Automorphisms of graphs corresponding to conjugacy classes of finite-rank self-adjoint operators

Abstract

We consider the graph whose vertex set is a conjugacy class C consisting of finite-rank self-adjoint operators on a complex Hilbert space H. The dimension of H is assumed to be not less than 3. In the case when operators from C have two eigenvalues only, we obtain the Grassmann graph formed by k-dimensional subspaces of H, where k is the smallest dimension of eigenspaces. Chow's theorem describes automorphisms of this graph for k>1. Under the assumption that operators from C have more than two eigenvalues we show that every automorphism of the graph is induced by a unitary or anti-unitary operator up to a permutation of eigenspaces with the same dimensions. In contrast to this result, Chow's theorem states that there are graph automorphisms induced by semilinear automorphisms not preserving orthogonality if C is formed by operators with precisely two eigenvalues.