${\mathcal{E}}^{\mathrm{\prime }}$ as an algebra by multiplicative convolution

We study the algebra ${\mathcal{E}}^{\mathrm{\prime }}({\mathbb{R}}^d)$ equipped with the multiplication $(T\star S)(f)=T_x(S_y(f(xy))$ where $xy=(x_1{y}_1,\dots ,x_d{y}_d)$. This allows us a very elegant access to the theory of Hadamard type operators on $C^{\mathrm{\infty }}(\mathrm{\Omega })$, $\mathrm{\Omega }$ open in ${\mathbb{R}}^d$, that is, of operators which admit all monomials as eigenvectors. We obtain a representation of the algebra of such operators as an algebra of holomorphic functions with classical Hadamard multiplication. Finally we study global solvability for such operators on open subsets of ${\mathbb{R}}_{+}^d$.