Exercise 5, § 4

Let $E$, $F$ be two Hilbert spaces, $u$ a continuous linear mapping from $E$ into $F$. We denote by $\ell(u)$ the set of the $x \in E$ such that $\|u(x)\| = \|u\|.\|x\|$. (a) Now prove that $\ell(u)$ is the closed vector subspace of E, kernel of $u^*u - \|u\|^2 1_E$, and is orthogonal to the kernel of $u$. (b) Now prove that the restriction of $u$ to $\ell(u)$ is a bijection from $\ell(u)$ onto $\ell(u^*)$, whose reciprocal bijection is the restriction of $\|u\|^{-2}.u^*$ to $\ell(u^*)$; moreover the image by $u$ of the orthogonal supplement $(\ell(u))^{\circ}$ is contained in $(\ell(u^*))^{\circ}$. If $u_1$ is the restriction of $u$ to $(\ell(u))^{\circ}$, considered as a mapping of $(\ell(u))^{\circ}$ into $(\ell(u^*))^{\circ}$, the adjoint $u_1^*$ is the restriction of $u^*$ to $(\ell(u^*))^{\circ}$; if $\ell(u) \neq E$, we denote by $\langle u \rangle$ and call the subnorm of $u$ the norm $\|u_1\|$; if $\ell(u) = E$, we put $\langle u \rangle = 0$. We have $\langle u^* \rangle = \langle u \rangle$.