(abridged) We investigate the kinematics of the central gas disk of the radio-loud elliptical galaxy NGC 4335, derived from HST/STIS long-slit spectroscopic observations of Halpha+[NII] along 3 parallel slit positions. The observed mean velocities are consistent with a rotating thin disk. We model the gas disk in the customary way. This sets a 3 sigma upper limit of 10^8 Msun on black hole mass, Mbh. The velocity dispersion at r <0.5 is in excess of that predicted by the thin rotating disk model. This does not invalidate the model, if the excess dispersion is caused by localized turbulent motion in addition to bulk circular rotation. However, if instead the dispersion is caused by the BH potential then a mass Mbh ~ 6x10^8 Msun is inferred by modeling the central gas dispersion as due to an isotropic spherical distribution of collisionless gas cloudlets. The stellar kinematics for NGC 4335 are derived from a ground-based (WHT/ISIS) long-slit observation along the galaxy major axis. A two-integral model of the stellar dynamics yields Mbh >= 3x10^9 Msun. However, there is reason to believe that this model overestimates Mbh. Reported correlations between black hole mass and inner stellar velocity dispersion sigma predict Mbh to be >= 5.4x10^8 Msun in NGC 4335. If our standard thin disk modeling of the gas kinematics is valid, then NGC 4335 has an unusually low Mbh for its velocity dispersion. If, on the other hand, this approach is flawed, and provides an underestimate of Mbh, then black hole masses for other galaxies derived from HST gas kinematics with the same assumptions should be treated with caution.
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