Abstract
Let A be an anti-selfadjoint operator on a Hilbert space H with ‖A‖≤12. 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=PQ−QP. Then we discuss the path components of the set CA={(P,Q):A=PQ−QP}. We prove that the action of unitary group U({A}′) is transitive in each path component of CA when A is in generic position. Moreover, we characterize the von Neumann algebra generated by all projections in CA. As an application, we obtain that the set of all commutators of pairs of orthogonal projections is connected.
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