polar decomposition
The polar decomposition^{} of an operator^{} is a generalization^{} of the familiar factorization of a complex number $z$ in a radial part $z$ and an angular part $z/z$.
Let $\mathscr{H}$ be a Hilbert space^{}, $x$ a bounded operator^{} on $\mathscr{H}$. Then there exist a pair $(h,u)$, with $h$ a bounded^{} positive operator and $u$ a partial isometry on $\mathscr{H}$, such that
$$x=uh.$$ 
If we impose the further conditions that $1{u}^{*}u$ is the projection to the kernel of $x$, and $\mathrm{ker}(h)=\mathrm{ker}(x)$, then $(h,u)$ is unique, and is called the polar decomposition of $x$. The operator $h$ will be $x$, the square root of ${x}^{*}x$, and $u$ will be the partial isometry, determined by

•
$u\xi =0$ for $\xi \in \mathrm{ker}(x)$

•
$u(x\xi )=x\xi $ for $\xi \in \mathscr{H}$.
If $x$ is a closed, densely defined unbounded operator on $\mathscr{H}$, the polar decomposition $(u,h)$ still exists, where now $h$ will be the unbounded^{} positive operator $x$ with the same domain $\mathcal{D}(x)$ as $x$, and $u$ still the partial isometry determined by

•
$u\xi =0$ for $\xi \in \mathrm{ker}(x)$

•
$u(x\xi )=x\xi $ for $\xi \in \mathcal{D}(x)$.
If $x$ is affiliated with a von Neumann algebra^{} $M$, both $u$ and $h$ will be affiliated with $M$.
Title  polar decomposition 

Canonical name  PolarDecomposition 
Date of creation  20130322 16:01:54 
Last modified on  20130322 16:01:54 
Owner  aube (13953) 
Last modified by  aube (13953) 
Numerical id  10 
Author  aube (13953) 
Entry type  Definition 
Classification  msc 47A05 