A multichannel theory for the positron annihilation above the positronium formation threshold based on the modified Faddeev equation is presented. Its relation with the one channel effective charge formula is discussed. Application to the positron-hydrogen scattering has been carried out. A number of positron-electron direct annihilation cross sections in the Ore gap between Ps(1s) and H (n = 2) thresholds are reported. A sharp increase in the calculated annihilation cross section near a Feshbach resonance and followed by an even more drastic decrease of the annihilation cross section across the resonant energy is found for all first S, P and D wave Feshbach resonances.