Definition 35.2.2 (Partial Isometry).label Let $H$ be a complex Hilbert space and $T \in B(H)$, then $T$ is a partial isometry if $T|_{\ker(T)^\perp}$ is an isometry. In which case, $\ker(T)^{\perp}$ is the initial space of $T$, and $T(H)$ is the final space of $T$.