On commutant of differentiation and translation operators in weighted spaces of entire functions

We describe continuous linear operators acting in a countable inductive limit $E$ of weighted Fréchet spaces of entire functions of several complex variables and commuting in these spaces with systems of partial differentiation and translation operators. Under the made assumptions, the commutants of the systems of differentiation and translation operators coincide. They consist of convolution operators defined by an arbitrary continuous linear functional on $E$. At that, we do not assume that the set of the polynomials is dense in $E$. In the space $E'$ topological dual to $E$, we introduce the natural multiplication. Under this multiplication, the algebra $E'$ is isomorphic to the aforementioned commutant with the usual multiplication, which is the composition of the operators. This isomorphism is also topological if $E'$ is equipped by the weak topology, while the commutant is equipped by the weak operator topology. This implies that the set of the polynomials of the differentiation operators is dense in the commutant with topology of pointwise convergence. We also study the possibility of representing an operator in the commutant as an infinite order differential operator with constant coefficients. We prove the immediate continuity of linear operators commuting with all differentiation operators in a weighted $\mathrm{(LF)}$-space of entire functions isomorphic via Fourier-Laplace transform to the space of infinitely differentiable functions compactly supported in a real multi-dimensional space.