Smoothing of operator semigroups under relatively bounded perturbations

Abstract

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.

27 pages. Compared to v2, Section 4 has been reorganised and split, and several technical corrections and clarifications have been incorporated

References in corpus (7)

• Regularity of Solutions of Linear Second Order Elliptic and Parabolic Boundary Value Problems on Lipschitz Domains
• Eventually Positive Semigroups of Linear Operators
• Eventually and asymptotically positive semigroups on Banach lattices
• The role of domination and smoothing conditions in the theory of eventually positive semigroups
• Local uniform convergence and eventual positivity of solutions to biharmonic heat equations
• Limit-case admissibility for positive infinite-dimensional systems
• Admissible operators for sun-dual semigroups