NEW CONDITIONS FOR THE REVERSE ORDER LAWS FOR {1,3} AND {1,4}-GENERALIZED INVERSES ∗

Abstract. In this note, the reverse order laws for {1,3} and {1,4}-generalized inverses of matrices are considered. New necessary and sufficient conditions for (AB){1,3} ⊆ B{1,3} ·A{1,3} and (AB){1,4} ⊆ B{1,4} ·A{1,4} are presented. Key words. Generalized inverses, {1,3}-Generalized inverses, {1,4}-Generalized inverses, MPinverse, Reverse order law. AMS subject classifications. 15A09. 1. Results. Let A be a complex matrix. We denote by R(A), N(A), r(A) and nul(A) the range, the null space, the rank, and the nullity of a matrix A, respectively. By PM, we denote the orthogonal projection (P = P 2 = P ∗ ) on the subspace M. The Moore–Penrose inverse of A ∈ C n×m is the unique matrix A † ∈ C m×n satisfying the four Penrose equations in [10]

REPRESENTATIONS FOR THE DRAZIN INVERSE OF BOUNDED OPERATORS ON BANACH SPACE ∗

Abstract. In this paper a representation is given for the Drazin inverse of a 2 × 2operator matrix, extending to Banach spaces results of Hartwig, Li and Wei [SIAM J. Matrix Anal. Appl., 27 (2006) pp. 757–771]. Also, formulae are derived for the Drazin inverse of an operator matrix M under some new conditions. Key words. Operator matrix, Drazin inverse, D-invertibility, GD-invertibility. AMS subject classifications. 47A52, 47A62, 15A24. 1. Introduction. Throughout this paper X and Y are Banac