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

TL;DR

This work proves that, under the Müntz–Szász-type conditions $\inf_n(\lambda_{n+1}-\lambda_n)>0$ and $\sum_n 1/\lambda_n<\infty$, the exponential system $E_\Lambda$ is hereditarily complete in the closed span ${\mathcal{H}_\Lambda}(a,b)$ within $L^2(a,b)$. It derives a Fourier-type Dirichlet-series framework and constructs the unique biorthogonal family $\{r_n\}$ with precise norm bounds, enabling a strong Markushevich basis property for $E_\Lambda$ in ${\mathcal{H}_\Lambda}(a,b)$. Building on this, the paper exhibits a class of compact, non-normal operators on ${\mathcal{H}_\Lambda}(a,b)$ that admit spectral synthesis, including a shift/translation example $T_\delta$ when $u_n=e^{-\delta\lambda_n}$. These results bridge exponential systems, Dirichlet-series representations, and spectral-synthesis theory, providing both a robust basis mechanism in the closed span and concrete operator models with spectral-synthesis behavior. This advances understanding of how hereditary completeness interacts with operator theory in spaces governed by Dirichlet-series expansions.

Abstract

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.

Theorems & Definitions (17)

• Theorem 1.1
• Remark 1.1
• Lemma 3.1
• proof
• Example 4.1
• Remark 4.1
• Theorem 4.1
• proof
• Corollary 4.1
• Lemma 4.1