The Weighted g-drazin inverse for operators

Abstract

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 gDMP inverse of Hilbert space operators

“…Then we can easily verify this result. [4]. For the weighted core-EP inverse, the situation is similar in the case of the operator W A, but it is a little different for AW [13].…”

“…We now present some special classes of outer inverses. For a fixed operator W ∈ B(Y, X)\{0}, an operator A ∈ B(X, Y ) is called Wg-Drazin invertible [4] if there exists a unique operator B ∈ B(X, Y ) (denoted by A d,w ) such that AW B = BW A, BW AW B = B and A − AW BW A is quasinilpotent.…”

Weighted binary relations for operators on Banach spaces

“…[2] Let A ∈ B(X, Y ) and W ∈ B(Y, X)\{0}. Then A is W gDrazin invertible if and only if there exist topological direct sums X = X 1 ⊕X 2 ,…”

“…2 W 2 2 and B 2 2 W 2 2 quasinilpotent in B(Y 2 2 ), W 2 2 A 2 2 and W 2 2 B 2 2 quasinilpotent in B(X 2 2 ). We also prove the next theorem.…”

Análisis de la perturbación de la inversa de Drazin de matrices, elementos en anillos y operadores acotados en espacios de Banach

“…En [72, (1992) En [57, (2001)], J. J. Koliha y T. D. Tran, definen la g-Drazin inversa de un operador cerrado T en un espacio de Banach en el caso que el 0 sea un punto aislado del espectro de T . Recientemente, A. Dajić y J. J. Koliha, [26], han extendido el concepto de g-Drazin inversa a operadores lineales acotados entre espacios de Banach distintos, W-g-Drazin inversa.…”