point spectrum

The point spectrum is the 'classical' part of the spectrum — the actual eigenvalues, the values at which the operator has a genuine eigenvector that it merely rescales. These are the spectral points you can detect by solving T v = lambda v with v nonzero, exactly as in finite-dimensional linear algebra.

Formally, the point spectrum of an operator T is the set of scalars lambda for which T - lambda*I fails to be injective, equivalently for which there exists a nonzero vector v with T v = lambda v. It is one piece of the standard decomposition of the spectrum: the point spectrum (non-injectivity), the continuous spectrum (injective with dense but not surjective range), and the residual spectrum (range not even dense).

In finite dimensions the spectrum is exhausted by the point spectrum, so every spectral value is an eigenvalue. In infinite dimensions this fails badly: an operator can have empty point spectrum yet a large spectrum (the shift operator is the standard example). Conversely the point spectrum need not be closed, so it is not always all of what one sees in the limit.