Invariant subspaces for Banach space operators with an annular spectral set

Consider an annulus $\Omega=\{z\in\mathbb{C}:r_ {0}<\vert z\vert<1\}$ for some $0<r_{0}<1$, and let $T$ be a bounded invertible linear operator on a Banach space $X$ whose spectrum contains $\partial\Omega$. Assume there exists a constant $K>0$ such that $\Vert p(T)\Vert~\leq~ K \sup\{\vert p(\lambda)\vert:\vert\lambda\vert\leq 1\}$ and $\Vert p(r_0T^{-1})\Vert\leq K \sup\{\vert p(\lambda)\vert:\vert\lambda\vert\leq 1\}$ for all polynomials $p$. Then there exists a nontrivial common invariant subspace for $T^{*}$ and ${T^{*}}^{-1}$.