Additive perturbation results for the Drazin inverse

AbstractAdditive perturbation results for the generalized Drazin inverse of Banach space operators are presented. Precisely, if Ad denotes the generalized Drazin inverse of a bounded linear operator A on an arbitrary complex Banach space, then in some special cases (A + B)d is computed in terms of Ad and Bd. Thus, recent results of Hartwig, Wang and Wei (Linear Algebra Appl. 322 (2001), 207–217) are extended to infinite dimensional settings with simplified proofs.

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The Drazin inverse matrix modification formulae with Peirce corners

Filomat ◽

10.2298/fil2108605z ◽

2021 ◽

Vol 35 (8) ◽

pp. 2605-2616

Author(s):

Daochang Zhang ◽

Dijana Mosic ◽

Jianping Hu

Keyword(s):

Inverse Matrix ◽

Drazin Inverse ◽

Special Cases ◽

Matrix Modification ◽

Additive Perturbation ◽

Special Matrix

Our motivation is to derive the Drazin inverse matrix modification formulae utilizing the Drazin inverses of adequate Peirce corners under some special cases, and the Drazin inverse of a special matrix with an additive perturbation. As applications, several new results for the expressions of the Drazin inverses of modified matrices A ?? CB and A ?? CDdB are obtained, and some well known results in the literature, as the Sherman-Morrison-Woodbury formula and Jacobson?s Lemma, are generalized.

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Cline’s formula for g-Drazin inverses in a ring

Filomat ◽

10.2298/fil1908249c ◽

2019 ◽

Vol 33 (8) ◽

pp. 2249-2255

Author(s):

Huanyin Chen ◽

Marjan Abdolyousefi

Keyword(s):

Operator Theory ◽

Spectral Properties ◽

Associative Ring ◽

Linear Operators ◽

Drazin Inverse ◽

Bounded Linear Operators ◽

Bounded Linear ◽

Cline’S Formula

It is well known that for an associative ring R, if ab has g-Drazin inverse then ba has g-Drazin inverse. In this case, (ba)d = b((ab)d)2a. This formula is so-called Cline?s formula for g-Drazin inverse, which plays an elementary role in matrix and operator theory. In this paper, we generalize Cline?s formula to the wider case. In particular, as applications, we obtain new common spectral properties of bounded linear operators.

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Acute perturbation of Drazin inverse and oblique projectors

Frontiers of Mathematics in China ◽

10.1007/s11464-018-0731-y ◽

2018 ◽

Vol 13 (6) ◽

pp. 1427-1445 ◽

Cited By ~ 2

Author(s):

Sanzheng Qiao ◽

Yimin Wei

Keyword(s):

Drazin Inverse

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Expressions for the g-Drazin inverse of additive perturbed elements in a Banach algebra

Linear Algebra and its Applications ◽

10.1016/j.laa.2009.01.017 ◽

2010 ◽

Vol 432 (8) ◽

pp. 1885-1895 ◽

Cited By ~ 9

Author(s):

N. Castro-González ◽

M.F. Martínez-Serrano

Keyword(s):

Banach Algebra ◽

Drazin Inverse

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New results on the W-weighted Drazin inverse

10.1063/1.4882482 ◽

2014 ◽

Author(s):

M. Nikuie ◽

M. Z. Ahmad

Keyword(s):

Drazin Inverse ◽

Weighted Drazin Inverse

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Frequency-Domain Robust Performance Condition for Controller Uncertainty in SISO LTI Systems: A Geometric Approach

Journal of Control Science and Engineering ◽

10.1155/2009/746762 ◽

2009 ◽

Vol 2009 ◽

pp. 1-9

Author(s):

Vahid Raissi Dehkordi ◽

Benoit Boulet

Keyword(s):

Frequency Domain ◽

Linear Time ◽

Order Reduction ◽

Geometric Approach ◽

Geometrical Approach ◽

Robust Performance ◽

Linear Time Invariant ◽

Additive Perturbation ◽

Plant Uncertainty ◽

Invariant Control

This paper deals with the robust performance problem of a linear time-invariant control system in the presence of robust controller uncertainty. Assuming that plant uncertainty is modeled as an additive perturbation, a geometrical approach is followed in order to find a necessary and sufficient condition for robust performance in the form of a bound on the magnitude of controller uncertainty. This frequency domain bound is derived by converting the problem into an optimization problem, whose solution is shown to be more time-efficient than a conventional structured singular value calculation. The bound on controller uncertainty can be used in controller order reduction and implementation problems.

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