Characterization of the K2-19 Multiple-Transiting Planetary System via High-Dispersion Spectroscopy, AO Imaging, and Transit Timing Variations

K2-19 (EPIC201505350) is an interesting planetary system in which two transiting planets with radii ~ 7 $R_{Earth}$ (inner planet b) and ~ 4 $R_{Earth}$ (outer planet c) have orbits that are nearly in a 3:2 mean-motion resonance. Here, we present results of ground-based follow-up observations for the K2-19 planetary system. We have performed high-dispersion spectroscopy and high-contrast adaptive-optics imaging of the host star with the HDS and HiCIAO on the Subaru 8.2m telescope. We find that the host star is relatively old (>8 Gyr) late G-type star ($T_{eff}$ ~ 5350 K, $M_s$ ~ 0.9 $M_{Sun}$, and $R_{s}$ ~ 0.9 $R_{Sun}$). We do not find any contaminating faint objects near the host star which could be responsible for (or dilute) the transit signals. We have also conducted transit follow-up photometry for the inner planet with KeplerCam on the FLWO 1.2m telescope, TRAPPISTCAM on the TRAPPIST 0.6m telescope, and MuSCAT on the OAO 1.88m telescope. We confirm the presence of transit-timing variations, as previously reported by Armstrong and coworkers. We model the observed transit-timing variations of the inner planet using the synodic chopping formulae given by Deck & Agol (2015). We find two statistically indistinguishable solutions for which the period ratios ($P_{c}/P_{b}$) are located slightly above and below the exact 3:2 commensurability. Despite the degeneracy, we derive the orbital period of the inner planet $P_b$ ~ 7.921 days and the mass of the outer planet $M_c$ ~ 20 $M_{Earth}$. Additional transit photometry (especially for the outer planet) as well as precise radial-velocity measurements would be helpful to break the degeneracy and to determine the mass of the inner planet.