  We propose that the shape of the upper-end X-ray luminosity function observed
in elliptical galaxies for point sources is a footprint of the black-hole (BH)
mass spectrum among old X-ray transients formed in the galaxies. We show that
this underlying BH mass spectrum is modified by a weighting factor that is
related to the transient duty cycle and it generally depends on the BH mass and
XRB donor type (main-sequence, red-giant, or white-dwarf donors). A duty cycle
that depends on the binary mass-transfer rate relative to the critical rate for
transient behavior is most probably favored. We also find that the derived BH
mass spectrum slope depends on the strength of angular momentum loss due to
magnetic braking for main-sequence donors. More specifically, we find that, for
``standard'' magnetic braking, BH XRBs with red-giant donors dominate the
upper-end XLF; for weaker magnetic braking prescriptions main-sequence donors
are found to be dominant. In both cases the BH mass spectrum has a differential
slope of ~2.5 and an upper BH mass cut-off at ~20 Msun is needed to understand
the very brightest of the BH XRBs in elliptical galaxies. We expect that our
result will help to constrain binary population synthesis models and the
adopted relations between black holes and the masses of their progenitors.
