Remark 2

— Let $E$ be a hilbertian space, which is the hilbertian sum of a family $(E_i)_{i \in I}$ of hilbertian subspaces. Let $u$ be an element of $\mathscr{L}(E)$ such that $u(E_i) \subset E_i$ for all $i \in I$; let $u_i$ be the element of $\mathscr{L}(E_i)$ which coincides with $u$ on $E_i$. Then $\operatorname{Tr}(u) = \sum_{i \in I} \operatorname{Tr}(u_i)$ when $u$ is positive, or belongs to $\mathscr{L}^1(E)$; this relation follows from formula (25) of V, p. 50 applied to an orthonormal basis of $E$ which is the union of orthonormal bases of each of the $E_i$.