Automatic continuity of separating and biseparating isomorphisms

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Abstract

Let p be a fixed number with 1 ≤ p < ∞. It is shown that every surjective and biseparating linear map between Lp-spaces is continuous when the underlying measure space is non-atomic. We also prove that a separating isomorphism on lp 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.

Original languageEnglish
Pages (from-to)1569-1577
Number of pages9
JournalProceedings of the American Mathematical Society
Volume151
Issue number4
DOIs
Publication statusPublished - 1 Apr 2023

Keywords

  • Automatic continuity
  • Lebesgue spaces
  • biseparating maps
  • separating maps
  • weighted composition operators

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