Definition 8

— Let E be a complex hilbertian space. We write $\mathscr{L}^1(E)$ for the vector subspace of $\mathscr{L}(E)$ generated by all continuous, positive endomorphisms of E with finite trace.

By formula (25) of V, p. 50, the trace extends to a linear form on $\mathscr{L}^1(E)$, again denoted by Tr, and satisfying the relation $\operatorname{Tr}(u) = \sum_{i \in I} \langle e_i | u(e_i) \rangle$ for all $u$ in $\mathscr{L}^1(E)$ and for every orthonormal basis $(e_i)_{i \in I}$ of E. For every $u \in \mathscr{L}^1(E)$, we have $u^* \in \mathscr{L}^1(E)$ and $\operatorname{Tr}(u^*) = \overline{\operatorname{Tr}(u)}$. Formula (28) of V, p. 50 extends to the case where $u$ belongs to $\mathscr{L}^1(E)$. Let F be a hilbertian subspace of E; by formula (30), the orthoprojector $p_F$ belongs to $\mathscr{L}^1(E)$ if and only if F is finite dimensional. For every $a$ and $b$ in E, we have $4ab^* = \sum_{\varepsilon^4 = 1} \varepsilon(a + \varepsilon b)(a + \varepsilon b)^*$ and $cc^*$ is a positive operator with finite trace for all $c \in E$; consequently, if $u$ is a finite rank, continuous endomorphism of E, then $u \in \mathscr{L}^1(E)$ and $\operatorname{Tr}(u) = \tau(u)$.

Let E be a real hilbertian space, and let $E_{(c)}$ be its complexification (V, p. 5). We identify E with a subset of $E_{(c)}$. Then $\mathscr{L}(E)$ can be identified with a real vector subspace of $\mathscr{L}(E_{(c)})$ consisting of all continuous linear mappings $u$ from $E_{(c)}$ into $E_{(c)}$ such that $u(E) \subset E$. In this case we write $\mathscr{L}^1(E) = \mathscr{L}(E) \cap \mathscr{L}^1(E_{(c)})$. For every $u \in \mathscr{L}^1(E)$, the trace $\operatorname{Tr}(u)$ is real and is equal to $\operatorname{Tr}(u^*)$. Formulas (25) and (28) are again valid, $\mathscr{L}_f(E) \subset \mathscr{L}^1(E)$ and $\operatorname{Tr}(u) = \tau(u)$ for all $u \in \mathscr{L}_f(E)$. Finally, a closed vector subspace F of E is finite dimensional if and only if $p_F$ belongs to $\mathscr{L}^1(E)$.

\* Remark 1. — We shall later define the notion of a nuclear mapping from a Banach space E into a Banach space F. We shall show that when $\mathscr{L}^1(E)$ consists of all nuclear mappings from E into E, then E is a real or complex hilbertian space. \*