Spectrum of Weighted Composition Operators. Part XII: The Kamowitz–Scheinberg Theorem Revisited

The well-known Kamowitz–Scheinberg theorem states that if $ U $ is an automorphism of a commutative semisimple Banach algebra and $ U^{n}\neq I $ for all $ n\in 𝕅 $ , then the spectrum of $ U $ contains the unit circle. In this paper we present some results on the spectrum of weighted automorphisms of unital commutative semisimple Banach algebras that considerably strengthen the statement of the Kamowitz–Scheinberg theorem.