Space of holomorphic functions of polynomial growth as local algebra

Let $G$ be a domain in the complex plane, star-shaped with respect to the point 0, $H^{-\infty}(G)$ be the space of holomorphic functions in $G$ of polynomial growth near the boundary of $G$. The Duhamel product $\ast$ is introduced in it. This product is used in operational and operator calculus, in the spectral theory, in the problem of the spectral multiplicity of a linear operator, in boundary value problems. It is shown that $H^{-\infty}(G)$ with it is a unital topological algebra. The integration operator $J(f)(z)=\int\nolimits_0^z f(t) dt$ acts linearly and continuously in $H^{-\infty}(G)$. It is proved that all linear continuous operators in $H^{-\infty}(G)$ that commute with $J$, are represented as $S_g(f)=f\ast g$, where $g$ is a fixed function from $H^{-\infty}(G)$. In the case where $G$ is strictly star-shaped with respect to zero, a criterion for the invertibility of an element of the algebra $H^{-\infty}(G)$ and a criterion for the operator $S_g$ to have the continuous linear inverse are proved. It is shown that every nonzero operator from the commutator subgroup $J$ is a composition of the power of the operator $J$ and some isomorphism from the aforementioned commutator subgroup. In the proving of $\ast$-invertibility the Neumann series is used, usually applied in Banach spaces. In non-normable locally convex spaces of functions it was previously used by L. Berg, N. Wigley, and M. T. Karaev. All closed ideals of the algebra $(H^{-\infty}(G),\ast)$, closed invariant subspaces and cyclic vectors of $J$ in $H^{-\infty}(G)$ are described. From the obtained results it follows that the operator $J$ is unicellular and the algebra $(H^{-\infty}(G),\ast)$ is local. The only maximal ideal in it is the set of all $\ast$-irreversible elements.