A Dynamical System Approach to the Inverse Spectral Problem for Hankel Operators: A Model Case

We present an alternative proof of the result by P. G\'erard and S. Grellier, stating that given two real sequences $(\lambda_n)_{n=1}^\infty$, $(\mu_n)_{n=1}^\infty$ satisfying the intertwining relations \[ |\lambda_1| > |\mu_1| > |\lambda_2| > |\mu_2| > ...> |\lambda_n| > |\mu_n|>\ldots >0 , \qquad \lambda_n\to 0, \] there exists a unique compact Hankel operator $\Gamma$ such that $( \lambda_n )_{n=1}^{\infty}$ are the (simple) eigenvalues of $\Gamma$ and $( \mu_n )_{n=1}^{\infty}$ are the simple eigenvalues of its truncation $\Gamma_1$ obtained from $\Gamma$ by removing the first column. We use the dynamical system approach originated stated in a paper by A. V. Megretski, V. V. Peller, and S. R. Treil' in 1995, and the proof is split into three independent parts. The first part is an abstract operator-theoretic statement reducing the problem to the asymptotic stability of some contractions. The second part is the proof of asymptotic stability, which is usually the hardest part, but in our case of compact operators it is almost trivial. And the third part is an abstract version of the Borg's two spectra theorem, which is essentially a simple exercise in graduate complex analysis