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 (2022). We provide a complete characterization of when these spaces are ultrabornological and barrelled in terms of the defining weight system, thereby improving the main result of our above mentioned work. In particular, we obtain that the multiplier space of the Gelfand–Shilov space \Sigma^r_s(\mathbb{R}^d) of Beurling type is ultrabornological, whereas the one of the Gelfand–Shilov space \mathcal{S}^r_s(\mathbb{R}^d) of Roumieu type is not barrelled.