Extending previous work by Meise and Vogt, we characterize those convolution operators, defined on the space ${}_{\left(\omega \right)}\left(ℝ\right)$ of (ω)-quasianalytic functions of Beurling type of one variable, which admit a continuous linear right inverse. Also, we characterize those (ω)-ultradifferential operators which admit a continuous linear right inverse on ${}_{\left(\omega \right)}[a,b]$ for each compact interval [a,b] and we show that this property is in fact weaker than the existence of a continuous linear right inverse on ${}_{\left(\omega \right)}\left(ℝ\right)$.

We characterize stability under composition of ultradifferentiable classes defined by weight sequences M, by weight functions ω, and, more generally, by weight matrices , and investigate continuity of composition (g,f) ↦ f ∘ g. In addition, we represent the Beurling space ${}^{\left(\omega \right)}$ and the Roumieu space ${}^{\omega }$ as intersection and union of spaces ${}^{\left(M\right)}$ and ${}^M$ for associated weight sequences, respectively.