Commutativity preserving transformations on conjugacy classes of compact self-adjoint operators

Let H be a complex Hilbert space of dimension not less than 3 and let C be a conjugacy class of compact self-adjoint operators on H. Suppose that the dimension of the kernels of operators from C is not less than the dimension of their ranges. In the case when C is formed by operators of finite rank k and dim⁡H=2k, we require that k≥4. We show that every bijective transformation of C preserving the commutativity in both directions is induced by a unitary or anti-unitary operator up to a permutation of eigenspaces of the same dimension.