Maps on idempotent operators
The set of all bounded linear idempotent operators on a Banach space $X$ is a poset with the partial order defined by $P\le Q$ if $PQ = QP = P$. Another natural relation on the set of idempotent operators is the orthogonality relation defined by $P\perp Q \iff PQ=QP=0$. We briefly survey known theorems on maps on idempotents preserving order or orthogonality. We discuss some related results and open problems. The connections with physics, geometry, theory of automorphisms, and linear preserver problems will be explained. At the end we will prove a new result concerning bijective maps on idempotent operators preserving comparability.