论文标题
汉克尔操作员的反光谱问题的动态系统方法：模型案例
A Dynamical System Approach to the Inverse Spectral Problem for Hankel Operators: A Model Case
论文作者
论文摘要
我们为P. Gerard和S. Grellier提供了结果的替代证明，并指出给定两个真实序列$（λ_n）_ {n = 1}^\ infty $，$（μ_n）_ {n = 1}^\ infty $满足相互交织的关系\ [|λ_1|λ_1| > |μ_1| > |λ_2| > |μ_2| > ...> |λ_n| > |μ_n|> \ ldots> 0，\qquadλ_n\ to 0，\]存在独特的紧凑型汉克尔操作员$γ$，因此$λ_n$是（简单的）特征的$γ$和$μ_n$的（$μ_n$）是简单的$ geenvalues $ co $ $γ___1$ coul $γ__1$ coul $γ__1$ coul $γ__1$ coul $γ__1$γ_1$ cove $γ_1$ cove $γ__1$ cove。 我们使用的动态系统方法源自A. V. Megretski，V.V。的论文。佩勒。 S. R. Treil于1995年，证明分为三个独立部分。第一个是对该论文中结果的稍作修改，是一种抽象的操作者理论陈述，将问题降低为某些操作员的渐近稳定性。第二个是渐近稳定性的证明，这通常是最困难的部分，但是在我们的紧凑型操作员的情况下，这几乎是微不足道的。第三部分是Borg的两个光谱定理的抽象版本，这实际上是研究生复杂分析中的一个简单练习。
We present an alternative proof of the result by P. Gerard and S. Grellier, stating that given two real sequences $(λ_n)_{n=1}^\infty$, $(μ_n)_{n=1}^\infty$ satisfying the intertwining relations \[ |λ_1| > |μ_1| > |λ_2| > |μ_2| > ...> |λ_n| > |μ_n|>\ldots >0 , \qquad λ_n\to 0, \] there exists a unique compact Hankel operator $Γ$ such that $λ_n$ are the (simple) eigenvalues of $Γ$ and $μ_n$ are the simple eigenvalues of its truncation $Γ_1$ obtained from $Γ$ by removing the first column. We use the dynamical systems approach originated in a paper by A. V. Megretski, V.V. Peller. S. R. Treil in 1995, and the proof is split into three independent parts. The first one, which is a slight modification of a result in that paper, is an abstract operator-theoretic statement reducing the problem to the asymptotic stability of some operators. The second one is the proof of the asymptotic stability, which is usually the hardest part, but in our case of compact operators it is almost trivial. And the third part is an abstract version of the Borg's two spectra theorem, which is essentially a simple exercise in graduate complex analysis.
