Alternating projections on a Hilbert space

Question

Let $P_1, P_2$ be the orthoprojections onto $S_1, S_2$, closed subspaces of a Hilbert space $H$. It is straightforward to show that if $(P_1P_2)^nx \to z$ then $z \in S_1 \cap S_2$ (I can post a quick proof if anyone wants it). But under what conditions on $P_1$ and $P_2$ or $x$ does $(P_1P_2)^nx$ converges at all?

Answer 1

One can prove strong convergence following this article:

The product of projection operators

I. Halperin

http://acta.fyx.hu/acta/showCustomerArticle.action?id=7164&dataObjectType=article

The original result is due to Neumann. Halperin's note generalizes this to $m$ projections:

There it is proven that $(P_1P_2\dots P_m)^n x \to z$ with $z$ being the orthogonal projection of $x$ onto $S_1\cap S_2\cap \dots \cap S_m$.