Invariant subspaces for Banach space operators with a multiply connected spectrum (Q2470879)

Consider a multiply connected domain in the complex plane \(\Omega={\mathbb D}\setminus\bigcup_{j=1}^{n}\overline{B}(\lambda_{j},r_{j})\), where \(\mathbb D\) denotes the unit disk and \(\overline{B}(\lambda_{j},r_{j})\subset\mathbb D\) denotes the closed disk centered at the point \(\lambda_{j}\) with the radius \(r_{j}\), \(j=1,\dots,n\). Let \(T\) be a bounded linear operator on a Banach space \(X\) whose spectrum contains \(\partial\Omega\) and does not contain the points \(\lambda_{j}\), \(j = 1,\dots,n\). Suppose that the operators \(T\) and \(r_{j}(T-\lambda_{j}I)^{-1}\) (\(j = 1,\dots,n\)) are polynomially bounded. Then there exists a nontrivial common invariant subspace for the operators \(T^{\ast}\) and \((T-\lambda_{j}I)^{\ast -1}\), \(j=1,\dots,n\).