On the linear inverse from right operator for the convolution operator on the spaces of germs of analytical functions on convex compacts in \(\mathbb{C}\) (Q1390475)

Let \(G\) and \(K\) be convex compact subsets of the complex plane, and \(A(G)\) and \(A(G+ K)\) denote the spaces of germs of analytic functions on \(G\) and \(G+K\), resp. Every analytic functional \(\mu\) determined by \(K\) generates a linear continuous convolution operator \(T_\mu: A(G+K)\to A(G)\) by \(T_\mu(g)(z)= \mu(g(\cdot+ z))\), \(g\in A(G+ K)\). The problem of existence of a linear continuous right inverse operator to \(T_\mu\) is under consideration. Let \(G\) differ from a point, \(T_\mu\) be surjective, and let the zero set of the function \(\mu(\exp(\cdot z))\) be infinite. Under these assumptions, two necessary and sufficient conditions for existence of a linear continuous right inverse to \(T_\mu\) are obtained in terms of properties of the support function of the set \(G\).