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

Andreas Debrouwere, Michiel Huttener, Jasson Vindas

Abstract

We prove that the spaces of smooth and ultradifferentiable vectors associated with a representation of a real Lie group on a Fréchet space $E$ are quasinormable if $E$ is so. A similar result is shown to hold for the linear topological invariant $(Ω)$. 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 $(Ω)$ for a broad class of Fréchet spaces of smooth and ultradifferentiable functions on Lie groups globally defined via families of weight functions.

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

Abstract

We prove that the spaces of smooth and ultradifferentiable vectors associated with a representation of a real Lie group on a Fréchet space are quasinormable if is so. A similar result is shown to hold for the linear topological invariant . 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 for a broad class of Fréchet spaces of smooth and ultradifferentiable functions on Lie groups globally defined via families of weight functions.
Paper Structure (12 sections, 29 theorems, 160 equations)

This paper contains 12 sections, 29 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Spaces of vector-valued ultradifferentiable functions on open subsets of Rⁿ
  3. Spaces of vector-valued smooth and ultradifferentiable functions on manifolds
  4. Spaces of smooth and ultradifferentiable vectors associated with Lie group representations
  5. Some auxiliary results
  6. An approximation result
  7. The parametrix method
  8. Main results
  9. Quasinormability
  10. The condition (Ω)
  11. Right-invarant spaces of smooth and ultradifferentiable functions
  12. Examples of right-invarant Fréchet spaces of smooth and ultradifferentiable functions

Key Result

Theorem 1.1

Let $\pi$ be a locally equicontinuous representation of a Lie group on a Fréchet space $E$. Let $E^\infty$ be the space of smooth vectors associated with $\pi$.

Theorems & Definitions (70)

  • Theorem 1.1
  • Theorem 1.2
  • Example 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 60 more