César Ruiz

Domination by Positive Narrow Operators

We prove that each positive operator from a Köthe function-space E(μ) to a Banach lattice F with a narrow majorant is itself narrow provided the norm on F is order continuous. We also prove that every l2-strictly singular regular operator from Lp[0,1], 1≤p < ∞, to a Banach lattice F is narrow, provided F has an order continuous norm.

Disjointly strictly singular operators and interpolation

Interpolation properties of the class of disjointly strictly singular operators on Banach lattices are studied. We also give some applications to compare the lattice structure of two rearrangement invariant function spaces. In particular, we obtain suitable analytic characterisations of when the inclusion map between two Orlicz function spaces is disjointly strictly singular.

Norm Attaining Operators on Some Classical Banach Spaces

We show that for the Kothe space X = c0 + 1(w), equipped with the Luxemburg norm, the set of norm attaining operators from X into any infinite-dimensional strictly convex Banach space Y is not dense in the space of all bounded operators. The same assertion holds for any infinite-dimensional L1(μ). This gives the first example of a classical space X satisfying the previous property. We also prove that all the spaces c0 + 1(w) are isomorphic for a large class of weights w.