Sufficient conditions for absolute asymptotic stability of linear equations in a Banach space

For a linear differential equation of the type
\begin{equation} \frac{dx}{dt}=A_0x(t)+A_1x(t-\Delta_1)+\dots+A_nx(t-\Delta_n)\tag{1} \end{equation}
we establish the following
\underline {THEOREM}. If
$$ \overline{\bigcup_{|z_1|=\dots=|z_n|=1}\sigma\Bigl(A_0+\sum_{k=1}^nz_kA_k\Bigl)}\subset\{\lambda:\operatorname{Re}\lambda<0\}, $$
then system (1) is absolutely asymptotically stable.