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Spectral inverse problems for compact Hankel operators

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.