  Superconductor$-$ferromagnetic-insulator heterostructures are paradigmatic
systems for studying the mutual influence of superconductivity and magnetism
via proximity effects. In particular, spin-split superconductivity is realized
in such structures. Recent experiments and theories demonstrate a rich variety
of transport phenomena occurring in devices based on such heterostructures that
suggest direct applications in thermoelectricity, low-dissipative spintronics,
radiation detection, and sensing. In this work we investigate the influence of
the electron-magnon interaction at the superconductor$-$ferromagnetic-insulator
interface on the spin-split superconductivity. It is predicted that due to the
magnon-mediated electron spin-flip processes the spin-split quasiparticle
branches are partially mixed and reconstructed, and the BCS-like spin-split
shape of the superconducting density of states, which is typical for
superconductors in the effective exchange field, is strongly modified. An
odd-frequency superconducting order parameter admixture to the leading singlet
order parameter is also found. These findings expand the physical picture of
spin-split superconductivity beyond the mean-field description of the
ferromagnet exchange field.
