Surjectivity of Hadamard type operators on spaces of smooth functions

Abstract

We obtain a complete characterization of surjective Hadamard type operators \(H_T,T\in C^\infty (\mathbb {R}^d)'\) (i.e. of multiplicative convolution operators) on \(C^\infty (\mathbb {R}^d)\) using a restrictive slowly decreasing condition and a division property both new and valid for the Mellin transform \(\mathscr {M}(T)\). We also characterize bijectivity and calculate the spectrum of Hadamard type operators on \(C^\infty (\mathbb {R}^d)\). We prove a Theorem of Supports for the multiplicative convolution. The Mellin transform \(\mathscr {M}\) is defined on the space of all distributions with compact support, providing a topological isomorphism onto a certain weighted space of holomorphic germs.