Exercise 9, Appendix 0

Let E and F be two real Hilbert spaces.

a) Show that there exists a continuous linear mapping $\varphi$ of $E \otimes_2 F$ into $\mathscr{L}(E; F)$ characterized by $\varphi(x \otimes y)(x') = (x|x') \cdot y$ for $x, x'$ in E and $y$ in F. Show that $\varphi$ has norm 1, and that $\varphi(E \otimes F)$ is the set of continuous linear mappings of finite rank of E into F.

b) Show that $\varphi$ is a bijection of $E \otimes_2 F$ onto the set HS(E, F) of Hilbert–Schmidt linear mappings of E into F (make use of Exer. 8 c) and Prop. 3). Equip HS(E, F) with the Hilbert space structure deduced from that of $E \otimes_2 F$ by means of the bijection $\varphi$; the corresponding norm is denoted $\|u\|_2$. Let $u \in \mathrm{HS}(E, F)$; one defines positive quadratic forms $Q_u$ and H on E by $Q_u(x) = \|u(x)\|^2$ and $H(x) = \|x\|^2$. Show that $\|u\|_2^2 = \mathrm{Tr}(Q_u/H)$.

c) Let $E_1$ and $F_1$ be two real Hilbert spaces, and $u : E_1 \to E, v : F \to F_1$ continuous linear mappings. Let $\varphi_1$ be the isomorphism of $E_1 \otimes_2 F_1$ onto $\mathrm{HS}(E_1, F_1)$ defined in a manner analogous to $\varphi$. Show that $v \circ \varphi(t) \circ u = \varphi_1((u^* \otimes v)(t))$ for all $t \in E \otimes_2 F$. From this, deduce that for every Hilbert–Schmidt mapping w of E into F, the linear mapping $v \circ w \circ u$ of $E_1$ into $F_1$ is Hilbert–Schmidt, and that $\|v \circ w \circ u\|_2 \leq \|v\| \cdot \|w\|_2 \cdot \|u\|$.