partial isometry on Hilbert spaces

Definition 1.

Let and 𝒦 be Hilbert spacesMathworldPlanetmath. An operator WL(,𝒦) is called a partial isometry if W is an isometry on M=(kerW). We then call M=(kerW) 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.

WL(,𝒦) is a partial isometry iff W*W is a projection from H to M.


We have:

W  partial isometry with initial spaceM
Wf,Wg =f,gf,gM
W*Wf,g =f,gfM,g
W*Wf =f,fM
andW*Wf =0,fM=kerW

Remark 1.

If WL(H,K) is a partial isometry with initial space M and final space N we have:

W*(Wf) =ffM
kerW* =(ranW)=N

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