Partial isometry

Remarks

Example: the backward shift $V(x_0, x_1, x_2, \ldots) = (x_1, x_2, \ldots)$ on $\ell^2(\N)$ has kernel spanned by $(1, 0, 0, \ldots)$. A vector orthogonal to the kernel has $x_0 = 0$, and then $\norm{Vx}^2 = \sum_{n \ge 1} \abs{x_n}^2 = \norm{x}^2$; so $V$ is a partial isometry.

Non-example: $T = 2I$ on $\C$ has kernel $\set{0}$, so its initial space is $\C$, but $\norm{T 1} = 2 \ne 1$.

Equivalent forms: $V$ is a partial isometry if and only if $V^*V$ is an orthogonal projection. Then $V^*V$ is the projection onto the initial space and $VV^*$ is the projection onto the final space. Another equivalent condition is $VV^*V = V$. The final space of a partial isometry is closed.