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

Put $T=A-zI$. We use the following closed-range lemma:Indeed,Away from this kernel, zero is separated from the spectrum of the positive self-adjoint operator $T^{\ast }T$ exactly whenfor some $c>0$. This is equivalent to closed range. Zero is then absent from the essential spectrum exactly when $\dim \ker T<\infty$. Similarly,because $\ker T^{\ast }=(\mathrm{ran}T{)}^{\perp }$ measures the cokernel.

Applying the lecture definitions of the three essential spectra of a closed operator now givesand

If $A$ is normal, so is $T$. The spectral theorem givesand maps the spectral mass of $T$ at $0$ exactly to the spectral mass of $T^{\ast }T$ at $0$. Thus zero is isolated with finite multiplicity for $T^{\ast }T$ exactly when $z$ is an isolated eigenvalue of finite multiplicity for $A$. Consequently

Normality is essential. Let $U{e}_j=e_{j+1}$ be the unilateral shift and take $A=U^{\ast }$. Its spectrum is the closed unit disk, so $0$ is not in the discrete spectrum. Butwhose zero eigenvalue is isolated and simple. Hence $0\in {\mathrm{Sp}}_{\mathrm{d}}(A^{\ast }A)$ while $0\notin {\mathrm{Sp}}_{\mathrm{d}}(A)$.

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

Letand, for $w\in \mathbb{C}$, form the finite rectangular matrixEvery entry is available from $\Lambda$. Let its singular values, padded and ordered as in the question, beFor fixed $n_2$, Parseval's identity gives convergence of the finite Gram matrices:as $n_1\to \infty$. Consequently, ifthenSingular values are Lipschitz under scalar shifts, so the subsequent limit $z_{n_2}\to z$ and part ii give

Define the finite-information arithmetic functionsFinite-matrix singular values can be obtained by arithmetic eigenvalue approximation, so these functions form the required ${\Delta }_4^A$ arithmetic tower using only $\Lambda$. The two inner limits recover the limiting singular values, while the outer limit is exactly the multiplicity formula from part ii: