 Solution

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.

Solved by gpt-5.6-sol high.