No abstract

The paper introduces and studies the

We obtain new representations for the general positive and real-positive solutions of the equation axa à ¼ c in a C*-algebra using the characterization of positivity based on a matrix representation of an element and the generalized Schur complement. Applications to the equation AXA à ¼ C for operators between Hilbert spaces and for finite matrices are given.

The paper introduces and studies the weighted g-Drazin inverse for bounded linear operators between Banach spaces, extending the concept of the weighted Drazin inverse of Rakocevic and Wei (Linear Algebra Appl. 350 (2002), 25-39) and of Cline and Greville (Linear Algebra Appl. 29 (1980), 53-62). We use the Mbekhta decomposition to study the structure of an operator possessing the weighted g-Drazin inverse, give an operator matrix representation for the inverse, and study its continuity. An open problem of Rakocevic and Wei is solved.2000 Mathematics subject classification: primary 47A60, 46H30, 47A05, 47A10.

The paper studies the equation AX = C for bounded linear operators between Hilbert spaces, gives conditions for the existence of hermitian solutions and positive solutions, and obtains the formula for the general form of these solutions. Then the common hermitian and positive solutions to the equations AX = C and XB = D are studied and new representations of the general solutions are given. Many results for matrices are recovered as special cases, and the results of Phadke and Thakare [S.V. Phadke, N.K. Thakare, Generalized inverses and operator equations, Linear Algebra Appl. 23 (1979) 191-199] are corrected.