Uniqueness under spectral variation in the socle of a Banach algebra

Let A be a complex semisimple Banach algebra with identity, and denote by sigma'(x) and rho(x) the nonzero spectrum and spectral radius of an element x is an element of A, respectively. We explore the relationship between elements a, b is an element of A that satisfy one of the following conditions: (1) sigma' (ax) subset of sigma' (bx) for all x is an element of A, (2) rho(ax) <= rho(bx) for all x is an element of A. The latter problem was identified by Bregar and Spenko 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. (C) 2016 Elsevier Inc. All rights reserved.