On an algebra of analytic functionals connected with a Pommiez operator

We study properties of a convolution algebra formed by the dual $E'$ of a countable inductive limit $E$ of weighted Fréchet spaces of entire funtions of one complex variable with the multiplication-convolution $\otimes$ which is defined with the help of the shift operator for a Pommiez operator. The algebra $(E',\otimes)$ is isomorphic to the commutant of a Pommiez operator in the ring of all continuous linear operators in $E$. We prove that this isomorphism is topological if $E'$ is endowed with the weak topology and the corresponding commutant is endowed with the weakly operator topology. This result we use for powers of a Pommiez operator series expansions for all continuous linear operators commuting with this Pommiez operator on $E$. We describe also all nonzero multiplicative functionals on the algebra $(E',\otimes)$.