Spectral synthesis in the space of infinitely differentiable functions

In the space of all infinitely differentiable functions on an interval $(a,b)$ consider a differentiation invariant subspace and assume that the restriction of differentiation onto this subspace has discrete spectrum. Is it true that in this case the subspace is generated by the exponential monoms it contains? In general, the answer is negative, since the subspace may have the so-called “residual” part (all functions vanishing on some subinterval). In 2007 A. Aleman and B. Korenblum posed the spectral synthesis question: is any invariant subspace generated by its residual part and the corresponding exponentials? We give complete description of subspaces which admit spectral synthesis in terms of their spectra. The talk is based on joint works with A. Aleman and Yu. Belov.