Perturbations of Strongly Continuous Operator Semigroups, and Matrix Muckenhoupt Weights

Let $A$ and $A_0$ be linear continuously invertible operators on a Hilbert space $\mathfrak{H}$ such that $A^{-1}-A_0^{-1}$ has finite rank. Assuming that $\sigma(A_0)=\varnothing$ and that the operator semigroup $V_+(t)=\exp\{iA_0t\}$, $t\ge0$, is of class $C_0$, we state criteria under which the semigroups $U_\pm(t)=\exp\{\pm iAt\}$, $t\ge0$, are of class $C_0$ as well. The analysis in the paper is based on functional models for nonself-adjoint operators and techniques of matrix Muckenhoupt weights.