Sharp Norm-Estimations for Moore–Penrose Inverses of Stable Perturbations of Hilbert $C^*$-Module Operators
Qingxiang Xu, Yimin Wei, Yangyang Gu · SIAM Journal on Numerical Analysis · 2010
It is well known that norm upper bound $\Vert\bar{T}^{\dag}\Vert\leq\frac{\Vert T^{\dag}\Vert}{1-\Vert T^{\dag}\Vert\cdot\Vert\Delta(T)\Vert}$ plays a fundamental role in stable (resp., rank-preserving) perturbation analysis for Moore–Penrose inverses of Hilbert space operators (resp., matrices). In this paper, in the general setting of Hilbert $C^*$-module operators, we provide a new approach to the study of norm upper bounds of $\bar{T}^{\dag}$ and obtain a sharp estimation for $\Vert\bar{T}^{\dag}\Vert$. This obtained estimation is applied to the study of the linear least squares problem and of perturbation analysis for the Schur complement of a positive semidefinite operator matrix. Some new norm upper bounds for Schur complements of positive semidefinite operator matrices are derived.