ON AUPETIT’S SCARCITY THEOREM

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.