Smoothing of operator semigroups under relatively bounded perturbations

We investigate a smoothing property for strongly-continuous operator semigroups, akin to ultracontractivity in parabolic evolution equations. Specifically, we establish the stability of this property under certain relatively bounded perturbations of the semigroup generator. This result yields a spectral perturbation theorem, which has implications for the long-term dynamics of evolution equations driven by elliptic operators of second and higher orders. In particular, a new perturbation theorem for so-called eventually positive semigroups is derived as a consequence of the general results.