  Differential cross sections for $K^-$ radiative capture in flight on the
proton, leading to the $\gamma\Lambda$ and $\gamma\Sigma^0$ final states, have
been measured at eight $K^-$ momenta between 514 and 750 MeV/$c$. The data were
obtained with the Crystal Ball multiphoton spectrometer installed at the
separated $K/\pi$ beam line C6 of the BNL Alternating Gradient Synchrotron. The
results substantially improve the existing experimental data available for
studying radiative decays of excited hyperon states. An exploratory theoretical
analysis is performed within the Regge-plus-resonance approach. According to
this analysis, the $\gamma\Sigma^0$ final state is dominated by
hyperonresonance exchange and hints at an important role for a resonance in the
mass region of 1700 MeV. In the $\gamma\Lambda$ final state, on the other hand,
the resonant contributions account for only half the strength, and the data
suggest the importance of a resonance in the mass region of 1550 MeV.
