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Consider an annulus Ω = {z ∈ C: r 0 0 such that ||p(T)|| < J<sup{|p(λ)|: |λ| < 1} and ∥p(r o T -1 )∥ < K sup{|p(λ)|: |λ| ≤ 1} for all polynomials p. Then there exists a nontrivial common invariant subspace for T* and T* -1 .

Invariant subspaces for Banach space operators with an annular spectral set