A note on the barrelledness of weighted $(PLB)$-spaces of ultradifferentiable functions

In this note we consider weighted $(PLB)$-spaces of ultradifferentiable functions defined via a weight function and a weight system, as introduced in our previous work [4]. We provide a complete characterization of when these spaces are ultrabornological and barrelled in terms of the defining weight system, thereby improving the main Theorem 5.1 of [4]. In particular, we obtain that the multiplier space of the Gelfand-Shilov space ${\mathrm{\Sigma }}_s^r({\mathbb{R}}^d)$ of Beurling type is ultrabornological, whereas the one of the Gelfand-Shilov space ${\mathcal{S}}_s^r({\mathbb{R}}^d)$ of Roumieu type is not barrelled.