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Inverse spectral problems for compact Hankel operators. (English) Zbl 1304.47038

For a sequence \(c=\{c_j\}_{j\geq0}\), the authors of the paper under review consider the Hankel operator on \(\ell^2\) with matrix \(\Gamma_c=\{c_{j+k}\}_{j,k\geq0}\). They study simultaneous spectral properties of the Hankel operators \(\Gamma_c\) and \(\Gamma_{\tilde c}\), where \(\tilde c=\{c_{j+1}\}_{j\geq0}\).
Let \(\{\lambda_j\}_{j\geq1}\) and \(\{\mu_j\}_{j\geq1}\) be sequences of real numbers such that \[ |\lambda_1|>|\mu_1|>|\lambda_2|>|\mu_2|>\cdots\quad\text{and}\quad\lim_{j\to\infty}\lambda_j=0. \] The main result of the paper is that there exists a unique sequence \(c=\{c_j\}_{j\geq0}\) of real numbers such that the Hankel operators \(\Gamma_c\) and \(\Gamma_{\tilde c}\) are compact self-adjoint operators on \(\ell^2\), and the sequences \(\{\lambda_j\}_{j\geq1}\) and \(\{\mu_j\}_{j\geq1}\) are the sequences of nonzero eigenvalues of \(\Gamma_c\) and \(\Gamma_{\tilde c}\).

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
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