The general reader should know that symbol $A\sp {(2)}\sb {T,S}$
denotes a (unique) generalized inverse $G$ of a bounded linear map
$A$ from Banach space $X$ (containing a closed subset $T$) to Banach
space $Y$ (containing a closed subset $S$) such that the range of
$G$ is $T$ and the kernel of $G$ is $S$ while one has $GAG=G$. In
the sense of certain perturbations of $T$, $S$ and $A$ the paper
(basically, a set of explicit formulae assuming that $X$ and $Y$ are
Hilbert spaces over $\mathbb{C}$) offers explicit upper bounds of
the norm of the perturbed $G'$ and of the difference $G'-G$, using
the methods and amending the respective results of the 2012
monograph by one of the authors.
MR3243384 Du, Fapeng; Xue, Yifeng Perturbation analysis of $A\sp
{(2)}\sb {T,S}$ on Hilbert spaces. Funct. Anal. Approx. Comput. 5
(2013), no. 1, 5--13. 47A55 (15A09)