A generalized Hermite–Biehler theorem and non-Hermitian perturbations of Jacobi matrices
Rostyslav Kozhan, Mikhail Tyaglov · Journal of Mathematical Analysis and Applications · 2024
The classical Hermite–Biehler theorem describes the zero configuration of a complex linear combination of two real polynomials whose zeros are real, simple, and strictly interlace. We provide the full characterization of the zero configuration for the case when this interlacing is broken at exactly one location. We apply this result to solve the direct and inverse spectral problem for non-Hermitian rank-one multiplicative perturbations and rank-two additive perturbations of finite Hermitian and Jacobi matrices.