An inverse spectral theory for finite CMV matrices

TL;DR

This paper addresses inverse spectral problems for finite CMV matrices, providing methods to reconstruct these matrices from various spectral data, advancing understanding in spectral theory and matrix analysis.

Contribution

It introduces new solutions for reconstructing finite CMV matrices from Weyl functions, dual spectra, and truncated spectra, expanding inverse spectral theory.

Findings

Reconstruction from Weyl's function established

Reconstruction from two spectra demonstrated

Reconstruction from spectrum and truncated spectrum achieved

Abstract

For finite dimensional CMV matrices the classical inverse spectral problems are considered. We solve the inverse problem of reconstructing a CMV matrix by its Weyl's function, the problem of reconstructing the matrix by two spectra of CMV matrices with different "boundary conditions", and the problem of reconstructing the CMV matrix by its spectrum and the spectrum of the CMV matrix obtained from it by truncation.