A reflexivity result concerning banach space operators with a multiply connected spectrum

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Official URL: http://dx.doi.org/10.1007/s00020-010-1818-3

We consider a multiply connected domain Ω which is obtained by removing n closed disks which are centered at λ j with radius r j 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, . . . , λ n , and the operators T and r j (T − λ j I)−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.