COMPACT ENDOMORPHISMS OF BANACH ALGEBRAS OF INFINITELY DIFFERENTIABLE FUNCTIONS

Abstract

Let $(M_n)$ be a sequence of positive numbers satisfying $M_0\,{=}\,1$ and \[\displaystyle \frac{M_{n+m}}{M_n M_m}\,{\geq}\,{{n+m}\choose{m}} \] for all non-negative integers $m$, $n$. Let \[D([0,1], M)=\left\{f\,{\in}\,C^{\infty}([0,1]):\|f\|_{D}=\sum_{n=0}^{\infty} \frac{\|f^{(n)}\|_{\infty}}{M_n}\,{<}\,\infty\right\}.\] With pointwise addition and multiplication, $D([0,1],M)$ is a unital commutative semisimple Banach algebra. If $\lim_{n\to\infty} (n!/M_n)^{1/n}\,{=}\,0,$ then the maximal ideal space of the algebra is $[0,1]$, and every non-zero endomorphism $T$ has the form $Tf(x)\,{=}\,f(\phi(x))$ for some selfmap $\phi$ of the unit interval. The authors have previously shown for a wide class of $\phi$ mapping the unit interval to itself that if $\|\phi'\|_\infty\,{<}\,1$, then $\phi$ induces a compact endomorphism. The paper investigates the extent to which this condition is necessary, and the spectra of all compact endomorphisms of $D([0,1],M)$ are determined. Some of the authors' earlier results on general endomorphisms of $D([0,1],M)$ are simplified and strengthened.