On the differentiable vectors for contragredient representations

TL;DR

This paper explores the properties of contragredient representations of Lie groups to develop an abstract framework for understanding spaces of pseudo-differential operators, including their norm closures.

Contribution

It introduces a new abstract approach to characterize spaces of pseudo-differential operators using contragredient representations of Lie groups.

Findings

Provides simple results on contragredient representations

Offers an abstract method for describing pseudo-differential operator spaces

Relates to Nourrigat's description of the norm closure of order zero operators

Abstract

We establish a few simple results on contragredient representations of Lie groups, with a view toward applications to the abstract characterization of some spaces of pseudo-differential operators. In particular, this method provides an abstract approach to J.Nourrigat's recent description of the norm closure of the pseudo-differential operators of order zero.