On the Minimum Property of the Pseudo $x$-Condition Number for a Linear Operator

Abstract

It is well known that the $x$-condition number of a linear operator is a measure of ill condition with respect to its generalized inverses and a relative error bound with respect to the generalized inverses of operator $T$ with a small perturbation operator E, namely,$$\frac{\|(T+E)^+-T^+\|}{\|T^+\|}\leq \frac{x(T)\frac{\|E\|}{\|T\|}}{1-x(T)\frac{\|E\|}{\|T\|}},$$ where $x{T}=\|T\|\dot\|T^+\|$. The problem is whether there exists a positive number $μ(T)$ independent of $E$ but dependent on $T$ such that the above relative error bound holds and $μ(T)＜x(T)$.
In this paper, an answer is given to this problem.