Subspaces of C∞C∞ invariant under the differentiation

Let L   be a proper differentiation invariant subspace of C∞(a,b)C∞(a,b) such that the restriction operator ddx|L has a discrete spectrum Λ (counting with multiplicities). We prove that L   is spanned by functions vanishing outside some closed interval I⊂(a,b)I⊂(a,b) and monomial exponentials xkeλxxkeλx corresponding to Λ   if its density is strictly less than the critical value |I|2π, and moreover, we show that the result is not necessarily true when the density of Λ equals the critical value. This answers a question posed by the first author and B. Korenblum. Finally, if the residual part of L is trivial, then L is spanned by the monomial exponentials it contains.