  Studies of massive black holes (BHs) in nearby galactic centers have revealed
a tight correlation between BH mass and galactic velocity dispersion. In this
paper we investigate how the BH mass versus velocity dispersion relation and
the nuclear luminosity versus velocity dispersion relation in QSOs/active
galactic nuclei (AGNs) are connected with the BH mass versus velocity
dispersion relation in local galaxies, through the nuclear luminosity evolution
of individual QSOs/AGNs and the mass growth of individual BHs. In the study we
ignore the effects of BH mergers and assume that the velocity dispersion does
not change significantly during and after the nuclear activity phase. Using the
observed correlation in local galaxies and an assumed form of the QSO/AGN
luminosity evolution and BH growth, we obtain the simulated observational
appearance of the BH mass versus velocity dispersion relation in QSOs/AGNs. The
simulation results illustrate how the BH accretion history (e.g., the lifetime
of nuclear activity and the possibility that QSOs/AGNs accrete at a
super-Eddington accretion rate at the early evolutionary stage) can be inferred
from the difference between the relation in QSOs/AGNs and that in local
galaxies. We also show how the difference may be weakened by the flux limit of
telescopes. We expect that a large complete sample of QSOs/AGNs with accurate
BH mass and velocity dispersion measurements will help to quantitatively
constrain QSO/AGN luminosity evolution and BH growth models.
