Approximate eigenvalue ($\lambda \in {\sigma }_{\mathrm{ap}}(T)$)

An approximate eigenvalue of a bounded operator $T$ is a scalar $\lambda$ admitting unit vectors $x_n$ with $\Vert (T-\lambda I)x_n\Vert \to 0$. Every approximate eigenvalue belongs to the spectrum; for a normal operator, all spectral points are approximate eigenvalues.

The unit vectors realizing an approximate eigenvalue are approximate eigenvectors: their residual tends to zero, although they need not converge to any genuine eigenvector.