= Solution
The operator $A^*A$ is positive, self-adjoint, and compact. Because $\operatorname{ran}A$ is infinite dimensional, the <spectral theorem for compact Hermitian operators> gives positive spectral values tending to zero. Even if $\ker A=\{0\}$, zero remains in the spectrum as a limit point. Hence for every $\tau$,
$$
\|I-\tau A^*A\|
=\sup_{\lambda\in\sigma(A^*A)}|1-\tau\lambda|
\geq1.
$$
The strict inequality $\|I-\tau K\|<1$ needed for the operator-norm <Neumann series> is impossible, so formula (3) cannot be applied directly to invert $A^*A$.
Back to article page
