Many-Dimensional Duhamel Product in the Space of Holomorphic Functions and Backward Shift Operators

The system $\mathcal D_0$ of partial backward shift operators in a countable inductive limit $E$ of weighted Banach spaces of entire functions of several complex variables is studied. Its commutator subgroup $\mathcal K(\mathcal D_0)$ in the algebra of all continuous linear operators on $E$ operators is described. In the topological dual of $E$, a multiplication $\circledast$ is introduced and studied, which is determined by shifts associated with the system $\mathcal D_0$. For a domain $\Omega$ in $\mathbb C^N$ polystar-shaped with respect to 0, Duhamel product in the space $H(\Omega)$ of all holomorphic functions on $\Omega$ is studied. In the case where, in addition, the domain $\Omega$ is convex, it is shown that the operation $\circledast$ is realized by means of the adjoint of the Laplace transform as Duhamel product.