  States in the $N=35$ and 37 isotopes $^{55,57}$Ca have been populated by
direct proton-induced nucleon removal reactions from $^{56,58}$Sc and $^{56}$Ca
beams at the RIBF. In addition, the $(p,2p)$ quasi-free single-proton removal
reaction from $^{56}$Ca was studied. Excited states in $^{55}$K, $^{55}$Ca, and
$^{57}$Ca were established for the first time via in-beam $\gamma$-ray
spectroscopy. Results for the proton and neutron removal reactions from
$^{56}$Ca to states in $^{55}$K and $^{55}$Ca for the level energies, excited
state lifetimes, and exclusive cross sections agree well with state-of-the-art
theoretical calculations using different approaches. The observation of a
short-lived state in $^{57}$Ca suggests a transition in the calcium isotopic
chain from single-particle dominated states at $N=35$ to collective excitations
at $N=37$.
