Convolution operators on $G$-holomorphic functions in infinite dimensions

Abstract:

For a complex vector space E, let ${H_G}(E)$ denote the space of G (Gateaux)-holomorphic functions on $E\;(f:E \to C$ is G-holomorphic if the restriction of f to every finite dimensional subspace of E is holomorphic in the usual sense). The most natural topology on ${H_G}(E)$ is that of uniform convergence on finite dimensional compact subsets of E. A convolution operator A on ${H_G}(E)$ is a continuous linear mapping $A:{H_G}(E) \to {H_G}(E)$ such that A commutes with translations. The concept of a convolution operator generalizes that of a differential operator with constant coefficients. We prove that if A is a convolution operator on ${H_G}(E)$, then the kernel of A is the closed linear span of the exponential polynomials contained in the kernel. In addition, we show that any nonzero convolution operator on ${H_G}(E)$ is a surjective mapping.