On the stability and boundedness properties of operators in the Gelfand–Shilov spaces of Roumieu type

This article presents an extensive study of Gelfand–Shilov spaces of Roumieu type defined via sequences of positive real numbers satisfying natural growth and convexity conditions. We introduce generalized anisotropic spaces $S_{\{N\}}^{\{M\}}\left({\mathbb{R}}^n\right)$ and explore their fundamental properties, including stability under differentiation, multiplication, translation, and dilatation operators. We establish boundedness results that demonstrate these spaces form differential subalgebras of the space of continuous functions. These results enrich the functional–analytic framework of ultradifferentiable spaces and provide a foundation for further studies in microlocal analysis and partial differential equations involving ultradistributions.