  It is well-known that characters classify linear representations of finite
groups, that is if characters of two representations of a finite group are the
same, these representations are equivalent. It is also well-known that, in
general, this is not true for representations of infinite groups, even if they
are finitely generated. The goal of this paper is to establish a
characterization of representations of finitely generated groups in terms of
projective joint spectra. This approach has a clear advantage compared to
character classification as it is valid for a much wider family of groups and
for both finite and infinite dimensional representations. The main tool in
establishing our spectral characterization is a reconstruction of an operator
acting on a separable Hilbert space from the proper projective joint spectrum
of the quadruple containing this operator along with a certain triple of
bounded operators acting on the same space.
