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

Abstract

We introduce and investigate a class of complex semi-infinite banded Toeplitz matrices satisfying the condition that the spectra of their principal submatrices accumulate onto a real interval when the size of the submatrix grows to $\infty$. We prove that a banded Toeplitz matrix belongs to this class if and only if its symbol has real values on a Jordan curve located in $\mathbb{C}\setminus \{0\}$. Surprisingly, it turns out that, if such a Jordan curve is present, the spectra of all the submatrices have to be real. The latter claim is also proven for matrices given by a more general symbol. Further, the limiting eigenvalue distribution of a real banded Toeplitz matrix is related to the solution of a determinate Hamburger moment problem. We use this to derive a formula for the limiting measure using a parametrization of the Jordan curve. We also describe a Jacobi operator, whose spectral measure coincides with the limiting measure. We show that this Jacobi operator is a compact perturbation of a tridiagonal Toeplitz matrix. Our main results are illustrated by several concrete examples; some of them allow an explicit analytic treatment, while some are only treated numerically. Update: The proof of Theorem 8 contains an error. An erratum is attached in the end

There is an error in the proof of Theorem 8. An erratum has been attached to the end of the paper

References in corpus (2)

• Perturbation theory of PT-symmetric Hamiltonians
• Polynomials with rational generating functions and real zeros