Spectrum of weighted composition operators. Part XII. Kamowitz - Scheinberg theorem revisited

The well-known Kamowitz - Scheinberg theorem states that if $U$ is an automorphism of a commutative semi-simple Banach algebra and $U^n\mathrm{\ne }I,n\in \text{mathds}N$, then the spectrum of $U$ contains the unit circle. In this paper we present some results about the spectrum of weighted automorphisms of unital commutative semi-simple Banach algebras that considerably strengthen the statement of the Kamowitz - Scheinberg theorem.