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 be a Hilbert space, $x$ a bounded operator on . Then there exist a pair $(h,u)$ , with $h$ a bounded positive operator and $u$ a partial isometry on , 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 .

If $x$ is a closed, densely defined unbounded operator on , the polar decomposition $(u,h)$ still exists, where now $h$ will be the unbounded positive operator $|x|$ with the same domain as $x$ , and $u$ still the partial isometry determined by

• $u\xi=0$ for $\xi \in \ker(x)$
• $u(|x|\xi)=x\xi$ for .

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