The Strong Property (B) for Lp Spaces

Abstract

Given a purely non-atomic, finite measure space (Ω, Σ, ν), it is proved that for every closed, infinite-dimensional subspace V of Lp(ν) (1 ≤ p < ∞) there exists a decomposition Lp(ν) = X1 ⊕ X2, such that both subspaces X1 and X2 are isomorphic to Lp(ν) and both V ∩X1 and V ∩ X2 are infinite-dimensional. Some consequences concerning dense, non-closed range operators on L1 are derived