Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 335 / 3 / ii / Solution

A singular system of a compact operator is a family $({\sigma }_i,u_i,v_i)$ with ${\sigma }_i>0$, ${\sigma }_i\to 0$, orthonormal $u_i\in Y$ and $v_i\in X$, andOn each singular vector, $A^{\ast }A+\alpha I$ acts by multiplication by ${\sigma }_i^2+\alpha$. HenceThe bounded filter $\sigma /({\sigma }^2+\alpha )$ replaces the unstable inverse factor $1/\sigma$.