Let $X$ be a complex Banach space, and let $T:X\longrightarrow X$ be a bounded linear operator. Let $\sigma_1$ and $\sigma_2$ be disjoint compact subsets of $\mathbb{C}$ for which $\sigma_1\cup \sigma_2 = \sigma(T)$. Let $P$ be the spectral projection for $T$ over $\sigma_1$ (say). Must $P$ necessarily preserve every $T$-invariant closed subspace of $X$?

I'm fairly sure the answer is yes if the polynomially convex hulls of $\sigma_1$ and $\sigma_2$ are disjoint, but suspect that the answer is no in general. Can anyone think of an example where something goes wrong?

Thanks for any suggestions!