  The Fano effect (U. Fano, Phys. Rev. \textbf{15},1866 (1961) shows that an
inelastic scattering process can be suppressed when the output channel (OC) is
coupled to an isolated bound state. In this paper we investigate the
application of this effect for the suppression of two-body collisional losses
of ultracold atoms. The Fano effect is originally derived via a first-order
perturbation treatment for coupling between the incident channel (IC) and the
OC. We generalize the Fano effect to systems with arbitrarily strong IC--OC
couplings. We analytically prove that, in a system with one IC and one OC, when
the inter-atomic interaction potentials are real functions of the inter-atomic
distance, the exact s-wave inelastic scattering amplitude can always be
suppressed to \emph{zero} by coupling between the IC or the OC (or both of
them) and an extra isolated bound state. We further show that when the
low-energy inelastic collision between two ultracold atoms is suppressed by
this effect, the real part of the elastic scattering length between the atoms
is still possible to be much larger than the range of inter-atomic
interaction.In addition, when open scattering channels are coupled to two bound
states, with the help of the Fano effect, independent control of the elastic
and inelastic scattering amplitudes of two ultracold atoms can be achieved.
Possible experimental realizations of our scheme are also discussed.
