Algebras of analytic functionals and the generalized Duhamel product

Let $\Omega$ be a simply connected domain in the complex plane containing the origin; $H(\Omega)$ be the Fréchet space of all holomorphic functions on $\Omega$. A holomorphic on $\Omega$ function $g_0$, such that $g_0(0)=1$, defines a continuous linear Pommiez operator in $H (\Omega)$. It is a one-dimensional perturbation of the backward shift operator and coincides with it if $g_0$ is the constant function one. Its commutant in the ring of all continuous linear operators in $H(\Omega)$ is isomorphic to the algebra formed by the dual $ H(\Omega)'$ of $ H(\Omega)$ with the multiplication $\otimes$ defined by the shift operators for the Pommiez operator according to the convolution rule. It is shown that this algebra is unital associative, commutative and topological. Its representations are obtained with the help of Laplace and Cauchy transformations. The focus in the article is the research of the representations with the help of the Laplace transformation. It leads to an isomorphic algebra, formed by some space $P_\Omega$ of entire functions of exponential type. The multiplication $\ast$ in it is the generalized Duhamel product. If $g_0$ is the identity unit, then this multiplication is the usual Duhamel product. The generalized Duhamel product is given by convolution operators, defined by the function $g_0$. In the case of the Cauchy transformation (for the function $g_0$ equal to the constant function one) the realization of $(H(\Omega)',\otimes)$ is the space of germs all holomorphic functions on the complement $\Omega$ in the extended complex plane, which are equal to zero at infinity, with multiplication, inverse to the usual product of functions and the independent variable. A description of all proper closed ideals $(P_\Omega, \ast)$ is obtained. It is based on the description of all proper closed $D_{0,g_0}$-invariant subspaces of $H(\Omega)$, obtained earlier by the authors. The set of all proper closed ideals $(P_\Omega,\ast)$ consists of two families. The one contains finite-dimensional ideals defined by subsets of the zero manifold of the function $g_0$. The other contains infinite ideals, defined, in particular, by a finite number of points outside of $\Omega$. A similar problem was solved earlier by the authors in the dual situation, namely, for the algebra of germs of all functions, holomorphic on a convex locally closed set in the complex plane. In this case, the function $g_0$ was considered, which is the product of а polynomial and an exponential function.