紧Hankel算子的谱反问题
Spectral inverse problems for compact Hankel operators
摘要：
给出了满足$$|λ_1|>|μ_1|>|λ_2|>|μ_2|>...>| λ_j| >| μ_j| \to 0\ ,$$的任意实数序列$(λ_j)_{j\ge 1}$和$(μ_j)_{j\ge 1}$，证明了存在唯一的实数序列$c=(c_n)_{n\in\Z_+}$，使得符号$c=(c_{n})_{n\ge 0}$和$\tilde c=(c_{n+1})_{n\ge 0}$的Hankel算子$Γ_c$和$Γ_{\tilde c}$是$\ell^2(\Z_+)$上的自伴紧算子，且序列$(λ_j)_{j\ge 1}$和$(μ_j)_{j\ge 1}$分别为非零特征值.给出了$c$的一个显式公式，并用序列$(λ_j)_{j\ge 1}$和$(μ_j)_{j\ge 1}$描述了$Γ_c$和$Γ_{\tilde c}$的核.更一般地，给定满足$$ρ_1>σ_1>ρ_2>σ_2>...> ρ_j> σ_j \to 0\ ,$$的任意两个正数序列$(ρ_j)_{j\ge 1}$和$(σ_j)_{j\ge 1}$，我们描述复数序列$c=(c_n)_{n\in\Z_+}$的集合，使得Hankel算子$Γ_c$和$Γ_{\tilde c}$在$\ell ^2(\Z_+)$上是紧的，并且分别具有序列$(ρ_j)_{j\ge 1}$和$(σ_j)_{j\ge 1}$作为非零奇异值。
英文摘要：
Given two arbitrary sequences $(λ_j)_{j\ge 1}$ and $(μ_j)_{j\ge 1}$ of real numbers satisfying $$|λ_1|>|μ_1|>|λ_2|>|μ_2|>...>| λ_j| >| μ_j| \to 0\ ,$$ we prove that there exists a unique sequence $c=(c_n)_{n\in\Z_+}$, real valued, such that the Hankel operators $Γ_c$ and $Γ_{\tilde c}$ of symbols $c=(c_{n})_{n\ge 0}$ and $\tilde c=(c_{n+1})_{n\ge 0}$ respectively, are selfadjoint compact operators on $\ell^2(\Z_+)$ and have the sequences $(λ_j)_{j\ge 1}$ and $(μ_j)_{j\ge 1}$ respectively as non zero eigenvalues. Moreover, we give an explicit formula for $c$ and we describe the kernel of $Γ_c$ and of $Γ_{\tilde c}$ in terms of the sequences $(λ_j)_{j\ge 1}$ and $(μ_j)_{j\ge 1}$. More generally, given two arbitrary sequences $(ρ_j)_{j\ge 1}$ and $(σ_j)_{j\ge 1}$ of positive numbers satisfying $$ρ_1>σ_1>ρ_2>σ_2>...> ρ_j> σ_j \to 0\ ,$$ we describe the set of sequences $c=(c_n)_{n\in\Z_+}$ of complex numbers such that the Hankel operators $Γ_c$ and $Γ_{\tilde c}$ are compact on $\ell ^2(\Z_+)$ and have sequences $(ρ_j)_{j\ge 1}$ and $(σ_j)_{j\ge 1}$ respectively as non zero singular values.
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