Definition 3

One says that $(u^*u)^{1/2}$ is the absolute value of $u$, and it is denoted by $|u|$.

In the case where $E_1= E_2$, this definition coincides with that given in Remark 9 of I, p. 119.

For an element $u$ of $\mathscr{L}(E_1; E_2)$, let us recall (EVT, V, p. 41, Def. 2) that the initial subspace of $u$ is the closed subspace Ker($u$)$^{\circ}$ of $E_1$ and the final subspace of $u$ is the closed subspace Im($u$) of $E_2$.