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We have obtained deep optical and near-IR images of a sample of radio-quiet QSOs at $z\sim2.3$ and of an additional sample at $z\sim1.0$. Unlike their radio-loud counterparts, the radio-quiet sample at high redshift have no detectable extended host galaxy ``fuzz'' in either the optical or the K-band images; our limits correspond to a total flux in fuzz of less than $\sim$5\% of the QSO flux. The low-redshift ($z\sim1.0$) sample of radio-quiet QSOs shows marginal evidence of extended fuzz in the $K-$band; the sizes and luminosities of the putative extended emission regions are consistent with current-day 2-3L$_*$~ galaxies.
Both the high-redshift upper limits and the low-redshift tentative detections of host galaxy fuzz indicate that radio-quiet QSOs live in objects significantly fainter than the powerful radio galaxies observed at high redshift; the radio-quiet fuzz is generally at least 1-2 magnitudes fainter than the tight $K-z$ (``$K$--Hubble'') relation defined by both the radio galaxies and the radio-loud QSO fuzz. This suggests that radio-loud and radio-quiet QSOs inhabit inherently different parent galaxies, rather than living in identical galaxies that can turn on and off the radio emission, since if the extended fuzz seen around radio-loud QSOs is starlight, then it cannot disappear on short timescales to become a radio-quiet QSO host.
These results are generally consistent with the findings of studies at lower redshift, which indicate that in the recent past radio-loud QSOs and AGNs inhabited more luminous, preferentially elliptical galaxies than their radio-quiet counterparts.
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