Remark 2

*It will be seen later (cf. IV, p. 149, theorem 1) that when $p= 2$, every normal compact endomorphism of the Hilbert space $\ell^2_{\mathbf{C}}(I)$ is of the form $u=w\circ v\circ w^{-1}$, where $v$ is a diagonal compact endomorphism and $w$ is a unitary endomorphism (EVT, V, p. 40) of the Hilbert space $\ell^2_{\mathbf{C}}(I).*$