Exercise 11, § 7

Let $u$ be an endomorphism of a complex Hilbert space E and let $(j,|u|)$ be its polar decomposition.

(a) In order that $j$ commute with $|u|$, it is necessary and sufficient that $u$ commute with $u^*u$.

(b) Prove that one then has $\|u^*(x)\|\leqslant \|u(x)\|$ for all $x\in E$, and that $\varrho (u) =\|u\|$.

(c) Let $E =\ell^2(\mathbf{N})$ and let $u$ be the endomorphism of E given by

It is not normal. Calculate the polar decomposition $(j,|u|)$ of $u$; $j$ commutes with $|u|$.