LAUSR

Abstract Let A be an anti-selfadjoint operator on a Hilbert space H with ‖ A ‖ ≤ 1 2 . We give a sufficient and necessary condition for A to be a commutator of a pair of orthogonal projections, and establish the general representation of all pairs ( P , Q ) of orthogonal projections such that A = P Q − Q P . Then we discuss the path components of the set C A = { ( P , Q ) : A = P Q − Q P } . We prove that the action of unitary group U ( { A } ′ ) is transitive in each path component of C A when A is in generic position. Moreover, we characterize the von Neumann algebra generated by all projections in C A . As an application, we obtain that the set of all commutators of pairs of orthogonal projections is connected.