  In some recent papers, the studies on biorthogonal Riesz bases has found a
renewed motivation because of their connection with pseudo-hermitian Quantum
Mechanics, which deals with physical systems described by Hamiltonians which
are not self-adjoint but still may have real point spectra. Also, their
eigenvectors may form Riesz, not necessarily orthonormal, bases for the Hilbert
space in which the model is defined. Those Riesz bases allow a decomposition of
the Hamiltonian, as already discussed is some previous papers. However, in many
physical models, one has to deal not with o.n. bases or with Riesz bases, but
just with biorthogonal sets. Here, we consider the more general concept of
$\mathcal{G}$-quasi basis and we show a series of conditions under which a
definition of non self-adjoint Hamiltonian with purely point real spectra is
still possible.
