Asymptotics of Individual Eigenvalues of a Class of Large Hessenberg Toeplitz Matrices

Abstract

Abstract. We study the asymptotic behavior of individual eigenvalues of the n-by-n truncations of certain infinite Hessenberg Toeplitz matrices as n goes to infinity. The generating function of the Toeplitz matrices is supposed to be of the form a(t) = t, where α is a positive real number but not an integer and f is a smooth function in H ∞ . The classes of generating functions considered here and in a recent paper by Dai, Geary, and Kadanoff are overlapping, and in the overlapping cases, our results imply in particular a rigorous justification of an asymptotic formula which was conjectured by Dai, Geary, and Kadanoff on the basis of numerical computations.MSC 2010. Primary 47B35. Secondary 15A15, 15A18, 47N50, 65F15

Asymptotics of eigenvalues of large symmetric Toeplitz matrices with smooth simple-loop symbols

“…The asymptotic expansions of all eigenvalues are constructed in the case of essentially complex-valued symbols having singularities of the Fisher-Hartwig type in the articles [23], [24], [25], [26]. We note that the complex-valued (non self-adjoint) case is more complicated than the real-valued one, because finding the location of the limit set of eigenvalues of Toeplitz matrices with n tending to infinity is a nontrivial question.…”

“…25), we get e 2i(n+1)s = e −2iηn(s) 1 + R n+1)s = e −2iηn(s)+R (n) 5 (s) .The last equation is equivalent to the following set of equations:2i(n + 1)s = −2iη n (s) + R ±is , s) and R (n) 2 (e ±is ), we get now the asymptotic expansion (5.29).6. Solvability Analysis of Equation (5.28)Rewrite the equation (5.28) in the form F n (s) + R…”

Spectral theory of large Wiener–Hopf operators with complex-symmetric kernels and rational symbols

“…This paper was motivated by the recent papers [3], [4], [8], [9] dedicated to the Fox-Li operator and it was encouraged by the recent investigations in [1], [5] on individual eigenvalues of certain Hessenberg or Hermitian Toeplitz matrices. The Fox-Li operator is (1•1) with the kernel k(t) = e it 2 .…”

“…k,20 are denoted by white circles, the first iteration z (1) k,20 by filled-in discs and the eigenvalues λ (n) k,20 by white stars. Figure 2.…”

Commentary on "Expansions for Eigenfunctions and Eigenvalues of large-$n$ Toeplitz Matrices"

“…After a preprint version of the paper appeared, Bogoya, Böttcher and Grudsky [2] gave a rigorous proof of the eigenvalue asymptotics in the special case of symbols a(z) with β = α − 1. The general case is (as of now) still open.…”