nLab polar decomposition

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

A unitary isomorphism

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

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

The positive operator

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

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.

Related entries

• matrix decomposition

  • Gauss decomposition

  • QR decomposition

  • LU decomposition

References

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

• Richard V. Kadison, John R. Ringrose, Fundamentals of the theory of operator algebras, Vol II Advanced Theory, Graduate Studies in Mathematics 16, AMS 1997 [ISBN:978-0-8218-0820-7]

See also:

• Wikipedia, Polar decomposition