Admissibility of retarded diagonal systems with one-dimensional input space

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