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<p>We prove that the spaces of smooth and ultradifferentiable vectors associated with a representation of a real Lie group on a Fréchet space <i>E</i> are quasinormable if <i>E</i> is so. A similar result is shown to hold for the linear topological invariant <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((\Omega )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. 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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\((\Omega )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for a broad class of Fréchet spaces of smooth and ultradifferentiable functions on Lie groups globally defined via families of weight functions.</p>

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Quasinormability and Property \((\Omega )\) for Spaces of Smooth and Ultradifferentiable Vectors Associated with Lie Group Representations

  • 期刊论文

  • 发表日期：

    2026年6月16日

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  • Andreas Debrouwere,
  • Michiel Huttener,
  • Jasson Vindas

摘要

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 \((\Omega )\) ( 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 \((\Omega )\) ( Ω ) for a broad class of Fréchet spaces of smooth and ultradifferentiable functions on Lie groups globally defined via families of weight functions.