## 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 A_{b} (B_{X} : X) be the Banach space of all bounded continuous functions f on the unit ball B_{X} 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 G_{δ}-subset of A_{b} (B_{X} : 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 A_{b} (B_{X} : X). In particular, when X = L_{p} (μ)(1 < p < ∞) or X = ℓ_{1}, 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 G_{δ}-subset of A_{b} (B_{X} : X). As an application, the existence and properties of numerical boundary of A_{b} (B_{X} : X) are studied. Finally, the numerical peak function in A_{b} (B_{X} : X) is characterized when X = C (K) and some negative results on the denseness of numerical (strong) peak holomorphic functions are given.

Original language | English |
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Pages (from-to) | 437-452 |

Number of pages | 16 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 364 |

Issue number | 2 |

DOIs | |

State | Published - 15 Apr 2010 |

## Keywords

- Numerical Shilov boundary
- Numerical boundary
- Numerical peak holomorphic functions