Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 358 / 2

Choose normalized eigenvectors $u_n$ of the finite compressions and embed them as $v_n=P_n^{\ast }{u}_n\in H$. ThenThe bounded sequence $(v_n)$ has weakly convergent subsequences. If $v_{n_j}\rightharpoonup v$, then for every $y\in H$, strong convergence $P_n^{\ast }{P}_{n}y\to y$ and the compressed eigenvalue equation giveBecause $\lambda \notin \sigma (A)$, this forces $v=0$. Every weak cluster point is zero, so $v_n\rightharpoonup 0$. Since $⟨A{v}_n,v_n⟩={\lambda }_n\to \lambda$, the weak-null characterization givesThus finite-section spectral pollution of a bounded operator can occur only in its essential numerical range.

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LetThe tail spaces decrease, so $C_{n+1}\subseteq C_n$. If $\lambda \in {\bigcap }_n{C}_n$, choose a unit vector $v_n$ supported after coordinate $n$ with $|⟨A{v}_n,v_n⟩-\lambda |<1/n$. Such vectors converge weakly to zero, hence $\lambda \in W_e(A)$.

Conversely, if $v_j\rightharpoonup 0$ and $⟨A{v}_j,v_j⟩\to \lambda$, then for every fixed $n$ the first $n$ coordinates of $v_j$ tend to zero. After normalizing $Q_n{v}_j$, its numerical values still tend to $\lambda$, so $\lambda \in C_n$. Therefore

When the intersection is nonempty, decreasing closed sets have distance functions increasing pointwise to the distance from their intersection; on each compact set this convergence is uniform. This is preciselyIf $W_e(A)=\varnothing$ and a compact $K$ met every $C_n$, nestedness and compactness would supply a convergent sequence whose limit belongs to all $C_n$, a contradiction. Hence $K\cap C_n=\varnothing$ for all sufficiently large $n$.

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Let $H_0=-d^2/d{x}^2+x^2$. The Hermite expansion givesSince $T=-d^2/d{x}^2+i{x}^2$,and therefore

Write $X=\Vert xf{\Vert }^2$ and $S={\sum }_{m=n+1}^{\infty }(2m+1)|c_m{|}^2$. Then $\mathrm{Re}⟨Tf,f⟩=S-X$, $\mathrm{Im}⟨Tf,f⟩=X$, andIf $|z|\le n$, then $|\mathrm{Re}z+\mathrm{Im}z|\le \sqrt{2}n<2n+3$, soEvery fixed compact set is eventually excluded from the tail numerical ranges. Part (b) therefore implies

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