partial isometry on Hilbert spaces

Definition 1.

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

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

Proof.

We have:

$W$$\mathrm{ }\text{partial isometry with initial space}M$

$\iff ⟨Wf,Wg⟩$$=⟨f,g⟩\forall f,g\in M$

$\iff ⟨W^{*}Wf,g⟩$$=⟨f,g⟩\forall f\in M,g\in \mathscr{H}$

$\iff W^{*}Wf$$=f,f\in M$

$\text{and}{W}^{*}Wf$$=0,f\in M^{⟂}=\ker W$

∎

Remark 1.

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

$W^{*}(Wf)$$=f\forall f\in M$

$\ker W^{*}$$={(\mathrm{ran}W)}^{⟂}=N^{⟂}$

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