Admissibility of retarded diagonal systems with one-dimensional input space

We investigate infinite-time admissibility of a control operator $B$ in a Hilbert space state-delayed dynamical system setting of the form $\dot{z}(t)=Az(t)+A_{1}z(t-\tau )+Bu(t)$, where $A$ generates a diagonal $C_0$-semigroup, $A_1\in \mathcal{L}(X)$ is also diagonal and $u\in L^2(0,\mathrm{\infty };\mathbb{C})$. Our approach is based on the Laplace embedding between $L^2$ and the Hardy space $H^2({\mathbb{C}}_{+})$. The results are expressed in terms of the eigenvalues of $A$ and $A_1$ and the sequence representing the control operator.