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 Frechet 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).