Hereditary completeness of Exponential systems $\{e^{λ_n t}\}_{n=1}^{\infty}$ in their closed span in $L^2 (a, b)$ and Spectral Synthesis
Suppose that $\{λ_n\}_{n=1}^{\infty}$ is a sequence of distinct positive real numbers satisfying the conditions inf$\{λ_{n+1}-λ_n \}>0,$ and $\sum_{n=1}^{\infty}λ_n^{-1}<\infty.$ We prove that the exponential system $\{e^{λ_n t}\}_{n=1}^{\infty}$ is hereditarily complete in the closure of the subspace spanned by $\{e^{λ_n t}\}_{n=1}^{\infty}$ 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.
math.FA↗