Powers of a product of two orthogonal projections
= Powers of a product of two orthogonal projections
{title2=$\|(PQ)^r\|\leq q^{2r-1}$}
For <orthogonal projections> $P,Q$ with $\|PQ\|\leq q<1$, the identity $(PQ)^r=PQ(QPQ)^{r-1}$ and $\|QPQ\|=\|PQ\|^2$ imply $\|(PQ)^r\|\leq q^{2r-1}$ for $r\geq1$. When the projections have a common fixed subspace, first remove its <orthogonal projection>. This estimate is stronger than generic submultiplicativity and is useful for alternating local projection layers.