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

We show that irreducible strongly continuous representations of $\mathrm{S}\mathrm{L}(2,\mathbb{R})$ 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.