Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 335 / 3 / i / Solution

For admissible data, the Moore–Penrose inverse of an operator selects the minimum-norm least-squares solution:Its domain is $\mathrm{ran}A\oplus (\mathrm{ran}A{)}^{\perp }$. When the equation is consistent it is the solution of $Ax=y$ orthogonal to the kernel; for inconsistent admissible data it solves the normal equation for a linear inverse problem and projects away the component orthogonal to the range.

Take a singular system of a compact operator satisfying $A{v}_j={\sigma }_j{u}_j$, $A^{\ast }{u}_j={\sigma }_j{v}_j$. The generalized inverse isThe Picard criterion requires the squared coefficients to be summable. For an infinite-rank compact operator, ${\sigma }_j\to 0$, so $\Vert A^{†}{u}_j\Vert =1/{\sigma }_j\to \infty$ even though $\Vert u_j\Vert =1$. Hence $A^{†}$ is unbounded. Finite-rank compact operators do not have this obstruction. Data failing the Picard condition need not have a least-squares minimizer at all, even when the infimum residual is zero.