Let $V$ be a Banach space equipped with a continuous linear action of $S^1$ (meaning, the map $S^1\times V\to V$ is continuous). Assume that all the eigenspaces of the $S^1$-action are finite dimensional.

Does $V$ then admit an unconditional basis consisting of eigenvectors for the $S^1$-action?

If the answer is yes, then I'm also interested in the corresponding question when $V$ is Frechet, or when $V$ is a general complete locally convex topological vector space.