Does there exists a triple $(G, X, \pi)$, where $G$ is a compact group, $X$ an infinite dimensional Banach space over $\mathbf{C}$, and $\pi : G \to B(X)$ a strongly continuous representation of $G$, such that if $Y$ is a non-zero closed invariant subspace of $X$ in the sense that $\pi(g)Y \subseteq Y$ for all $g \in G$, then $Y = X$ (we call nonzero strongly representations of $G$ satisfying this property topologically irreducible)?


I am not sure that if this is settled already, as the question seems natural, yet I am ignorant of the relevant literature, nor could I come up with a proof or disproof. Let me be (perhaps overly) precise about the question asked in the title. By compact group, I mean Hausdorff plus quasi-compact, and topologically irreducible means that the only invariant closed subspaces are either $0$ or the whole space. More generally, can we find such an example by replacing infinite dimensional Banach spaces by complete infinite dimensional locally convex Hausdorff spaces instead? While we are on the subject, I'd also appreciate some recommendation of books, papers/surveys on representations of topological groups on Banach spaces that is not focused on unitary representations on Hilbert spaces?



Existence of a strongly continuous topologically irreducible representation of a compact group on an infinite dimensional Banach space?
For Banach spaces, the question is no : there is a form of the Peter-Weyl theorem, due to Shiga, which implies that in every Banach space representation of a compact group, the finite-dimensional sub-representations span a dense subspace. In particular, strongly continuous irreducible representations on a Banach space are finite-dimensional. I am not sure about arbitrary locally convex topological vector space.


Shiga's paper is [here](https://projecteuclid.org/journals/journal-of-the-mathematical-society-of-japan/volume-7/issue-3/Representations-of-a-compact-group-on-a-Banach-space/10.2969/jmsj/00730224.full). If the link does not work, the precise reference is:


K. Shiga, Representations of a compact group on a Banach space, Journal of the Mathematical Society of Japan 7 (1955), 224–248.


