Compact Endomorphisms of Banach Algebras of Infinitely Differentiable Functions

Let ( M n ) be a sequence of positive numbers satisfying M 0 = 1 andM n + mM n M m⩾ (n + mm)for all non‐negative integers m, n . LetD ( [ 0 , 1 ] , M ) = { f ∈ C ∞ ( [ 0 , 1 ] ) :‖ f ‖ D = ∑ n = 0 ∞‖f ( n )‖ ∞M n< ∞} .With pointwise addition and multiplication, D ([0,1], M ) is a unital commutative semisimple Banach algebra. If lim n →∞ ( 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 (ϕ( x )) for some selfmap ϕ of the unit interval. The authors have previously shown for a wide class of ϕ mapping the unit interval to itself that if ‖ϕ′‖ ∞ <1, then ϕ 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.