2017
DOI: 10.1007/s00365-017-9408-0
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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 … Show more

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Cited by 10 publications
(13 citation statements)
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“…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.…”
Section: Examplesupporting
confidence: 82%
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“…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.…”
Section: Examplesupporting
confidence: 82%
“…We note that, which is equivalent to (41) in [24,Example 3. ] with z = e iθ , a = −1, r = 3, and s = 1.…”
Section: Motivation and Illustrative Examplesmentioning
confidence: 97%
“…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]}.…”
Section: Branch Pointsmentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%
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