On the perturbation of the reduced minimum modulus of bounded linear operators.
In the Banach space \(L(X,Y)\) of all bounded linear operators between two Banach spaces \(X\) and \(Y\), the notion of the reduced minimum modulus \(\gamma(T), T\in L(X,Y),\) is introduced by the formula \(\gamma(T)=\inf_{x\in X}\frac{Tx}{d(x,N(T))}\). Here \(N(T)\) is the zero subspace of \(T\), \(d(x,N(T))\) is the distance between the element \(x\) and the set \(N(T)\) and 0/0 is defined to be \(\infty\). The author obtains some perturbation results for the reduced minimum modulus \(\gamma(T)\) for Hilbert and Banach spaces. These results imply perturbation bounds for generalized inverses and least squares problems in Hilbert spaces. Reviewer's remarks: The obtained results correct some mistakes in the works of \textit{G.-L. Chen} and \textit{Y.-M. Wei} [Chin. J. Contemp. Math. 19, 245--252 (1998; Zbl 1015.47001); Appl. Math. Comput. 121, 177--183 (2001; Zbl 1038.47006)].
scientific article; zbMATH DE number 1978064