Truncation and spectral variation in Banach algebras

Let a and b be elements of a semisimple, complex and unital Banach algebra A  . Using subharmonic methods, we show that if the spectral containment σ(ax)⊆σ(bx)σ(ax)⊆σ(bx) holds for all x∈Ax∈A, then ax belongs to the bicommutant of bx   for all x∈Ax∈A. Given the aforementioned spectral containment, the strong commutation property then allows one to derive, for a variety of scenarios, a precise connection between a and b. The current paper gives another perspective on the implications of the above spectral containment which was also studied, not long ago, by J. Alaminos, M. Brešar et al.