Seminar at SMU Delhi

There are several important relations on the set I(H) of idempotent operators on a Hilbert space. We will be concerned by the following: partial order ($P \leq Q \iff PQ = QP = P$), orthogonality ($P \perp Q \iff PQ = QP = 0$), commutativity ($PQ = QP$), and image/kernel equality (${\rm im} P = {\rm im} Q$ or $\ker P = \ker Q$). We study bijective maps on $I(H)$ or its subsets which preserve these relations. The most famous result of this kind is Ovchinnikov's theorem (1993) which describes poset automorphism of $I(H)$ and was motivated by problems in quantum mechanics.