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 $\ker (h)=\ker (x)$, 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 $\xi \in \ker (x)$

• $u(|x|\xi )=x\xi$ 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 $𝒟(x)$ as $x$, and $u$ still the partial isometry determined by

• $u\xi =0$ for $\xi \in \ker (x)$

• $u(|x|\xi )=x\xi$ for $\xi \in 𝒟(x)$.

If $x$ is affiliated with a von Neumann algebra $M$, both $u$ and $h$ will be affiliated with $M$.