  Coupled-channel $\pi K$ and $\eta K$ scattering amplitudes are determined by
studying the finite-volume energy spectra obtained from dynamical lattice QCD
calculations. Using a large basis of interpolating operators, including both
those resembling a $q\bar{q}$ construction and those resembling a pair of
mesons with relative momentum, a reliable excited-state spectrum can be
obtained. Working at ${m_\pi=391\,\mathrm{MeV}}$, we find a gradual increase in
the $J^P=0^+$ $\pi K$ phase-shift which may be identified with a broad scalar
resonance that couples strongly to $\pi K$ and weakly to $\eta K$. The
low-energy behavior of this amplitude suggests a virtual bound-state that may
be related to the $\kappa$ resonance. A bound state with $J^P=1^-$ is found
very close to the $\pi K$ threshold energy, whose coupling to the $\pi K$
channel is compatible with that of the experimental $K^\star(892)$. Evidence is
found for a narrow resonance in $J^P=2^+$. Isospin--3/2 $\pi K$ scattering is
also studied and non-resonant phase-shifts spanning the whole elastic
scattering region are obtained.
