  We utilize the local velocity dispersion function (VDF) of spheroids,
together with their inferred age--distributions, to predict the VDF at higher
redshifts (0<z<6), under the assumption that (i) most of the stars in each
nearby spheroid formed in a single episode, and (ii) the velocity dispersion
sigma remained nearly constant afterward. We assume further that a supermassive
black hole (BH) forms concurrently with the stars, and within ~1 Gyr of the
formation of the potential well of the spheroid, and that the relation between
the mass of the BH and host velocity dispersion maintains the form M_BH ~
sigma^{beta} with beta~4, but with the normalization allowed to evolve with
redshift as ~(1+z)^{alpha}. We compute the BH mass function associated with the
VDF at each redshift, and compare the accumulated total BH mass density with
that inferred from the integrated quasar luminosity function (LF; the
so--called Soltan argument). This comparison is insensitive to the assumed duty
cycle or Eddington ratio of quasar activity, and we find that the match between
the two BH mass densities favors a relatively mild redshift evolution, with
alpha ~ 0.26, with a positive evolution as strong as alpha>1.3 excluded at the
99% confidence level. A direct match between the characteristic BH mass in the
VDF--based and quasar LF--based BH mass functions also yields a mean Eddington
ratio of lambda ~ 0.5-1 that is roughly constant within 0<z<3. A strong
positive evolution in the M_BH-sigma relation is still allowed by the data if
galaxies increase, on average, their velocity dispersions since the moment of
formation, due to dissipative processes. If we assume that the mean velocity
dispersion of the host galaxies evolves as sigma(z)=sigma(0)*(1+z)^{-gamma}, we
find a lower limit of gamma>0.23 for alpha>1.5. abridged
