Some Banach Space Characterizations Of The δ₂ Condition

Abstract

In [1] J. Alexopoulos has shown that if Φ ∈ δ₂ and if its complement ψ satisfies lim_{t ψ∞}ψ(ct) ψ(t) = ∞ for some c > 0 then a bounded set K ⊂ L _Φ is relatively weakly compact if and only if K has equi-absolutely continuous norms. Even though all such Φ fail the₂ condition we do not know whether the theorem is applicable to all Φ ∈ δ₂\₂. In this paper we make some progress towards a generalization of this theorem. In particular we show that an N-function Φ/∈₂ if and only if every weakly null sequence (c _n χE_n) in E ^Φ (μ) has equi-absolutely continuous norms. Other characterizations of the δ₂ condition are given.