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Let $H$ be a complex Hilbert space and let ${\mathcal C}$ be a conjugacyclass of finite rank self-adjoint operators on $H$ with respect to the actionof unitary operators. We suppose that ${\mathcal C}$ is formed by operators ofrank $k$ and for every $A\in {\mathcal C}$ the dimensions of distinct maximaleigenspaces are distinct. Under the assumption that $\dim H\ge 4k$ we establishthat every bijective transformation $f$ of ${\mathcal C}$ preserving thecommutativity in both directions is induced by a unitary or anti-unitaryoperator, 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 thecondition concerning the dimensions of maximal eigenspaces cannot be omitted.

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