On Aupetit’s Scarcity Theorem
Tom 171 / 2023
Streszczenie
Let be a complex and unital Banach algebra, D a domain in \mathbb {C} , and f\colon D\to 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(\lambda ) 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(\lambda )f(\kappa )=f(\kappa )f(\lambda ) for all \kappa ,\lambda \in 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.
