Functional models of non-selfadjoint operators, strongly continuous semigroups, and matrix Muckenhoupt weights

We consider unbounded continuously invertible operators $A$, $A_0$ on a Hilbert space $\mathfrak{H}$ such that the operator $A^{-1}-A^{-1}_0$ has finite rank. Assuming that $\sigma(A_0)=\varnothing$ and the semigroup $V_+(t):=\exp\{iA_0t\}$, $t\geqslant0$, is of class $C_0$, we state criteria under which the semigroups $U_\pm(t):=\exp\{\pm iAt\}$, $t\geqslant0$, are also of class $C_0$. We give applications to the theory of mean-periodic functions. The investigation is based on functional models of non-selfadjoint operators and on the technique of matrix Muckenhoupt weights.