Abstract: In general, property (ω) holding for T is not transmitted to the perturbation T+K, even when K is a finite-rank operator or compact operator. We discussed the sufficient and necessary conditions for property (ω) and its stability by means of the variant of the essential approximate point spectrum and the induced spectrum of consistency invertibility. Using these results, we studied the case of algebraically quasi-class A operators.

Key words: property (ω), perturbation, consistent in invertibility, power finite-rank operators, algebraically quasi-class A