Polar Decomposition

Abstract

Begin with the construction of P. Since A*A is a positive operator on H, it has a (unique) positive square root; call it P. Since

$$ {\left\| {Pf} \right\|^2} = \left( {Pf,Pf} \right) = \left( {{P^2}f,f} \right) = \left( {A*Af,f} \right) = {\left\| {Af} \right\|^2} $$

for all f in H, it follows that the equation

$$ UPf = Af $$

unambiguously defines a linear transformation U from the range R of P into the space K, and that U is isometric on R. Since U is bounded on R, it has a unique bounded extension to the closure R̄, and, from there, a unique extension to a partial isometry from H to K with initial space R̄. The equation A = UP holds by construction. The kernel of a partial isometry is the orthogonal complement of its initial space, and the orthogonal complement of the range of a Hermitian operator is its kernel. This implies that ker U = ker P and completes the existence proof.