Effective (i.e., subspace-constrained) Hamiltonians become, by construction, energy-dependent while all the energy-dependent forces prove non-linear because the energy itself is merely an eigenvalue of the Hamiltonian H. One of the most natural resolutions of such a puzzle is proposed via an introduction of the two separate linear representatives of the respective right and left action of H=H(E). Both the new energy-independent operators are quasi-Hermitian so that the formalism admits a natural extension to all the quasi-Hermitian initial H(E)s.