Category:Partial Isometries

This category contains results about Partial Isometries.
Definitions specific to this category can be found in Definitions/Partial Isometries.

Let $\struct {\HH_1, \innerprod \cdot \cdot_1}$ and $\struct {\HH_2, \innerprod \cdot \cdot_2}$ be Hilbert spaces.

Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ be the inner product norms

Let $T : \HH_1 \to \HH_2$ be a bounded linear transformation.

Let $\paren {\map \ker T}^\bot$ be the orthocomplement of the kernel $\map \ker T$ in $\HH_1$.

We say that $T$ is a partial isometry if and only if:

$\norm {T x}_2 = \norm x_1$ for each $x \in \paren {\map \ker T}^\bot$.