Stronger Constraints on the Evolution of the $M_{\rm{BH}}-\sigma_*$
  Relation up to $z\sim0.6$
  We revisit the possibility of redshift evolution in the
$M_{\rm{BH}}-\sigma_*$ relation with a sample of 22 Seyfert 1 galaxies with
black holes (BHs) in the mass range $10^{6.3}-10^{8.3}~M_\odot$ and redshift
range $0.03<z<0.57$ with spectra obtained from spatially resolved
Keck/Low-Resolution Imaging Spectrometer observations. Stellar velocity
dispersions were measured directly from the Mg Ib region, taking into
consideration the effect of Fe II contamination, active galactic nucleus (AGN)
dilution, and host-galaxy morphology on our measurements. BH masses are
estimated using the H$\beta$ line width, and the luminosity at 5100
$\overset{\lower.5em\circ}{\mathrm{A}}$ is estimated from surface brightness
decomposition of the AGN from the host galaxy using high-resolution imaging
from the Hubble Space Telescope. Additionally, we investigate the use of the [O
III]$\lambda5007$ emission line width as a surrogate for stellar velocity
dispersion, finding better correlation once corrected for Fe II contamination
and any possible blueshifted wing components. Our selection criteria allowed us
to probe lower-luminosity AGNs and lower-mass BHs in the non-local universe
than those measured in previous single-epoch studies. We find that any offset
in the $M_{\rm{BH}}-\sigma_*$ relation up to $z\sim0.6$ is consistent with the
scatter of local BH masses, and address the sources of biases and uncertainties
that contribute to this scatter.
