Endomorphisms of Banach Algebras of Infinitely Differentiable Functions on Compact Plane Sets

Let X be a perfect, compact subset of the complex plane, let (Mn) be a sequence of positive numbers satisfying M0=1 and ([formula])⩽Mm+n/MmMn, and let D(X, M)={f ∈D∞(X): ‖f‖ =∑∞n=0 (‖f(n)‖∞/Mn)<∞{. With pointwise addition and multiplication, D(X, M) is a commutative normed algebra. In this note we study the endomorphisms of such algebras. In a previous paper this problem was solved when X=[0, 1] for many weights (Mn). Here we investigate the extent to which the methods used previously apply to general X and introduce some new methods. In particular, we obtain many results when X=Δ where Δ is the open unit disc.