Orthogonality preserving transformations on the set of \(n\)-dimensional subspaces of a Hilbert space (Q1765979)

The author describes the general form of all bijective transformations on the set of all \(n\)-dimensional subspaces (\(n\) being a fixed positive integer) of a real or complex infinite-dimensional Hilbert space which preserve the orthogonality in both directions. It turns out that every such transformation is induced by either a unitary or an antinuitary operator on the underlying Hilbert space. The result is a remarkable generalization of Uhlhorn's well-known theorem (covering the case when \(n=1\)) that plays a rather important role in some parts of quantum mechanics.