Perturbations and expressions for generalized inverses in Banach spaces and Moore-Penrose inverses in Hilbert spaces of closed linear operators |
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Authors: | Qianglian Huang Wenxiao Zhai |
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Affiliation: | College of Mathematics, Yangzhou University, Yangzhou 225002, China |
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Abstract: | The problems of perturbation and expression for the generalized inverses of closed linear operators in Banach spaces and for the Moore-Penrose inverses of closed linear operators in Hilbert spaces are studied. We first provide some stability characterizations of generalized inverses of closed linear operators under T-bounded perturbation in Banach spaces, which are exactly equivalent to that the generalized inverse of the perturbed operator has the simplest expression T+(I+δTT+)-1. Utilizing these results, we investigate the expression for the Moore-Penrose inverse of the perturbed operator in Hilbert spaces and provide a unified approach to deal with the range preserving or null space preserving perturbation. An explicit representation for the Moore-Penrose inverse of the perturbation is also given. Moreover, we give an equivalent condition for the Moore-Penrose inverse to have the simplest expression T†(I+δTT†)-1. The results obtained in this paper extend and improve many recent results in this area. |
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Keywords: | 15A09 47A05 65F20 |
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