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# Contents

## Idea

The polar decomposition of bounded operators (on some Hilbert space) generalizes the decomposition of complex numbers $z$ as products $z = r e^{i \phi}$ 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 $z$.

## Definition

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

###### Definition
$U \,\colon\, S_1 \longrightarrow S_2$

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

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

###### Definition

$|T| \,\coloneqq\, \big(T^* T\big)^{\frac{1}{2}}$

is called the modulus of T.

## Properties

### Existence

###### Proposition

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

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

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

###### Remark

Beware that this statement does not generalize from algebras of bounded operators $\mathcal{B}(\mathcal{H})$ to general $C^\ast$-algebras $C$, because the partial isometry $U$ need not be contained in $C$.

However, the analog statement is again true for von Neumann algebras.

## References

Most textbooks on operator algebra and Hilbert spaces mention the polar decomposition, for example it can be found in the beginning of