A Criterion for the Fundamental Principle to Hold for Invariant Subspaces on Bounded Convex Domains in the Complex Plane

Let $D$ be a bounded convex domain of the complex plane. We study the problem of whether the fundamental principle holds for analytic function spaces on $D$ invariant with respect to the differentiation operator and admitting spectral synthesis. Earlier this problem was solved under a restriction on the multiplicities of the eigenvalues of the differentiation operator. In the present paper, we lift this restriction. Thus, we present a complete solution of the fundamental principle problem for arbitrary nontrivial closed invariant subspaces admitting spectral synthesis on arbitrary bounded convex domains.