Polar decomposition - Definition and Overview

In mathematics, particularly in linear algebra and functional analysis, the polar decomposition is a canonical factorization of any linear mapping T between complex Hilbert spaces as the product of a partial isometry and a non-negative self-adjoint operator. It generalizes the polar decomposition of a complex number z in the form

<math> z = r e^{i \theta}. \quad <math>

More precisely, if T is a bounded linear mapping then there is a unique partial isometry U defined on the closure of the range of |T| such that

<math> T \phi = U |T| \phi. \quad<math>

where

<math> |T|=\sqrt{T^* T} <math>

For the meaning of the square root of the non-negative operator T* T see functional calculus. Indeed

<math> \|T \phi\|^2 = \langle \phi \mid T^* T \phi \rangle = \langle \sqrt{T^* T}\phi \mid \sqrt{T^* T} \phi \rangle = \| |T| \phi\|^2 <math>

so there is an isometry U defined uniquely on the closure of the rangle of (T* T)^1/2 with the rquired properties.