A Reflexivity Result Concerning Banach Space Operators with a Multiply Connected Spectrum

Abstract

We consider a multiply connected domain Ω which is obtained by removing n closed disks which are centered at λj with radius rj for j = 1, . . . , n from the unit disk. We assume that T is a bounded linear operator on a separable reflexive Banach space whose spectrum contains ∂Ω and does not contain the points λ1, λ2, . . . , λn, and the operators T and rj(T − λjI)−1 are polynomially bounded. Then either T has a nontrivial hyperinvariant subspace or the WOT-closure of the algebra {f(T) : f is a rational function with poles off \({\overline\Omega}\)} is reflexive.