Uniqueness under spectral variation in the socle of a Banach algebra

Let A   be a complex semisimple Banach algebra with identity, and denote by σ′(x)σ′(x) and ρ(x)ρ(x) the nonzero spectrum and spectral radius of an element x∈Ax∈A, respectively. We explore the relationship between elements a,b∈Aa,b∈A that satisfy one of the following conditions: (1) σ′(ax)⊆σ′(bx)σ′(ax)⊆σ′(bx) for all x∈Ax∈A, (2) ρ(ax)≤ρ(bx)ρ(ax)≤ρ(bx) for all x∈Ax∈A. The latter problem was identified by Brešar and Špenko in [7]. In particular, we use these conditions to spectrally characterize prime Banach algebras amongst the class of Banach algebras with nonzero socles, as well as to obtain spectral characterizations of socles which are minimal two-sided ideals.