Symmetry between model space and target space in effective-Hamiltonian theory

General relations concerning wave operators are proposed that exhibit symmetry between the model space and the target space. The basic Bloch and des Cloizeaux effective Hamiltonians operating in the model space are shown to be equivalent to the exact Hamiltonian operating on well-defined vectors belonging to the target space. For the Bloch formalism, a new perturbative development, involving the usual wave operator, is proposed. It has improved convergence properties and makes possible the calculation, at each perturbation order, of an estimated error in the usual Bloch perturbative development. A numerical test on the simple Mathieu equation, performed at orders 2, 3, and 4, gives promising results.