Polar Decomposition

Any invertible bounded operator $T$ on a Hilbert space has a polar decomposition

where $u(T)$ is unitary (complex case) or orthogonal (real case).

Key properties (paper use):

• Used to define rGL(H) via $|T|\in GL(H)_2$.
• Lets one reduce many proofs to the positive cone (e.g. $Sp(K)^+$).

Example: If $T$ is positive selfadjoint, then $u(T)=I$ and $|T|=T$.