Geometric version of Wigner's theorem for Hilbert Grassmannians.

Let H be a complex Hilbert space of dimension not less than 3 and let G k ( H ) be the Grassmannian formed by k -dimensional subspaces of H . Suppose that dim ⁡ H ≥ 2 k > 2 . We show that the transformations of G k ( H ) induced by linear or conjugate-linear isometries can be characterized as transformations preserving some of principal angles (corresponding to the orthogonality, adjacency and ortho-adjacency relations). As a consequence, we get the following: if the dimension of H is finite and greater than 2 k , then every transformation of G k ( H ) preserving the orthogonality relation in both directions is a bijection induced by a unitary or anti-unitary operator. [ABSTRACT FROM AUTHOR]