The Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B

Richard Aron, Yun Sung Choi, Sun Kwang Kim, Han Ju Lee, Miguel Martín

Research output: Contribution to journalArticlepeer-review

46 Scopus citations

Abstract

We study a Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B. For domain spaces, we study Banach spaces X such that (X, Y) has the Bishop-Phelps-Bollobás property (BPBp) for every Banach space Y . We show that in this case, there exists a universal function ηX(ε) such that for every Y , the pair (X, Y) has the BPBp with this function. This allows us to prove some necessary isometric conditions for X to have the property. We also prove that if X has this property in every equivalent norm, then X is one-dimensional. For range spaces, we study Banach spaces Y such that (X, Y) has the Bishop-Phelps-Bollobás property for every Banach space X. In this case, we show that there is a universal function ηY (ε) such that for every X, the pair (X, Y) has the BPBp with this function. This implies that this property of Y is strictly stronger than Lindenstrauss property B. The main tool to get these results is the study of the Bishop-Phelps-Bollobás property for c0-, ℓ1- and ℓ-sums of Banach spaces.

Original languageEnglish
Pages (from-to)6085-6101
Number of pages17
JournalTransactions of the American Mathematical Society
Volume367
Issue number9
DOIs
StatePublished - 2015

Keywords

  • Approximation
  • Banach space
  • Bishop-Phelps-Bollobás theorem
  • Norm-attaining operators

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