Convolutors on $mathcal{S}_\omega(mathbb{R}^N)$

In this paper we continue the study of the spaces $O_{M,ω}(R^N)$ and $O_{C,ω}(R^N)$ undertaken in Albanese and Mele (J Pseudo-Differ Oper Appl, 2021). We determine new representations of such spaces and we give some structure theorems for their dual spaces. Furthermore, we show that $O'_{C,ω}(R^N)$ is the space of convolutors of the space $S_ω(R^N)$ of the ω-ultradifferentiable rapidly decreasing functions of Beurling type (in the sense of Braun, Meise and Taylor) and of its dual space $S'_ω(R^N)$. We also establish that the Fourier transform is an isomorphism from $O'_{C,ω}(R^N)$ onto $O_{M,ω}(R^N)$. In particular,we prove that this isomorphism is topological when the former space is endowed with the strong operator lc-topology induced by $L_b(S_ω(R^N))$ and the last space is endowed with its natural lc-topology.