Properties of a Product of Commuting Projections

Question

Let $\Pi_1$ and $\Pi_2$ be two positive-semidefinite projections which commute, and let $\Pi=\Pi_1\Pi_2=\Pi_2\Pi_1$.
Are the following statements correct?

1. $\quad\Pi\:\preceq\:\Pi_1\,$ and $\,\Pi\:\preceq\:\Pi_2$, where $A\preceq B$ means $\,B-A\,$ is positive-semidefinite.
2. $\quad I-\Pi\:\preceq\: I-\Pi_1+I-\Pi_2$, where $I$ is the identity matrix.

Since $\Pi_1$ and $\Pi_2$ commute we can show that $\Pi_1-\Pi$, $\Pi_2-\Pi$, and $\Pi-\Pi_1-\Pi_2+I$ are also projections.

Answer 1

The central assumption "$\Pi_1$ and $\Pi_2$ commute" implies that $\Pi=\Pi_1\Pi_2=\Pi_2\Pi_1$ is an orthogonal projection as well. It satisfies $\,\operatorname{Im}\Pi=\operatorname{Im}\Pi_1\cap\operatorname{Im}\Pi_2$, cf Range of Idempotent matrices . In particular we have $\Pi_i\Pi=\Pi=\Pi\,\Pi_i\,$.

As in general "something self-adjoint when squared is PSD" one has

1. $\quad 0\:\preceq\:\big(\Pi_i-\Pi\big)^2 \:=\: \Pi_i -\Pi_i\Pi -\Pi\,\Pi_i +\Pi \:=\: \Pi_i -\Pi$

Since $1-\Pi_1$ and $1-\Pi_2$ are commuting projections their product is alike, thus,

2. $\quad 0\:\preceq\:\big(1-\Pi_1\big)\big(1-\Pi_2\big) \:=\: 1 -\Pi_1 -\Pi_2 +\Pi$
   $\qquad\implies 1 -\Pi\:\preceq\:1 -\Pi_1 +1 -\Pi_2$