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

TL;DR

This paper studies automorphisms of graphs formed by conjugacy classes of finite-rank self-adjoint operators on complex Hilbert spaces, revealing conditions under which automorphisms are induced by unitary or anti-unitary operators.

Contribution

It extends classical results by characterizing automorphisms of these graphs for operators with multiple eigenvalues, generalizing Chow's theorem.

Findings

Automorphisms are induced by unitary or anti-unitary operators when operators have more than two eigenvalues.

For operators with exactly two eigenvalues, automorphisms can be induced by semilinear automorphisms not preserving orthogonality.

The results depend on the number of eigenvalues of the operators in the conjugacy class.

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.