Abstract

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 TD = DT, D = TD²) and T^k = T^k+1D 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.