On invertibility of Duhamel operator in spaces of ultradifferentiable functions

Let $\Delta$ be a non-point segment or an (open) interval on the real line containing the point $0$. In the space of entire functions realized by the Fourier-Laplace transform of the dual space to the space of ultradifferentiable or of all infinitely differentiable functions on $\Delta$, we study the operators from the commutator subgroup of the one-dimensional perturbation of the backward shift operator. We prove a criterion of their invertibility. In this case, the Riesz-Schauder theory is applied, the use of which in such a situation goes back to the works by V.A. Tkachenko. In the topological dual space to the original space, the multiplication $\circledast$ is introduced and we show that its dual space endowed with a strong topology is a topological algebra. Using the mapping associated with Fourier-Laplace transform, the introduced multiplication $\circledast$ is implemented as a generalized Duhamel product in the corresponding space of ultradifferentiable or infinitely differentiable functions on $\Delta$. We prove a criterion for the invertibility of the Duhamel operator in this space. The multiplication $\circledast$ is used to extend the Duhamel's formula to classes of ultradifferentiable functions. It represents the solution of an inhomogeneous differential equation of finite order with constant coefficients, satisfying zero initial conditions at the point $0$, in the form of Duhamel's product of the right-hand side and a solution to this equation with the right-hand side identically equalling to $1$. The obtained results cover both the non-quasianalytic and quasianalytic cases.