Harish-Chandra's admissibility problem for Banach space representations of $\mathrm{SL}(2,\mathbb{R})$

TL;DR

This paper investigates the admissibility of irreducible Banach space representations of SL(2,R) and explores its connection to the invariant subspace problem.

Contribution

It establishes conditions under which such representations are admissible and links this to the invariant subspace problem.

Findings

Irreducible strongly continuous Banach space representations of SL(2,R) are admissible.

Admissibility of Banach space representations relates closely to the invariant subspace problem.

Abstract

We show that irreducible strongly continuous representations of $\mathrm{SL}(2,\mathbb{R})$ on certain Banach spaces are admissible and that the admissibility of Banach space representations of SL(2,R) and the invariant subspace problem are intimately related.