Spectral inverse problems for compact Hankel operators
Résumé
Given two arbitrary sequences $(\lambda_j)_{j\ge 1}$ and $(\mu_j)_{j\ge 1}$ of real numbers satisfying $$|\lambda_1|>|\mu_1|>|\lambda_2|>|\mu_2|>\dots>\vert \lambda _j\vert >\vert \mu _j\vert \to 0\ ,$$ we prove that there exists a unique sequence $c=(c_n)_{n\in\Z_+}$, real valued, such that the Hankel operators $\Gamma_c$ and $\Gamma_{\tilde c}$ 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 $(\lambda_j)_{j\ge 1}$ and $(\mu_j)_{j\ge 1}$ respectively as non zero eigenvalues. Moreover, we give an explicit formula for $c$ and we describe the kernel of $\Gamma_c$ and of $\Gamma_{\tilde c}$ in terms of the sequences $(\lambda_j)_{j\ge 1}$ and $(\mu_j)_{j\ge 1}$. More generally, given two arbitrary sequences $(\rho _j)_{j\ge 1}$ and $(\sigma _j)_{j\ge 1}$ of positive numbers satisfying $$\rho _1>\sigma _1>\rho _2>\sigma _2>\dots> \rho _j> \sigma _j \to 0\ ,$$ we describe the set of sequences $c=(c_n)_{n\in\Z_+}$ of complex numbers such that the Hankel operators $\Gamma_c$ and $\Gamma_{\tilde c}$ are compact on $\ell ^2(\Z _+)$ and have sequences $(\rho _j)_{j\ge 1}$ and $(\sigma _j)_{j\ge 1}$ respectively as non zero singular values.
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