How to choose the signature operator such that the periodic pseudo-Jacobi inverse eigenvalue problem is solvable?

Abstract

Using a discrete version of Floquet theory in a space with indefinite metric we reconstruct periodic pseudo-Jacobi matrices from given spectral data. We use two methods to characterize the signature operator H so that the periodic pseudo-Jacobi inverse eigenvalue problem is solvable. As a by-product of the obtained results, we precisely polish some non-rigorous assumptions in the existing literature.