On the commutant of differetiation and translation operators in weighted spaces of entire functions
We describe linear continuous operators acting in a countable inductive limit E of weighted Fréchet spaces of entire functions of many complex variables and commuting in these spaces with systems of operators of partial differentiation and translation. Under the made assumptions the commutants of the systems of differentiation and translation operators coincide. They consist of convolution operators defined by a linear continuous functional on E. At that we do not assume that the set of the polynomials is dense in E. In the space E′ topologically dual to E, we naturally introduce the multiplication. Under this multiplication, the algebra E′ is isomorphic to the aforementioned commutant with the usual mulitplication, which the composition of the operators. This isomorphism is also topological if E′ is equipped by a 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 studied 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 (LF)-space of entire functions isomorphic via Fourier-Laplace transform to the space of infinitely differentiable functions compactly supported in a multi-dimensional real space.