Automatic continuity of separating and biseparating isomorphisms

Ching On Lo; WKA Loh

Automatic continuity of separating and biseparating isomorphisms

Division of Science, Engineering and Health Studies (SEHS)

Research output: Contribution to journal › Article › peer-review

Abstract

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

Original language	English
Pages (from-to)	1569-1577
Number of pages	9
Journal	Proceedings of the American Mathematical Society
Volume	151
Issue number	4
Publication status	Published - 1 Apr 2023

Keywords

Automatic continuity
Lebesgue spaces
biseparating maps
separating maps
weighted composition operators

Cite this

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N2 - 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.

AB - 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.

KW - Automatic continuity

KW - Lebesgue spaces

KW - biseparating maps

KW - separating maps

KW - weighted composition operators

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JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

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ER -

Automatic continuity of separating and biseparating isomorphisms

Abstract

Keywords

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