(Abridged) We have obtained radial velocities of a sample of 18 ultracool dwarfs (M6.5-T8) using high-resolution, near-infrared spectra obtained with NIRSPEC and the Keck II telescope. We have confirmed that the radial velocity of Gl 570 D is coincident with that of the K-type primary star Gl 570 A, thus providing additional support for their true companionship. The presence of planetary-mass companions around 2MASS J05591914-1404488 (T4.5V) has been analyzed using five NIRSPEC radial velocity measurements obtained over a period of 4.37 yr. We have computed UVW space motions for a total of 21 L and T dwarfs within 20 pc of the Sun. This population shows UVW velocities that nicely overlap the typical kinematics of solar to M-type stars within the same spatial volume. However, the mean Galactic (44.2 km/s) and tangential (36.5 km/s) velocities of the L and T dwarfs appear to be smaller than those of G to M stars. A significant fraction (~40%) of the L and T dwarfs lies near the Hyades moving group (0.4-2 Gyr), which contrasts with the 10-12% found for earlier-type stellar neighbors. Additionally, the distributions of all three UVW components (sigma_{UVW} = 30.2, 16.5, 15.8 km/s) and the distributions of the total Galactic (sigma_{v_tot} = 19.1 km/s) and tangential (sigma_{v_t} = 17.6 km/s) velocities derived for the L and T dwarf sample are narrower than those measured for nearby G, K, and M-type stars, but similar to the dispersions obtained for F stars. This suggests that, in the solar neighborhood, the L- and T-type ultracool dwarfs in our sample (including brown dwarfs) is kinematically younger than solar-type to early M stars with likely ages in the interval 0.5-4 Gyr.
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