Exercise 34, § 5

Let E be a complex Hilbert space. For all endomorphisms $u$ and $v$ of E, write $[u, v] =uv-vu$ and define $E_{u,v}=$ Im($u$)$\cap$ Ker($v$).

Let $p$ and $q$ be orthoprojectors of E. Put $a=p-q$ and $b= 1_E-p-q$. a) If there exists a unitary endomorphism $u$ of E such that $upu^{-1}$ = $q$ and $uqu^{-1}=p$, then the Hilbert dimensions of $E_{p,q}$ and $E_{1-p,1-q}$ are equal.

b) We have the relations

c) Up to question f ), inclusive, it will be supposed that $E_{p,q}= E_{1-p,1-q}=\{0\}$. The endomorphism $b$ is injective; let $s$ be the function from $\mathbf{R}$ into $\mathbf{R}$ such that

1 if $x >0$

$s(x) =$ 0 if $x= 0$

$-1$ if $x <0$.

The endomorphism $u=s(b)$ is unitary.

d) For every $\varepsilon >0$, one has $(|b|+\varepsilon )a=a(|b|+\varepsilon )$.

e) One has $uau^{-1}=-a$ (verify that $s(b) =$ lim$_{\varepsilon\rightarrow 0}b(|b|+\varepsilon )^{-1}$) and $ubu^{-1}=b$. f) One has $upu^{-1}=q$ and $uqu^{-1}=p$.

g) Suppose that the Hilbert dimension of $E_{p,q}$ is equal to that of $E_{1-p,1-q}$. Then there exists a unitary endomorphism $u$ of E such that $upu^{-1}=$ $q$ and $uqu^{-1}=p$. (The spaces $E_{p,q}$ and $E_{1-p,1-q}$ are orthogonal; consider the Hilbert space $F = E_{p,q}\oplus E_{1-p,1-q}$ and apply the preceding case to the orthoprojectors of $F^{\circ}$ deduced from $p$ and $q$ by passing to subspaces.)

h) Suppose that $\|p-q\|<1$. There exists a unitary endomorphism $u$ of E such that $upu^{-1}=q$ and $uqu^{-1}=p$.

i) Suppose that $a=p-q$ is compact. The endomorphism $k=qp$ induces by passing to the subspaces a Fredholm linear mapping $\widetilde{k}$ of the image of $p$ into the image of $q$. There exists a unitary endomorphism $u$ of E such that $upu^{-1}=q$ and $uqu^{-1}=p$ if and only if the index of $\widetilde{k}$ is zero.