Invariant subspaces for Banach space operators with an annular spectral set

Abstract

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