Definition 9

Let $E$ and $F$ be two hilbertian spaces. A continuous linear mapping $u$ from $E$ into $F$ is called a Hilbert-Schmidt mapping if the trace of the positive endomorphism $u^*u$ of $E$ is finite. The set of all Hilbert-Schmidt mappings from $E$ into $F$ is denoted by $\mathscr{L}^2(E, F)$.

When $E = F$, we write $\mathscr{L}^2(E)$ for $\mathscr{L}^2(E; E)$.

For every $u \in \mathscr{L}(E, F)$, let $\|u\|_2 = \operatorname{Tr}(u^*u)^{1/2}$, so that $u$ belongs to $\mathscr{L}^2(E; F)$ if and only if $\|u\|_2$ is finite. By the definition of the trace, we get

where $(x_1, \ldots, x_m)$ range over the set of finite orthonormal sequences in $E$. In particular, taking $m = 1$ in formula (32), we have

Let $(e_i)_{i \in I}$ be an orthonormal basis of $E$ and $(f_j)_{j \in J}$ an orthonormal basis of $F$. By formula (25) of V, p. 50 and the Parseval’s relation (V, p. 22), we have

Since $|\langle f_j|u(e_i)\rangle| = |\langle e_i|u^*(f_j)\rangle|$, formula (34) implies that

hence, the adjoint of a Hilbert-Schmidt mapping is a Hilbert-Schmidt mapping. Let $E_1, F_1$ be hilbertian spaces and $v : E_1 \to E, w : F \to F_1$ continuous linear mappings. From (32), we deduce immediately that By (35), (36) and the relation $uv = (v^*u^*)^*$, we get In particular, if $u$ belongs to $\mathscr{L}^2(E, F)$ then $wuv$ belongs to $\mathscr{L}^2(E_1, F_1)$.