Spectral synthesis on a system of unbounded domains starlike in a common direction (Q1317563)

A domain \(G \subset \mathbb{C}\) is starshaped in the direction \(\alpha \in [0,2\pi)\) if \(z \in G\) implies \(z + t \text{ exp }i\alpha \in G\) for all \(t > 0\). Let \(\Omega_ 1, \dots,\Omega_ n\) be a system of domains in \(\mathbb{C}\) starshaped in the common direction \(\alpha\) and \(H(\Omega_ i)\) \((1 \leq i\leq n)\) be the topological space of all holomorphic functions on \(\Omega_ i\) equipped with the topology of uniform convergence on compact sets. The author considers the linear operator of componentwise differentiation \(Df:=(Df_ 1,\dots,Df_ n)\) acting on the topological product \(H = H(\Omega_ 1) \times \dots \times H(\Omega_ n)\). It is proved that each closed subspace \(W \subset H\) invariant with respect to the operator \(D\) admits spectral synthesis (i.e. \(W\) is the closed linear hull of the root elements of the operator \(D\) contained in \(W\)).