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 ∞. 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 C\{0}. Surprisingly, it turns out that, if such a Jordan curve is present, the spectra of all the principal submatrices have to be real. The latter claim is also proved for matrices given by a more general symbol. The special role of the Jordan curve is further demonstrated by a new formula for the limiting density of the asymptotic eigenvalue distribution for banded Toeplitz matrices from the studied class. Certain connections between 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.

A matrix-less method to approximate the spectrum and the spectral function of Toeplitz matrices with real eigenvalues

“…The "perfect" grid ξ j,n is computed using data from Example 8. Numerically, we have λ j (T 1000 (f )) ∈ [−22.0912, 14.9641] in agreement with [24]. In Fig.…”

“…We note that, which is equivalent to (41) in [24,Example 3. ] with z = e iθ , a = −1, r = 3, and s = 1.…”

Eigenvalue Clusters of Large Tetradiagonal Toeplitz Matrices

“…So consider the curve Remark 4.6. The referee kindly pointed out that part of assertion (d), namely that Λ(f ) ⊂ R, may also be derived from the main result of [11], according to which Λ(f ) is contained in R if and only if the pre-image f −1 (R) contains a Jordan curve. To see this, let q(t) = sin(3t)/ sin(2t) and consider the set K = {κ(t) := 1 − qe it : t ∈ [−π/3, π/3]}.…”

“…Recently, Shapiro andŠtampach [11] very beautifully characterized the Laurent polynomials for which Λ(a) is a subset of the real line R: this happens if and only if the pre-image a −1 (R) contains a Jordan curve. We refer to the books [2,3,6,8] for more on the eigenvalue cluster problem for Toeplitz matrices.…”

“…As also −1/2 < Re b < 0, it is only b ∈ (−1/2, 0) which can give oppositely directed tangents. But these b correspond just to ±c ∈ (3 √ 3, ∞).The referee again kindly turned attention to the fact that the "if portion" of Theorem 6.2 can also be deduced from the more general result of[11]. Indeed, let c ∈ [3 √ 3, ∞).…”