point/continuous/residual spectrum

In finite dimensions T - lambda I either is invertible or it is not, and not means an eigenvector exists — one clean failure mode. In infinite dimensions there are three distinct ways T - lambda I can fail to be invertible, and they carve the spectrum into three named pieces. Only one of them, the point spectrum, corresponds to actual eigenvalues; the other two are new phenomena that have no finite-dimensional shadow.

Precisely, lambda lies in the spectrum and falls into exactly one part. Point spectrum: T - lambda I is not injective, so there is a genuine eigenvector — these are the eigenvalues. Continuous spectrum: T - lambda I is injective with dense range, but the range is not all of the space and the inverse is unbounded. Residual spectrum: T - lambda I is injective but its range is not even dense. Together these three exhaust the spectrum.

What each part feels like: point spectrum is the familiar eigenvalue case, where you can solve T v = lambda v with a nonzero v. Continuous spectrum is the 'almost-eigenvalue' case — there are approximate eigenvectors, unit vectors u_n with ||(T - lambda I) u_n|| -> 0, but no exact eigenvector. Residual spectrum signals an asymmetry that often disappears once you pass to the adjoint, where it tends to show up as point spectrum.

Why this classification matters: it explains how an operator can have a large spectrum yet no eigenvalues at all. For self-adjoint and normal operators the residual spectrum is empty and the spectrum is real, which is what makes quantum observables well behaved — their continuous spectrum models scattering states while their point spectrum gives bound states with sharp energy levels.