Bilateral shift operator
= Bilateral shift operator
{title2=$T:\ell^2(\mathbb Z)\to\ell^2(\mathbb Z)$}
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 $\sqrt{2/N}$.
It is the two-sided sequence case of a <shift operator>.