Equilibrium problem for the eigenvalues of banded block Toeplitz matrices

Abstract

We consider banded block Toeplitz matrices Tn with n block rows and columns. We show that under certain technical assumptions, the normalized eigenvalue counting measure of Tn for n → ∞ weakly converges to one component of the unique vector of measures that minimizes a certain energy functional. In this way we generalize a recent result of Duits and Kuijlaars for the scalar case. Along the way we also obtain an equilibrium problem associated to an arbitrary algebraic curve, not necessarily related to a block Toeplitz matrix.For banded block Toeplitz matrices, there are several new phenomena that do not occur in the scalar case: (i) The total masses of the equilibrium measures do not necessarily form a simple arithmetic series but in general are obtained through a combinatorial rule; (ii) The limiting eigenvalue distribution may contain point masses, and there may be attracting point sources in the equilibrium problem; (iii) More seriously, there are examples where the connection between the limiting eigenvalue distribution of Tn and the solution to the equilibrium problem breaks down. We provide sufficient conditions guaranteeing that no such breakdown occurs; in particular we show this if Tn is a Hessenberg matrix.

High-Order Three-Term Recursions, Riemann–Hilbert Minors and Nikishin Systems on Star-Like Sets

“…Moreover, z j+ (λ) and z j− (λ) are the boundary values of z j (λ) obtained from the +-side and −-side respectively of Γ k , where the +-side (−-side) is the side that lies on the left (right) when moving through Γ k according to its orientation. It turns out that µ k is a positive measure (obviously independent of the orientation given to Γ k ) with total mass [7,Sec. 4]…”

“…, ν p−1 ) admissible if ν k has finite logarithmic energy, ν k is supported on Γ k , and ν k has total mass ν k (Γ k ) = p−k p , k ∈ ( 1.21) The (vector) equilibrium problem is to minimize the energy functional (1.21) over all admissible vectors of positive measures ν. The equilibrium problem has a unique solution which is given by the measures µ k in ( 1.19), see [7].…”

“…Remark 5. 7. The proof of Lemma 5.1 simplifies considerably when k = 0 or k = p. This is because in the former case we deal with the polynomials Q n (x), whose zeros are uniformly bounded on S + , while in the latter case the numerator and denominator in (5.3) are simply constants.…”

“…The collection of functions det F r−1,0 (z k (x), x), k ∈ [0 : p], can be seen as a single meromorphic function defined on R. (It has poles at infinity, see also [29]). Now [7, Lemma 5 .5] (see also [7,Lemma 5.6]) shows that R is connected. Hence the above meromorphic function cannot be identically zero since this would imply by (6.4) and (6.6) that Q rn ≡ 0, clearly contradictory.…”

“…It is easy to see (see e.g. [7,Sec. 4] or [29, p. 102]) that for x → ∞, More precisely, for any x ∈ C there is a permutation (z k (x)) p k=1 of the set (z k (x)) p k=1 so that It turns out that Γ k is a finite union of line segments on the star S + if k is even and S − if k is odd: see Fig.…”