Hereditary completeness of Exponential systems $\{e^{\lambda_n t}\}_{n=1}^{\infty}$ in their closed span in $L^2 (a, b)$ and Spectral Synthesis

Suppose that $\{\lambda_n\}_{n=1}^{\infty}$ is a sequence of distinct positive real numbers satisfying the conditions inf$\{\lambda_{n+1}-\lambda_n \}>0,$ and $\sum_{n=1}^{\infty}\lambda_n^{-1}<\infty.$ We prove that the exponential system $\{e^{\lambda_n t}\}_{n=1}^{\infty}$ is hereditarily complete in the closure of the subspace spanned by $\{e^{\lambda_n t}\}_{n=1}^{\infty}$ in the space $L^2 (a,b)$. We also give an example of a class of compact non-normal operators defined on this closure which admit spectral synthesis.