Eigenvalues of Hessenberg Toeplitz matrices generated by symbols with several singularities
Manuel Bogoya, Albrecht Böttcher, Sergei M. Grudsky, Egor A. Maksimenko · 2011
In a recent paper, we established asymptotic formulas for the eigenvalues of the n×n truncations of certain infinite Hessenberg Toeplitz matrices as n goes to infinity. The symbol of the Toeplitz matrices was of the form a(t) = t−1(1−t)α f (t) (t ∈T), where α is a positive real number but not an integer and f is a smooth function in H∞. Thus, a has a single power singularity at the point 1. In the present work we extend the results to symbols with a finite number of power singularities. To be more precise, we consider symbols of the form a(t) = t−1 f (t)∏Kk=1(1 − t/tk)αk (t ∈ T), where tk = eiθk, the arguments θk are all different, and the exponents αk are positive real numbers but not integers.