The spectra of certain Toeplitz matrices
Isidore Isaac Hirschman · Illinois Journal of Mathematics · 1967
IntroductionLet f(eio) e LX(T) where T is the reals modulo 2r.f ( e) f"( r)e r If then the Toeplitz matrix of order n + 1 associated with f is Mn[f] [f^(r-s)]r,8=0,1,...,.Let {,,} =0 be the eigen values of Mn[f], that is the zeros of det [M.kI.] O.For each n we define a measure a on the Borel sets in the )-plane by a,( E) n + 1) -1 _.x,,s 1.An important topic in the theory of Toeplitz matrices is the study of the asymptotic behavior of the measures a as n -.If f(ei) is real, in which case the matrices M[f] are Hermitian, there is a simple and elegant solution.The support of each a is contained in the interval of the real line whose end points are ess inf f and ess sup f, and as n --+ the a converge wekly to the mesure a defined by (2)-/ o, (eiO)E see [1, 7.5].When f(e) is not real, however, the problem is very dicult and the only results are those obtained by P. Schmidt and F. Spitzer in [5].They assumed that f is a Laurent polynomial,where h, lc > 0, (otherwise the problem in question is trivial) and showed that there then exists a compact set C in the X-plane, which can be described pre- cisely, such that if N is any neighborhood of C the support of a must be in N provided n is sufficiently large.Moreover, no smaller closed set C has this property.In this paper we will complete the investigation of Schmidt and Spitzer by showing that as n -+ the a converge weakly to a measure a with support in C.This is rather easy.What is of greater interest is that we will obtain an explicit formula for a.