Proposition 16

Let $E_1$ and $F_1$ be Hilbert spaces. Let $u$ be in $\mathscr{L}_1(E; F),v$ be in $\mathscr{L}(E_1; E)$ and $v_2$ be in $\mathscr{L}(F; F_1)$. Then $v_2uv_1\in \mathscr{L}_1(E_1; F_1)$ and $\|v_2uv_1\|_1\leqslant \|v_2\| \|v_1\| \|u\|_1$.

Let $w\in \mathscr{L}_2(E; F_1)$ such that $\|w\|\leqslant 1$. One has $v_2u\in \mathscr{L}_2(E; F_1)$ (EVT, V, p. 52, formula (36)). Since $v^*_2w\in \mathscr{L}_2(E; F) ($loc. cit.), it follows that

(formula (11)), whence $v_2u\in \mathscr{L}_1(E; F_1)$ and $\|v_2u\|_1\leqslant \|v_2\|\|u\|_1$ (Prop. 14).

Let $v_1\in \mathscr{L}(E_1; E)$; since $uv_1= (v^*_1u^*)^*$, one has $uv_1\in \mathscr{L}_1(E_1; F)$ and $\|uv_1\|_1\leqslant \|v^*_1\| \|u^*\|_1=\|v_1\| \|u\|_1$ (Lemma 8).

The proposition follows at once from these inequalities.