Representations for the Drazin inverse of bounded operators on Banach space

Abstract

Abstract. In this paper a representation is given for the Drazin inverse of a 2 × 2 operator 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.

The generalized Drazin inverse of operator matrices

“…AA d B = 0 and C(I − AA d ) = 0, we have A d B = 0 and CA i B = CA i (I − AA d )B = 0, for any nonnegative integer i. So M satisfies the condition of Corollary 3.The following result is a direct corollary of Theorem 3.2, which extends [18, Theorem 2.2] to bounded linear operators on a Banach space, and generalizes the results in[9,13,16]. If A and D are generalized Drazin invertible and BC = 0, BDC = 0 and BD 2 = 0, then M is generalized Drazin invertible and…”

“…In Section 4, we give a new additive result of the generalized Drazin inverse for two bounded linear operators P, Q ∈ B(X) with P Q d = 0 and P Q i P = 0, for any integer i ≥ 1. As corollaries, many results in [4,5,9,13,14,16,18] are generalized.…”

“…Finding an explicit representation for the generalized Drazin inverse of an operator matrix M = A B C D in terms of A, B, C, D and related generalized Drazin inverses has been studied by several authors [4,5,9,[11][12][13][14][15][16]26,32,33,36,37]. Djordjević and Stanimirović [16] generalize the well-known result in [24,31] concerning the Drazin inverse of block 2 × 2 upper triangular matrices to the generalized Drazin inverse for block triangular operator matrices, and further consider the case that BC = 0, BD = 0 and DC = 0.…”

“…Djordjević and Stanimirović [16] generalize the well-known result in [24,31] concerning the Drazin inverse of block 2 × 2 upper triangular matrices to the generalized Drazin inverse for block triangular operator matrices, and further consider the case that BC = 0, BD = 0 and DC = 0. These requirements are relaxed and new conditions are presented in [4,5,9,[12][13][14], for example, the condition ABC = 0 is dealt with in [4,5,14] under some extra assumptions. This paper is inspired by [4,14,18].…”

Some results on the generalized Drazin inverse of operator matrices

“…The GD-inverse formula M d of a 2 Â 2 operator matrix appears frequently in many topics and has long been studied [4,7,[19][20][21]24,25,28,29,[33][34][35][36][37]. To find explicit representation for the GD-inverse of a general 2 Â 2 block matrix in terms of A d and D d with arbitrary A, B, C and D, posed by Campbell and Meyer in [8,10], appears to be difficult.…”