Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 358 / 2 / ii / Solution

Set $T=A-zI$. The squares of the singular values of $T{P}_n^{\ast }$ are the eigenvalues of the finite-dimensional positive operatorThe Rayleigh-Ritz variational principle and its min-max characterization show that, as the trial space grows, its $(j+1)$-st eigenvalue counted upward cannot increase. Therefore, for each fixed $j\ge 0$,once $n>j$. Being nonnegative, it has a limitThe core assumption ensures that these Ritz limits are the min-max values of $T^{\ast }T$, rather than values for a smaller closed restriction.

For a normal operator, the spectral theorem gives three possibilities.

vanish, while ${\sigma }_{\inf +m}(T)>0$.

For $k\in \mathbb{N}$, defineIn the first case $H_k=0$ for all sufficiently large $k$. In the second case its first $m$ summands tend to one and all remaining summands eventually vanish, so $H_k\to m$. In the third case $H_k=k+1\to +\infty$. These are exactly the three values in the definition of $h$, and hence