A projector $P$ is a Hermitian matrix satisfying $P^2=P$. For any two projectors, it is easy to show that there exists a unitary matrix $U$ such that both $U^*PU$ and $U^*QU$ are block-diagonal matrices, each block is of size at most $2$.
For arbitrary $k$ projectors, there is no simultaneous decomposition such that the sizes of all blocks are upper bounded by a function of $k$. The answer is given here
Simultaneous Block decomposition of a set of orthogonal projections.
My question is whether there are some special cases we can generalize. The following is a special case I am interested.
Given three projectors $\Pi_1, \Pi_2, \Delta$, where $Range(\Pi_1)\subseteq Range(\Pi_2)$, is there a similar block decomposition such that the size each block is constant?