1. Introduction. The main theme of this paper can be described as a study of the Drazin inverse for bounded linear operators in a Banach space X when 0 is an isolated spectral point of the operator. This inverse is useful for instance in the solution of differential equations formulated in a Banach space X. Since the elements of X rarely enter into our considerations, the exposition seems to gain in clarity when the operators are regarded as elements of the Banach algebra L(X).An element a of a complex Banach algebra A is called regular (or relatively regular) if there is x e A such that axa = a. Relatively regular elements have been extensively studied in the case that A is the Banach algebra L{X) of all bounded linear operators on a complex Banach space X\ they have been shown to generalize in certain aspects invertible operators.If a is relatively regular, then it has a generalized inverse, which is an element b e A satisfying the equations aba = a and bab = b. (See [18] for a comprehensive account of generalized inverses.) A relation between a relatively regular element and its generalized inverse is reflexive in the sense that if b is a generalized inverse of a, then a is a generalized inverse of b.In 1958, Drazin [7] introduced a different kind of a generalized inverse in associative rings and semigroups that does not have the reflexivity property but commutes with the element.

In this paper we define and study a generalized Drazin inverse x D for ring elements x, and give a characterization of elements a, b for which aa D = bb D . We apply our results to the study of EP elements in a ring with involution.2000 Mathematics subject classification: primary 16A32, 16A28, 15A09; secondary 46H05, 46L05.

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