polar decomposition

The polar decompositionMathworldPlanetmath of an operatorMathworldPlanetmath is a generalizationPlanetmathPlanetmath of the familiar factorization of a complex number z in a radial part |z| and an angular part z/|z|.

Let be a Hilbert spaceMathworldPlanetmath, x a bounded operatorMathworldPlanetmathPlanetmath on . Then there exist a pair (h,u), with h a boundedPlanetmathPlanetmathPlanetmathPlanetmath positive operator and u a partial isometry on , such that


If we impose the further conditions that 1-u*u is the projection to the kernel of x, and ker(h)=ker(x), then (h,u) is unique, and is called the polar decomposition of x. The operator h will be |x|, the square root of x*x, and u will be the partial isometry, determined by

  • uξ=0 for ξker(x)

  • u(|x|ξ)=xξ for ξ.

If x is a closed, densely defined unbounded operator on , the polar decomposition (u,h) still exists, where now h will be the unboundedPlanetmathPlanetmath positive operator |x| with the same domain 𝒟(x) as x, and u still the partial isometry determined by

  • uξ=0 for ξker(x)

  • u(|x|ξ)=xξ for ξ𝒟(x).

If x is affiliated with a von Neumann algebraMathworldPlanetmathPlanetmathPlanetmath M, both u and h will be affiliated with M.

Title polar decomposition
Canonical name PolarDecomposition
Date of creation 2013-03-22 16:01:54
Last modified on 2013-03-22 16:01:54
Owner aube (13953)
Last modified by aube (13953)
Numerical id 10
Author aube (13953)
Entry type Definition
Classification msc 47A05