Derivation-Invariant Subspaces of C∞

Abstract

Let C∞(a,b) be the Fréchet space of all complex-valued infinitely differentiable functions on a (finite or infinite) interval (a, b) ⊂ ℝ. Let L ⊂ C∞(a,b) be a closed subspace such that DL ⊂ L, where $D = {d \over dx}$. Then the spectrum σ L of D on L is either the whole complex plane, or a discrete possibly void set of eigenvalues λ, each one with some finite multiplicity mλ ∈ N such that the monomial exponentials e λ,j(x) = x j exp(λx), 0 ≤ j ≤ mλ − 1 belong to L. If the spectrum is void there is a relatively closed interval I ⊂ (a, b) such that L consists of those functions from C∞ (a,b) which vanish identically on I. The interval may reduce to a point in which case L consists of the functions that vanish together with all their derivatives at that point.