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
Canonical name	PartialIsometryOnHilbertSpaces
Date of creation	2013-03-22 18:35:00
Last modified on	2013-03-22 18:35:00
Owner	karstenb (16623)
Last modified by	karstenb (16623)
Numerical id	7
Author	karstenb (16623)
Entry type	Definition
Classification	msc 47C10