Perturbation analysis of bounded homogeneous generalized inverses on Banach spaces

Let $X, Y$ be Banach spaces and $T : X \to Y$ be a bounded linear operator. In this paper, we initiate the study of the perturbation problems for bounded homogeneous generalized inverse $T^h$ and quasi--linear projector generalized inverse $T^H$ of $T$. Some applications to the representations and perturbations of the Moore--Penrose metric generalized inverse $T^M$ of $T$ are also given. The obtained results in this paper extend some well--known results for linear operator generalized inverses in this field.