Approximation in Banach space representations of compact groups

Let $\pi \colon G \longrightarrow \mathcal B(E)$ be a continuous representation of a compact group $G$ on a Banach space $E$. We prove that the set of vectors $\pi (h)x$, as $h$ runs through the set $T(G)$ of all trigonometric polynomials on $G$, and $x$ runs through $E$, spans an invariant dense linear subspace of $E$. We prove the existence of a topological direct sum decomposition $E=\bigoplus _{\theta \in \widehat G}E_\theta$ for $E$, where each $E_\theta$ is a closed $\pi$-invariant subspace of $E$. If $\lambda _p\colon M(G)\longrightarrow \mathcal B(L^p(G))$, $p\in (1,\infty )$, is the left regular representation of the measure algebra $M(G)$ and $B\subset PM_p(G)$ is a homogeneous Banach space, we show that $B\cap \lambda _p(T(G))$ is norm dense in $B$. Since Hilbert space techniques are not available, new machinery is developed in the paper for the proofs.