Perturbation analysis for the reduced minimum modulus of bounded linear operator in Banach spaces. (Q1412421)

Let \(X\), \(Y\) be Banach spaces (both over \(\mathbb{C}\)) and let \(T: X\to Y\) be a bounded linear operator with the generalized inverse \(T^+\). Consider a bounded perturbation \(\overline T= T+\delta T\) with the property \(\| T^+\|\,\|\delta T\|< {1\over 2}\). It is shown in the paper under review that the reduced minimum modulus \(\gamma\) of \(T\) defined as \[ \gamma(T)= \inf\{\| T\|: \text{dist}(x,\text{ker\,}T)= 1,\,x\in X\} \] under appropriate conditions is upper semicontinuous (continuous). This implies a kind of stability of these perturbations.