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

Abstract

Let \(H\) be a complex Hilbert space of dimension not less than \(3\) and let \({\mathcal C}\) be a conjugacy class of compact self-adjoint operators on \(H\). Suppose that the dimension of the kernels of operators from \({\mathcal C}\) not less than the dimension of their ranges. In the case when \({\mathcal C}\) is formed by operators of finite rank \(k\) and \(\dim H=2k\), we assume that \(k\ge 4\). We describe all bijective transformations of \({\mathcal C}\) preserving the commutativity in both directions. As a consequence, we obtain that such transformations are induced by unitary or anti-unitary operators only in the case when the dimensions of eigenspaces of operators from \({\mathcal C}\) corresponding to different eigenvalues are mutually distinct.