Definition 7

Let u be a continuous and positive endomorphism of the hilbertian space E. Let

(upper bound in $\overline{\mathbf{R}}_+$), where $\mathscr{F}$ is the set of all finite rank orthoprojectors on E. We say that $\operatorname{Tr}(u)$ is the trace of u.

Let $p$ be the orthoprojector from $E$ onto a finite dimensional vector subspace of $E$, and let $(x_1, ..., x_m)$ be an orthonormal basis of $F$. We have established the relation

Consequently, we can define the trace by the formula where $(x_1, ..., x_m)$ ranges over the set of all finite orthonormal sequences of vectors of $E$.

By lemma 3 (V, p. 48), we have

for every orthonormal basis $(e_i)_{i \in I}$ of $E$. From this, we deduce for all continuous and positive endomorphisms $u$ and $v$ of $E$ and for every real number $\lambda \geq 0$ (we make the convention $0.(+\infty) = 0$ in (27)). Let $\phi$ be an isomorphism from $E$ onto a hilbertian space $F$; since $\phi$ transforms every orthonormal basis of $E$ into an orthonormal basis of $F$, we get from (25) that

Let $(u_\alpha)_{\alpha \in A}$ be a non-empty directed increasing and bounded family of continuous and positive endomorphisms of $E$; let $u = \sup_\alpha u_\alpha$, then $\langle x | u(x) \rangle = \sup_\alpha \langle x | u_\alpha(x) \rangle$ for all $x \in E$ (V, p. 46, prop. 13). We have $\text{Tr}(u) = \sup_{J \subset I} \sum_{i \in J} \langle e_i | u(e_i) \rangle$, where $J$ ranges over all finite subsets of $I$, hence

Let $p_F$ be the orthoprojector from $E$ onto the hilbertian subspace $F$; there exists an orthonormal basis $(e_i)_{i \in I}$ of $E$ and a subset $J$ of $I$, such that $(e_i)_{i \in J}$ is an orthonormal basis of $F$. We have $\text{Tr}(p_F up_F) = \sum_{i \in J} \langle e_i | u(e_i) \rangle$. This formula has two consequences : firstly, we have $\text{Tr}(p_F up_F) \leq \text{Tr}(u)$; secondly, taking $u = 1_E$, we get