Non-self-adjoint Toeplitz matrices whose principal submatrices have real spectrum (Q1731911)

The aim of this paper is to introduce and investigate a class of complex semi-infinite banded Toeplitz matrices such that the spectra of their principal submatrices accumulate onto a real interval when the size of the submatrix grows to infinite. The authors prove that a banded Toeplitz matrix belongs to this class iff its symbol has real values on a Jordan curve located in $\mathbb{C} \backslash \left\{0\right\} $. The particular role of the Jordan curve is further demonstrated by a new formula for the limiting density of the asymptotic eigenvalue distribution. Some specific connections among the problem under investigation, Jacobi operators, and the Hamburger moment problem are also discussed. The main results are illustrated by several concrete examples; some of them allow an explicit analytic treatment, while some are only treated numerically.