Commutativity preserving transformations on conjugacy classes of finite rank self-adjoint operators

Abstract

Let $H$ be a complex Hilbert space and let ${\mathcal C}$ be a conjugacy class of finite rank self-adjoint operators on $H$ with respect to the action of unitary operators. We suppose that ${\mathcal C}$ is formed by operators of rank $k$ and for every $A\in {\mathcal C}$ the dimensions of distinct maximal eigenspaces are distinct. Under the assumption that $\dim H\ge 4k$ we establish that every bijective transformation $f$ of ${\mathcal C}$ preserving the commutativity in both directions is induced by a unitary or anti-unitary operator, i.e. there is a unitary or anti-unitary operator $U$ such that $f(A)=UAU^{*}$ for every $A\in {\mathcal C}$. A simple example shows that the condition concerning the dimensions of maximal eigenspaces cannot be omitted.