Essential numerical range ($W_e(A)$)

The essential numerical range consists of limits $\lim ⟨A{v}_n,v_n⟩$ along unit vectors converging weakly to zero. It is stable under compact perturbations and may be empty for an unbounded operator.

Spectral pollution is convergence of finite-section eigenvalues to points outside the spectrum of the limiting operator. For bounded operators, such pollution lies in the essential numerical range.

Attouch--Wets convergence of closed sets is locally uniform convergence of their distance functions on bounded subsets. Decreasing closed sets converge to their nonempty intersection in this topology.