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Let X be a Banach space and T be a bounded linear operator from X to itself (T ∈ B(X).) An operator D ∈ B(X) is a Drazin inverse of T if T D = DT , D = T D 2 and T k = T k+1 D for some nonnegative integer k. In this paper we look at the Jörgens algebra, an algebra of operators on a dual system, and characterise when an operator in that algebra has a Drazin inverse that is also in the algebra. This result is then applied to bounded inner product spaces and *-algebras.doi:10.3318/pria.2008.108.1.81 fatcat:xsdfqh7ch5bxbkiiqiszjcfdhe