  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.
