Hollow core optical fibers are normally passive light transport components. In contrast, within this Letter, we numerically investigate the possibility of using them as optical amplifiers, through the adoption of a novel fiber structure. We show that optical amplification can be achieved in hollow core fibers, where the cladding region is partially doped and composed of both resonant and anti-resonant elements. A balance between loss and glass/optical mode overlap is obtained, which allows efficient amplification over a limited spectral bandwidth. We discuss the case of a thulium-doped optical amplifier based on this novel technological approach.