Quasinormability and property $(\Omega)$ for spaces of smooth and ultradifferentiable vectors associated with Lie group representations

TL;DR

This paper establishes that spaces of smooth and ultradifferentiable vectors linked to Lie group representations inherit quasinormability and property $( ext{Ω})$ from the underlying space, with applications to Gevrey vectors and function spaces on Lie groups.

Contribution

It proves quasinormability and property $( ext{Ω})$ transfer from a Fréchet space to associated vector spaces in Lie group representations, including Gevrey and weighted function spaces.

Findings

Spaces of vectors inherit quasinormability from the underlying space.

Spaces of vectors inherit property $( ext{Ω})$ from the underlying space.

Results apply to broad classes of function spaces on Lie groups.

Abstract

We prove that the spaces of smooth and ultradifferentiable vectors associated with a representation of a real Lie group on a Fr\'{e}chet space $E$ are quasinormable if $E$ is so. A similar result is shown to hold for the linear topological invariant $(\Omega)$. In the ultradifferentiable case, our results particularly apply to spaces of Gevrey vectors of Beurling type. As an application, we study the quasinormability and the property $(\Omega)$ for a broad class of Fr\'{e}chet spaces of smooth and ultradifferentiable functions on Lie groups globally defined via families of weight functions.