Exercise 15, § 1

Let E be a complex hilbertian space, $E_1, E_2$ two closed vector subspaces of E, $P_1, P_2$ the orthoprojectors from E onto $E_1, E_2$ respectively. a) Show that, in order that $P_1$ and $P_2$ commute, it is necessary and sufficient that E is the hilbertian sum of the four subspaces $E_1 \cap E_2, E_1^\circ \cap E_2^\circ, E_1^\circ \cap E_2, E_1 \cap E_2^\circ$ (where $M^\circ$ denotes the orthogonal complement of a vector subspace M of E).

b) Show that if E₁ is finite dimensional and $\|P_1 - P_2\| < 1$, then E₂ has the same dimension as E₁ (consider the intersection $E_1^\circ \cap E_2$). c) Show that the endomorphism $T = (P_1 - P_2)^2$ of E commutes with $P_1$ and $P_2$, and that the eigen subspace of T corresponding to the eigenvalue 0 (resp. 1) is the direct sum of the orthogonal subspaces $E_1 \cap E_2$ and $E_1^\circ \cap E_2^\circ$ (resp. $E_1^\circ \cap E_2$ and $E_1 \cap E_2^\circ$). d) Suppose E is finite dimensional and that $T = \hat{\lambda} I$ with $\hat{\lambda} \neq 0$. Then $\hat{\lambda} > 0$ and E is the hilbertian sum of subspaces of dimension $\leq 2$, each of which is stable under $P_1$ and $P_2$ (observe that $P_1 - P_2$ is hermitian and deduce that E is the hilbertian sum of two subspaces $E^+$, $E^-$ such that $P_1.x - P_2.x = \sqrt{\hat{\lambda}} x$ in $E^+$ and $P_1.x - P_2.x = -\sqrt{\hat{\lambda}} x$ in $E^-$; then show that for $x \in E^+$, we have $P_1.x = \frac{1 + \sqrt{\hat{\lambda}}}{2} x + z$ and $P_2.x = \frac{1 - \sqrt{\hat{\lambda}}}{2} x + z$, with $z \in E^-$. e) Suppose that E₁ and E₂ are finite dimensional. Show that there exists a family $(F_\alpha)_{\alpha \in A}$ of subspaces of E of dimension $\leq 2$ such that E, E₁ and E₂ are respectively the hilbertian sums of the families $(F_\alpha)_{\alpha \in A}$, $(F_\alpha \cap E_1)_{\alpha \in A}$ and $(F_\alpha \cap E_2)_{\alpha \in A}$ (use c) to reduce to the case when E is finite dimensional, then apply d)). \