Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 326 / 3 / d / Solution

The operator $A^{\ast }A$ is positive, self-adjoint, and compact. Because $\mathrm{ran}A$ 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 $\tau$,The strict inequality $\Vert I-\tau K\Vert <1$ needed for the operator-norm Neumann series is impossible, so formula (3) cannot be applied directly to invert $A^{\ast }A$.