Resonance properties of a stack consisting of a cholesteric liquid crystal and isotropic medium layers are investigated. Bragg's conditions for photonic band gaps are presented for the considered system. It is shown that a significant control of the system zone structure is possible in the case of a proper selection of the sublayer parameters. It is also shown that an effect of complete suppression of absorption is not usually observed in the case of anisotropic absorption in cholesteric liquid crystal sublayers. The possibility to achieve this effect in the considered system, imposing certain conditions on the thickness and the refraction index of the isotropic medium sublayers, is shown. Also, it is shown that in such conditions a large increase of the maximum of photonic density of states happens at the appropriate zone borders.