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

Abstract 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 commutant %$\mathcal K(\mathcal D_0)%$ in the algebra of all continuous linear operators on %$E%$ o...
Ausführliche Beschreibung

Abstract 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 commutant %$\mathcal K(\mathcal D_0)%$ 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. Ausführliche Beschreibung