Approximate point spectrum

Summary

The approximate point spectrum of an operator $T$ is the set of numbers $\lambda$ that are eigenvalues up to an arbitrarily small error. This means that there are unit vectors $x$ with $Tx - \lambda x$ as small as one likes. It contains every eigenvalue and the whole continuous spectrum, and also the boundary of the spectrum. For self-adjoint and normal operators on Hilbert spaces it is the whole spectrum.