A bounded operator \(S\) on a Hilbert space is a partial isometry if

\begin{align*} & S = SS^*S &\text{ or} \\ & S^*S &\text{ is a projection or} \\ & SS^* &\text{ is a projection} \end{align*}

Then \(S^*S\) is the projection on \((\ker)^\perp\) and \(SS^*\) is the projection on the range of \(S\).

If \(S\) is a partial isometry in a $C^*$-algebra \(A\) then we call \(S^*S\) the initial projection of \(S\) and \(SS^*\) the final projection of \(S\).