Exercise 5, § 4

Let $E, F$ be two Hilbert spaces, and $u$ a continuous linear mapping from $E$ into $F$. Let $\ell(u)$ denote the set of all $x \in E$ such that $\|u(x)\| = \|u\| \cdot \|x\|$. a) Show that $\ell(u)$ is the closed vector subspace of $E$ which is the kernel of $u^*u - \|u\|^2 1_E$, and is orthogonal to the kernel of $u$. b) Show that the restriction of $u$ to $\ell(u)$ is a bijection from $\ell(u)$ onto $\ell(u^*)$, whose inverse bijection is the restriction of $\|u\|^{-2} u^*$ to $\ell(u^*)$; moreover the image of the orthogonal complement $(\ell(u))^{\circ}$ under $u$ is contained in $(\ell(u^*))^{\circ}$. If $u_1$ is the restriction of $u$ to $(\ell(u))^{\circ}$, considered as a mapping from $(\ell(u))^{\circ}$ into $(\ell(u^*))^{\circ}$, the adjoint $u_1^*$ is the restriction of $u^*$ to $(\ell(u^*))^{\circ}$; if $\ell(u) \neq E$, let $\langle u \rangle$, called the sub-norm of $u$, denote the norm $\|u_1\|$; if $\ell(u) = E$, we put $\langle u \rangle = 0$. Then $\langle u^* \rangle = \langle u \rangle$.