Automatic continuity of separating and biseparating isomorphisms

Abstract

Let p p be a fixed number with 1 ≤ p > ∞ 1 \leq p > \infty . It is shown that every surjective and biseparating linear map between L p L^p -spaces is continuous when the underlying measure space is non-atomic. We also prove that a separating isomorphism on l p l^p is both continuous and biseparating. Furthermore, these (bi-)separating maps take the form of a weighted composition operator. Our proofs are direct, elementary and do not invoke deep results about Riesz spaces or Banach lattices.