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 sigma(ax) subset of sigma(bx) holds for all x is an element of A, then ax belongs to the bicommutant of bx for all x is an element of 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. Bresar et al. (C) 2016 Elsevier Inc. All rights reserved.