nLab
polar decomposition

Contents

Idea

This page is about the polar decomposition of bounded operators on Hilbert spaces. Any complex number zz has a representation as z=re iϕz = r e^{i \phi} with r,r0r \in \mathbb{R}, r \ge 0 being the absolute value of zz and the complex number e iϕe^{i \phi} of norm 11 being the modulus, or the complex sign, of zz. 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 2S_1, S_2 be closed linear subspaces.

Definition

An unitary isomorphism

U:S 1S 2 U: S_1 \to S_2

is called a partial isometry with initial space S 1S_1 and final space or range S 2S_2

Let TT be a bounded operator on \mathcal{H}

Definitions

The positive operator

|T|:=(T *T) 12 |T| := (T^*T)^{\frac{1}{2}}

is called the modulus of T.

The Theorem

For every bounded operator TT on \mathcal{H} there exists a unique partial isometry UU such that

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

  2. T=U|T|=U(T *T) 12T = 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 CC it need not hold because the partial isometry UU need not be contained in CC.

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

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