  Let $A$ be a semisimple Banach algebra with non-trivial, and possibly
infinite-dimensional socle. Addressing a problem raised by Harte and Hernandez,
we first define a characteristic polynomial for elements belonging to the
socle, and we then show that a Generalized Cayley-Hamilton Theorem holds for
the associated polynomial. The key arguments leading to the main result follow
from the observation that a purely spectral approach to the theory of the socle
carries alongside it an efficient method of dealing with relativistic problems
associated with infinite-dimensional socles.
