  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 $x \in A$, respectively. We explore the relationship between elements
$a, b \in A$ that satisfy one of the following conditions: (1) $\sigma' (ax)
\subseteq \sigma' (bx)$ for all $x \in A$, (2) $\rho (ax) \leq \rho (bx)$ for
all $x \in A$. The latter problem was identified by Bre\v{s}ar and \v{S}penko
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.
