| Popis: |
We study continuous linear operators, which commute with the generalized backward shift operator (a one-dimensional perturbation of the Pommiez operator) in a countable inductive limit E of weighted Banach spaces of entire functions. This space E is isomorphic with the help of the Fourier-Laplace transform to the strong dual of the Frechet space of all holomorphic functions on a convex domain Q in the complex plane, containing the origin. Necessary and sufficient conditions are obtained for an operator of the mentioned commutant to be a topological isomorphism of E. The problem of factorization of nonzero operators of this commutant is investigated. In the case when the function determining the generalized backward shift operator, has zeros in Q, the commutant is divided into two classes: the first one consists of isomorphisms and surjective operators with finite-dimensional kernels and the second one contains finite-dimensional operators. Using obtained results, we study the generalized Duhamel product in Frechet space of all holomorphic functions on Q. |
