Consider the simple operator on C given by multiplication by a complex number. It decomposes into a stretch and a rotation. What is the generalisation of this to operators on a Hilbert space?
Solution
The generalization of \(z=e^{i\theta}|z|\) is the polar decomposition. Let \(H\) be a complex Hilbert space, \(T\in B(H)\), and \(|T|=\sqrt{T^*T}\), the unique positive square root of the positive operator \(T^*T\). A partial isometry is an operator \(U\) that is isometric on \((\ker U)^\perp\).
1. \(\norm{\,|T|x\,}=\norm{Tx}\) for every \(x\in H\); hence \(\ker|T|=\ker T\) and \(\overline{\operatorname{im}|T|}=(\ker T)^\perp\).
\(\norm{\,|T|x\,}^2=\inner{|T|^2x}{x}=\inner{T^*Tx}{x}=\norm{Tx}^2\). Since \(|T|\) is self-adjoint, \(\overline{\operatorname{im}|T|}=(\ker|T|)^\perp\).
2. There is a partial isometry \(U\) with \(\ker U=\ker T\) and \(T=U|T|\).
1.It extends by continuity to an isometry \(U_1\colon\overline{\operatorname{im}|T|}\to\overline{\operatorname{im}T}\).Put \(U=U_1\) on \(\overline{\operatorname{im}|T|}\) and \(U=0\) on its orthogonal complement \(\ker|T|=\ker T\).Then \(U|T|x=Tx\) for every \(x\).
3. If \(T=WP\) with \(P\ge0\) and \(W\) a partial isometry with \(\ker W=\ker P\), then \(P=|T|\) and \(W=U\).
\(W^*W\) is the orthogonal projection onto \((\ker W)^\perp=(\ker P)^\perp=\overline{\operatorname{im}P}\), so \(T^*T=PW^*WP=P^2\), and \(P=|T|\) by uniqueness of positive square roots. Then \(W(|T|x)=Tx=U(|T|x)\), so \(W=U\) on \(\overline{\operatorname{im}|T|}\), and both vanish on \(\ker|T|\).
4. If \(T\) is invertible, then \(U\) is unitary.
Then \(\ker T=0\) and \(\operatorname{im}T=H\), so \(U\) is an isometry of \(H\) onto \(H\).