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Representations of 1,2,3-inverses, 1,2,4-inverses, and Drazin inverse of a partitioned matrix M=ABCD related to the generalized Schur complement are studied. First, we give the necessary and sufficient conditions under which 1,2,3-inverses, 1,2,4-inverses, and group inverse of a 2×2 block matrix can be represented in the Banachiewicz-Schur forms. Some results from the paper of Cvetković-Ilić, 2009, are generalized. Also, we expressed the quotient property and the first Sylvester identity in terms of the generalized Schur complement. [ABSTRACT FROM AUTHOR]

In this paper, we study partial orders in terms of the core-nilpotent decomposition. We derive some characterizations of the C-N partial ordering, create a new partial ordering (the G-Drazin partial ordering) which is a generalization of C-N partial ordering, and give some properties and characterizations of the G-Drazin partial ordering. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to compute the generalized inverse AT,S(2) over Banach spaces by using semi-iterative method and to present the error bounds of the semi-iterative method for approximating AT,S(2). [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we use the Drazin inverse to derive some new equivalences of the reverse order law for the group inverse in unitary rings. Moreover, if the ring has an involution, we present more equivalences when both involved elements are EP. [Copyright &y& Elsevier]

Abstract: In this paper, we give necessary and sufficient conditions for the absorption laws in terms of and -inverses. Also, we consider the various types of mixed absorption law for the generalized inverses. [Copyright &y& Elsevier]

Abstract: The reverse order laws for {1,3,4}-generalized inverses of a product of two operators have been studied by Wang et al. [J. Wang, H. Zhang, G. Ji, A generalized reverse order law for the products of two operators, Journal of Shaanxi Normal University, 38 (4) (2010), 13–17]. In this paper using a block-operator matrix technique we study mixed-type reverse-order laws for {1,3,4}-generalized inverses over Hilbert spaces. [Copyright &y& Elsevier]

In this paper, we present the explicit expressions of the perturbation of the Drazin inverse under different conditions. Also, we give the upper bounds of ‖(A+E)D−A D‖ P /‖A D‖ P for these cases. [ABSTRACT FROM PUBLISHER]

We consider the various representations of the outer inverse of an operator A on Banach spaces. [ABSTRACT FROM PUBLISHER]

In this article, we discuss the group inverse of aP + bQ + cPQ + dQP + ePQP + fQPQ + gPQPQ of idempotent matrices P and Q, where a, b, c, d, e, f, g ∈  and a ≠ 0, b ≠ 0, put forward its explicit expressions, and some necessary and sufficient conditions for the existence of the group inverse of aP + bQ + cPQ. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we establish some results relating star, left-star, right-star, minus ordering and the reverse order law under certain conditions on Moore–Penrose invertible elements of -algebras. [Copyright &y& Elsevier]

Based on the conditions a b 2 = 0 and b π (a b) ∈ A d , we derive that (a b) n , (b a) n , and a b + b a are all generalized Drazin invertible in a Banach algebra A , where n ∈ N and a and b are elements of A. By using these results, some results on the symmetry representations for the generalized Drazin inverse of a b + b a are given. We also consider that additive properties for the generalized Drazin inverse of the sum a + b . [ABSTRACT FROM AUTHOR]