A generalized Hermite-Biehler theorem
Classical Analysis and ODEs
2023-02-15 v1 Spectral Theory
Abstract
The classical Hermite-Biehler theorem describes possible zero sets of complex linear combinations of two real polynomials whose zeros strictly interlace. We provide the full characterization of zero sets for the case when this interlacing is broken at exactly one location. Using this we solve the direct and inverse spectral problem for rank-one multiplicative perturbations of finite Hermitian matrices. We also treat certain rank two additive perturbations of finite Jacobi matrices.
Cite
@article{arxiv.2302.07018,
title = {A generalized Hermite-Biehler theorem},
author = {Rostyslav Kozhan and Mikhail Tyaglov},
journal= {arXiv preprint arXiv:2302.07018},
year = {2023}
}