Projective Analytic Vectors and Infinitesimal Generators

TL;DR

This paper introduces the concept of projective analytic vectors, weaker than traditional analytic vectors, and uses it to characterize generators of one-parameter groups, with applications to Lie group representations on Banach spaces.

Contribution

It defines projective analytic vectors and proves their role in characterizing generators of one-parameter groups via projective limits, expanding the understanding of such operators.

Findings

Characterization of generators of one-parameter groups using projective analytic vectors

Application to Lie group representations on Banach spaces

Analysis of algebras of functions and pseudodifferential operators

Abstract

We establish the notion of a ``projective analytic vector'', whose defining requirements are weaker than the usual ones of an analytic vector, and use it to prove generation theorems for one-parameter groups on locally convex spaces. More specifically, we give a characterization of the generators of strongly continuous one-parameter groups which arise as the result of a projective limit procedure, in which the existence of a dense set of projective analytic vectors plays a central role. An application to strongly continuous Lie group representations on Banach spaces is given, with a focused analysis on concrete algebras of functions and of pseudodifferential operators.