This page is about the polar decomposition of bounded operators on Hilbert spaces. Any complex number $z$ has a representation as $z = r e^{i \phi}$ with $r \in \mathbb{R}, r \ge 0$ being the absolute value of $z$ and the complex number $e^{i \phi}$ of norm $1$ being the modulus, or the complex sign, of $z$. The polar decomposition of a bounded operator is a generalization of this representation.

Definition

Let $\mathcal{H}$ be a Hilbert space and $S_1, S_2$ be closed linear subspaces.

Definition

An unitary isomorphism

$U: S_1 \to S_2$

is called a partial isometry with initial space$S_1$ and final space or range $S_2$

Let $T$ be a bounded operator on $\mathcal{H}$

Definitions

The positive operator

$|T| := (T^*T)^{\frac{1}{2}}$

is called the modulus of T.

The Theorem

For every bounded operator $T$ on $\mathcal{H}$ there exists a unique partial isometry $U$ such that

U has initial space $\overline{R(|T|)}$ and range $\overline{R(T)}$

$T = U |T| = U (T^*T)^{\frac{1}{2}}$

Properties

We have stated the theorem for the operator algebra$\mathcal{B}(\mathcal{H})$ only, for a general C-star algebra$C$ it need not hold because the partial isometry $U$ need not be contained in $C$.