Exercise 4, § 5

Let $E$ be a complex hilbert space. For every invertible element $\xi$ of $\mathscr{C}$alk(E), the index of $\xi$ is called the index of any Fredholm endomorphism $u$ of $E$ such that $\pi (u) =\xi$. a) Now prove that for every $\xi \in \mathscr{C}$alk(E) hermitian, there exists $u\in \mathscr{L}(E)$ hermitian such that $\pi (u) =\xi$.

b) Now prove that for every $\xi \in \mathscr{C}$alk(E) unitary of index zero, there exists $u\in \mathscr{L}(E)$ unitary such that $\pi (u) =\xi$. (Let $v$ be invertible such that $\pi (v) =\xi$; consider the polar decomposition $v=j|v|$ of $v$ and verify that $|v| -1_E$ is compact, then deduce that $v-j$ is compact.)

c) Now prove that for every $\xi \in \mathscr{C}$alk(E) unitary such that $\xi^2= 1$, there exists $u\in \mathscr{L}(E)$ unitary such that $u^2= 1$ and $\pi (u) =\xi$.

d) For every element $p\in \mathscr{C}$alk(E) such that $p^2=p$, there exists a projector $P\in \mathscr{L}(E)$ such that $\pi (P) =p$.