Consistent invertibility and perturbations for property (ρ)

An operator T ϵ B(H) is said to be consistent in invertibility" provided that for each S ϵ B(H), TS and ST are either both or neither invertible. Using the induced spectrum, the paper investigates the permanence of property (ρ) under some commuting perturbations, which extends the corresponding results in P. Aiena et al. (2007) [3]. In addition, the stability of property (!) of the operators which are the products of finitely normal operators is considered.