On the injection of a Köthe function space into L 1 (μ)

Abstract

Operators from a Banach space X into a Banach space Y weaker than isomorphisms and, nevertheless, preserving some isomorphic property, i.e. such that a good property of Y lifts to X, have been studied by many authors. This note is also devoted to that study; more precisely, given a complete σ-finite measure space (Ω,Σ,μ) into L1(μ) and given necessary and sufficient conditions for such an operator to be a Tauberian or semi-Tauberian operator, a semi-embedding or a ∗∗-injection, in the case when X is an Orlicz space, it is also proved that these conditions are equivalent to the reflexivity of X. The results obtained are applied to deduce properties of the vector-valued Köthe function space X(E), E a Banach space, from the corresponding properties of L1(μ,E).