On the pointwise Bishop-Phelps-Bollobás property for operators

Sheldon Dantas, Vladimir Kadets, Sun Kwang Kim, Han Ju Lee, Miguel Martín

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

We study approximation of operators between Banach spaces X and Y that nearly attain their norms in a given point by operators that attain their norms at the same point. When such approximations exist, we say that the pair (X, Y) has the pointwise Bishop-Phelps-Bollobás property (pointwise BPB property for short). In this paper we mostly concentrate on those X, called universal pointwise BPB domain spaces, such that (X, Y) possesses pointwise BPB property for every Y, and on those Y, called universal pointwise BPB range spaces, such that (X, Y) enjoys pointwise BPB property for every uniformly smooth X. We show that every universal pointwise BPB domain space is uniformly convex and that Lp(µ) spaces fail to have this property when p > 2. No universal pointwise BPB range space can be simultaneously uniformly convex and uniformly smooth unless its dimension is one. We also discuss a version of the pointwise BPB property for compact operators.

Original languageEnglish
Pages (from-to)1421-1443
Number of pages23
JournalCanadian Journal of Mathematics
Volume71
Issue number6
DOIs
StatePublished - 2019

Keywords

  • Banach space
  • Bishop-Phelps-Bollobás property
  • Norm-attaining operator

Fingerprint

Dive into the research topics of 'On the pointwise Bishop-Phelps-Bollobás property for operators'. Together they form a unique fingerprint.

Cite this