Abstract
Self-adjoint operators represent bounded observables in mathematical foundations of quantum mechanics. The set of all self-adjoint operators can be equipped with several operations and relations having important interpretations in physics. Automorphisms with respect to these relations or operations are called symmetries. Many of them turn out to be real-linear up to a translation. We present a unified approach to the description of the general form of such symmetries based on adjacency preserving maps. We consider also symmetries defined on the set of all positive operators or on the set of all positive invertible operators. In particular, we will see that the structural result for adjacency preserving maps on the set of all positive invertible operators differs a lot from its counterpart on the set of all selfadjoint operators.
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