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

Suppose that $\{{\lambda }_n{\}}_{n=1}^{\mathrm{\infty }}$ is a sequence of distinct positive real numbers satisfying the conditions inf$\{{\lambda }_{n+1}-{\lambda }_n\}>0,$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\lambda }_n^{-1}<\mathrm{\infty }\mathrm{.}$ We prove that the exponential system $\{e^{{\lambda }_{n}t}{\}}_{n=1}^{\mathrm{\infty }}$ is hereditarily complete in the closure of the subspace spanned by $\{e^{{\lambda }_{n}t}{\}}_{n=1}^{\mathrm{\infty }}$ 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.