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

Abstract

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.

arXiv comment11 pages