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polar decomposition

Idea

This page is about the polar decomposition of bounded operators on Hilbert spaces. Any complex number $z$ has a representation as $z=r{e}^{i\varphi }$ with $r\in ℝ,r\ge 0$ being the absolute value of $z$ and the complex number $e^{i\varphi }$ 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 $\mathscr{H}$ be a Hilbert space and $S_1,S_2$ be closed linear subspaces.

Definition

An unitary isomorphism

$$U:S_1\to S_2$$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 $\mathscr{H}$

Definitions

The positive operator

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

is called the modulus of T.

The Theorem

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

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

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

Properties

We have stated the theorem for the operator algebra $ℬ(\mathscr{H})$ 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.

Examples

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