# Contents

## Idea

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

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

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

## 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 $C$.

This is true however for every von Neumann algebra.

## References

Most textbooks about operators on Hilbert spaces mention the polar decomposition, for example it can be found in the beginning of

• Kadison, Ringrose: Fundamentals of the Theory of Operator Algebras , volume 2, Advanced Theory

• wikipedia polar decomposition

Revised on April 20, 2011 15:05:39 by Zoran Škoda (109.227.26.208)