Spectral inverse problems for compact Hankel operators

TL;DR

This paper solves inverse spectral problems for compact Hankel operators, providing explicit formulas for symbols based on prescribed eigenvalues and singular values, and characterizing their kernels.

Contribution

It establishes existence, uniqueness, and explicit formulas for symbols of Hankel operators with given spectral data, extending inverse spectral theory.

Findings

Unique sequences of symbols correspond to prescribed eigenvalues.

Explicit formulas for symbols are derived.

Characterization of kernels of Hankel operators based on spectral data.

Abstract

Given two arbitrary sequences $(\lambda_j)_{j\ge 1}$ and $(\mu_j)_{j\ge 1}$ of real numbers satisfying $$|\lambda_1|>|\mu_1|>|\lambda_2|>|\mu_2|>...>| \lambda_j| >| \mu_j| \to 0\ ,$$ 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 $$\rho_1>\sigma_1>\rho_2>\sigma_2>...> \rho_j> \sigma_j \to 0\ ,$$ 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.