Invariant subspaces of analytic functions. I. Spectral analysis on convex regions

Let $G$ be a convex region in the complex plane and $H$ be the space of analytic functions on $G$ with the topology of uniform convergence on compacta of $G$. A closed subspace $W\subset H$ is said to be invariant if it is invariant with respect to the differentiation operator, i.e. if $f\in W$, then $f'\in W$. We say that $W$ admits a spectral synthesis if $W$ is the closed linear span of the exponential monomials contained in $W$. L. Schwartz in 1947 asked the question: Is it true that every invariant subspace admits a spectral synthesis? We find that the answer, generally speaking, is no. In this paper we formulate the precise criteria for the admissibility of spectral synthesis in terms of annihilator submodules of invariant subspaces.
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