In this paper, we first study the perturbations and expressions for the generalized inverses a (2) p,q , a (1,2) p,q , a (2,l) p,q and a (l)p,q with prescribed idempotents p and q. Then, we investigate the general perturbation analysis and error estimate for some of these generalized inverses when p, q and a also have some small perturbations.2010 Mathematics Subject Classification: 15A09; 46L05

Let and be Banach spaces, and let : → be a bounded linear operator. In this paper, we first define and characterize the quasi-linear operator (resp., out) generalized inverse (ℎ) , (resp., (2,ℎ) , ) for the operator , where ⊂ and ⊂ are homogeneous subsets. Then, we further investigate the perturbation problems of the generalized inverses (ℎ) , and (2,ℎ), . The results obtained in this paper extend some well-known results for linear operator generalized inverses with prescribed range and kernel.

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