Approximate point spectrum (${\sigma }_{\mathrm{ap}}(T)$)

The approximate point spectrum consists of the scalars $\lambda$ for which there are unit vectors $x_k$ satisfying $\Vert (T-\lambda I)x_k\Vert \to 0$. Equivalently, $T-\lambda I$ is not bounded below.

The bilateral shift $(Tx{)}_n=x_{n+1}$ is unitary. It has empty point spectrum, while its approximate point spectrum is the unit circle. For $|\lambda |=1$, normalized vectors with entries ${\lambda }^n$ on an interval of length $N$ fail to satisfy $Tx=\lambda x$ only at the two endpoints, giving residual norm $\sqrt{2/N}$.