# partial isometry on Hilbert spaces

###### Definition 1.

Let $\mathscr{H}$ and $\mathscr{K}$ be Hilbert spaces. An operator $W\in L(\mathscr{H},\mathscr{K})$ is called a partial isometry if $W$ is an isometry on $M=(\ker W)^{\perp}$. We then call $M=(\ker W)^{\perp}$ the initial space and $N=WM$ final space of $W$.

We need to show that the above definition is compatible with the general definition of partial isometry on rings. Indeed we have the following:

###### Proposition 1.

$W\in L(\mathscr{H},\mathscr{K})$ is a partial isometry iff $W^{*}W$ is a projection from $\mathscr{H}$ to $M$.

###### Proof.

We have:

 $\displaystyle W$ $\displaystyle\ \text{partial isometry with initial space}\ M$ $\displaystyle\Leftrightarrow\langle Wf,Wg\rangle$ $\displaystyle=\langle f,g\rangle\ \forall\ f,g\in M$ $\displaystyle\Leftrightarrow\langle W^{*}Wf,g\rangle$ $\displaystyle=\langle f,g\rangle\ \forall\ f\in M,g\in\mathscr{H}$ $\displaystyle\Leftrightarrow W^{*}Wf$ $\displaystyle=f,f\in M$ $\displaystyle\text{and}\ W^{*}Wf$ $\displaystyle=0,f\in M^{\perp}=\ker W$

###### Remark 1.

If $W\in L(\mathscr{H},\mathscr{K})$ is a partial isometry with initial space $M$ and final space $N$ we have:

 $\displaystyle W^{*}(Wf)$ $\displaystyle=f\ \forall\ f\in M$ $\displaystyle\ker W^{*}$ $\displaystyle=(\mathrm{ran}W)^{\perp}=N^{\perp}$

Thus $N$ is the initial space and $M$ the final space of $W^{*}$.

Title partial isometry on Hilbert spaces PartialIsometryOnHilbertSpaces 2013-03-22 18:35:00 2013-03-22 18:35:00 karstenb (16623) karstenb (16623) 7 karstenb (16623) Definition msc 47C10