Wandering-vector characterization of a unilateral shift
= Wandering-vector characterization of a unilateral shift
An operator $T$ is a unilateral shift exactly when it is an isometry, $(\operatorname{Im}T)^\perp$ is one-dimensional, and $\bigcap_{n\geq1}\operatorname{Im}(T^n)=\{0\}$. A unit vector $e_1$ in the first orthogonal complement is wandering, and $e_n=T^{n-1}e_1$ is a complete orthonormal basis.