Approximations to Generalized Inverses of Linear Operators
Robert H. Moore, M. Zuhair Nashed · SIAM Journal on Applied Mathematics · 1974
For a linear operator A on a Banach space $\mathcal{X}$, a generalized inverse $A^\dag $ is such that $E = AA^\dag $ and $P = A^\dag A$ are projectors onto the range of A and a topological complement $\mathcal{M}$ of the null space of A, respectively. It is shown that, if B approximates A so that $\delta = \| A^\dag (A - B)(I - EB) \| < 1$ (less generally, if $\| A^\dag (A - B) \| < 1$ then, while $B^\dag $ need not approximate $A^\dag $, still there is an operator $B^\phi $ mapping $\mathcal{X}$ onto $\mathcal{M}$ such that $F = BB^\phi $ and $Q = B^\phi B$ are projectors onto $B\mathcal{M}$ and $\mathcal{M}$, respectively, and $\| Q - P \|$, $\| B^\phi x - A^\dag x \|$, $\| Fx - Ex \|$ satisfy estimates tending to zero with $\delta $ when $\| Bx - Ax \| \to 0$, $x \in \mathcal{X}$. This is applied for $A = I - K,A_n = I - K_n $, where $\| K_n x - Kx \| \to 0$ and $\{ K_n ,n = 1,2, \cdots \}$ is collectively compact (e.g., K is an integral operator, $K_n $ is defined by numerical quadratures). Results are obtained for approximate least squares solutions of Fredholm integral equations of the second kind. Known results on the generalized inverse of perturbed matrices are also recovered.