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.
A unified treatment is given of generalizations of Farkas' theorem on linear inequalities to arbitrary convex cones and to dual pairs of real vector spaces of arbitrary dimension. Various theorems for locally convex spaces readily follow. The results are applied to duality and converse duality theory for linear programming and to a generalization of the Kuhn-Tucker theorem, both of these in spaces of arbitrary dimension and with inequalities involving arbitrary convex cones.1. Introduction. Farkas' theorem on systems of linear inequalities 11] (see also [ 10, Thm. 4-1) has been generalized to systems involving polyhedral cones [ 1], and to arbitrary cones in locally convex spaces [12]. A unified theory is presented here, based on arbitrary cones and dual pairs of real vector spaces. A necessary and sufficient condition is obtained (Theorem 2) for the solvability of a linear equation Ax b by a vector x in a given convex cone. A related necessary and sufficient condition (Theorem 1) is obtained as an inclusion relation between two cones. Although these results are closely related to the Hahn-Banach separation
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