Numerical peak holomorphic functions on Banach spaces

Sung Guen Kim, Han Ju Lee

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

We introduce the notion of numerical (strong) peak function and investigate the denseness of the norm and numerical peak functions on complex Banach spaces. Let Ab (BX : X) be the Banach space of all bounded continuous functions f on the unit ball BX of a Banach space X and their restrictions f |BX○ to the open unit ball are holomorphic. In finite dimensional spaces, we show that the intersection of the set of all norm peak functions and the set of all numerical peak functions is a dense Gδ-subset of Ab (BX : X). We also prove that if X is a smooth Banach space with the Radon-Nikodým property, then the set of all numerical strong peak functions is dense in Ab (BX : X). In particular, when X = Lp (μ)(1 < p < ∞) or X = ℓ1, it is shown that the intersection of the set of all norm strong peak functions and the set of all numerical strong peak functions is a dense Gδ-subset of Ab (BX : X). As an application, the existence and properties of numerical boundary of Ab (BX : X) are studied. Finally, the numerical peak function in Ab (BX : X) is characterized when X = C (K) and some negative results on the denseness of numerical (strong) peak holomorphic functions are given.

Original languageEnglish
Pages (from-to)437-452
Number of pages16
JournalJournal of Mathematical Analysis and Applications
Volume364
Issue number2
DOIs
StatePublished - 15 Apr 2010

Keywords

  • Numerical Shilov boundary
  • Numerical boundary
  • Numerical peak holomorphic functions

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