continuous spectrum

The continuous spectrum captures a way an operator can fail to be invertible that has no finite-dimensional counterpart. There is no eigenvector — you cannot solve T v = lambda v exactly — but you can get arbitrarily close: there are 'approximate eigenvectors' that the operator almost scales by lambda. The inverse formally exists on a dense set but cannot be made bounded.

In one standard decomposition, lambda lies in the continuous spectrum of T if T - lambda*I is injective (no eigenvector) and has dense range (so the inverse is defined on a dense set), but that inverse is unbounded (so T - lambda*I is not boundedly invertible, hence lambda is in the spectrum). Such a lambda has an approximate eigenvector sequence: unit vectors x_n with ||(T - lambda*I) x_n|| -> 0.

Continuous spectrum is generic for operators with no eigenvalues — multiplication operators and the shift are the prototypes. A caveat about terminology: there are several inequivalent conventions for splitting the spectrum (the injective-dense-range definition above versus the measure-theoretic 'continuous part' of a spectral measure used in the spectral theorem), so always check which decomposition an author means.