Abstract: Using the new spectrum set defined by the property of consistency in Fredholm and index, we give the judgements of property \left( \omega _1 \right) and property \left( \omega \right) for a operator T and its adjoint operator T^*. We also consider the stability of property \left( \omega \right) for a bounded linear operator acting on Hilbert space, under finite rank perturbations F commuting with T.