Approximations to Generalized Inverses of Linear Operators.

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 ar...