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 ${\mathcal C}$ 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 ${\mathcal C}$ have two eigenvalues only, we obtain the Grassmann graph formed by $k$-dimensional subspaces of $H$, where $k$ is the smallest dimension of eigenspaces. Classical Chow's theorem describes automorphisms of this graph for $k>1$. Under the assumption that operators from ${\mathcal C}$ 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 ${\mathcal C}$ is formed by operators with precisely two eigenvalues.