y}=(Y_{0},Y_{1})$$ through some suitable couples of weighted $$L_{{p}}$$ -spaces provided if $$X_{i},Y_{i}$$ are relatively s -decomposable for $$i=0,1$$ . Also, we undertake a detailed study of the properties of optimal upper and lower sequence spaces $$X_{U}$$ and $$X_{L}$$ , and, in particular, prove that these spaces are rearrangement invariant. In the Appendix, a description of the optimal upper sequence space for a separable Orlicz space as a certain intersection of some special Musielak-Orlicz sequence spaces is given.