## Abstract

Let A be a complex and unital Banach algebra, D a domain in C, and f: D → A an analytic function. A useful and remarkable result, due to B. Aupetit, is the Scarcity Theorem for elements with finite spectrum; the second part of the theorem classifies the spectrum of f(λ) under certain conditions, in terms of locally holomorphic functions. The first major result of this paper presents a raw improvement to this—with no further assumptions, it is possible to obtain functions which are (globally) holomorphic on a dense open subset M of D, which is not necessarily all of D. Under the additional assumption that f(λ)f(κ) = f(κ)f(λ) for all κ, λ ∈ D, we show that M = D can be achieved. We also give an easy example to illustrate that M = D is not always possible. The final part of the paper gives a simple proof of the Scarcity Theorem for rank.

Original language | English |
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Pages (from-to) | 321-330 |

Number of pages | 10 |

Journal | Colloquium Mathematicum |

Volume | 171 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2023 |

## Keywords

- Banach algebra
- scarcity of elements with finite spectra
- spectrum

## ASJC Scopus subject areas

- General Mathematics