On Aupetit’s Scarcity Theorem

Let $ A $ 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.