Stability Conditions for Perturbed Semigroups on a Hilbert Space via Commutators

Abstract

Let $A$ and $B$ be linear operators on a Hilbert space. Let $A$ and $A+B$ generate $C_0$-semigroups $e^{tA}$ and $e^{t(A+B)}$, respectively, and $e^{tA}$ be exponentially stable. We establish exponential stability conditions for $e^{t(A+B)}$ in terms of the commutator $AB-BA$, assuming that it has a bounded extension. Besides, $B$ can be unbounded.