An equilibrium problem for the limiting eigenvalue distribution of
  banded Toeplitz matrices
AN EQUILIBRIUM PROBLEM FOR THE LIMITING
EIGENVALUE DISTRIBUTION OF BANDED TOEPLITZ
MATRICES
MAURICE DUITS AND ARNO B.J. KUIJLAARS
Abstract. We study the limiting eigenvalue distribution of n×n banded
Toeplitz matrices as n → ∞. From classical results of Schmidt-Spitzer
and Hirschman it is known that the eigenvalues accumulate on a spe-
cial curve in the complex plane and the normalized eigenvalue counting
measure converges weakly to a measure on this curve as n → ∞. In this
paper, we characterize the limiting measure in terms of an equilibrium
problem. The limiting measure is one component of the unique vector
of measures that minimes an energy functional defined on admissible
vectors of measures. In addition, we show that each of the other com-
ponents is the limiting measure of the normalized counting measure on
certain generalized eigenvalues.
1. Introduction
For an integrable function a : {z ∈ C | |z| = 1} → C defined on the unit
circle in the complex plane, the n× n Toeplitz matrix Tn(a) with symbol a
is defined by
Tn(a)
= aj−k, j, k = 1, . . . , n, (1.1)
where ak is the kth Fourier coefficient of a,
a(eiθ)e−ikθ dθ. (1.2)
In this paper we study banded Toeplitz matrices for which the symbol has
only a finite number of non-zero Fourier coefficients. We assume that there
exist p, q ≥ 1 such that
a(z) =
k, ap 6= 0, a−q 6= 0. (1.3)
Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B,
3001 Leuven, Belgium. (maurice.duits@wis.kuleuven.be, arno.kuijlaars@wis.kuleuven.be).
The first author is a research assistant of the Fund for Scientific Research – Flanders.
The authors were supported by the European Science Foundation Program MISGAM.
The second author is supported by FWO-Flanders project G.0455.04, by K.U. Leuven
research grant OT/04/21, by Belgian Interuniversity Attraction Pole NOSY P06/02, and
by a grant from the Ministry of Education and Science of Spain, project code MTM2005-
08648-C02-01.
http://arxiv.org/abs/0704.0378v1
2 MAURICE DUITS AND ARNO B.J. KUIJLAARS
Thus Tn(a) has at most p + q + 1 non-zero diagonals. As in [1, p. 263], we
also assume without loss of generality that
g.c.d. {k ∈ Z | ak 6= 0} = 1. (1.4)
We are interested in the limiting behavior of the spectrum of Tn(a) as
n → ∞. We use spTn(a) to denote the spectrum of Tn(a):
spTn(a) = {λ ∈ C | det(Tn(a)− λI) = 0}
Spectral properties of banded Toeplitz matrices are the topic of the recent
book [1] by Böttcher and Grudsky. We will refer to this book frequently,
in particular to Chapter 11 where the limiting behavior of the spectrum is
discussed.
The limiting behavior of spTn(a) was characterized by Schmidt and Spitzer
[10]. They considered the set
lim inf
spTn(a), (1.5)
consisting of all λ ∈ C such that there exists a sequence {λn}n∈N, with
λn ∈ spTn(a), converging to λ, and the set
lim sup
spTn(a), (1.6)
consisting of all λ such that there exists a sequence {λn}n∈N, with λn ∈
spTn(a), that has a subsequence converging to λ. Schmidt and Spitzer
showed that these two sets are equal and can be characterized in terms of
the algebraic equation
a(z) − λ =
k − λ = 0. (1.7)
For every λ ∈ C there are p+q solutions for (1.7), which we denote by zj(λ),
for j = 1, . . . , p+ q. We order these solutions by absolute value, so that
0 < |z1(λ)| ≤ |z2(λ)| ≤ · · · ≤ |zp+q(λ)|. (1.8)
When all inequalities in (1.8) are strict then the values zk(λ) are unambigu-
ously defined. If equalities occur then we choose an arbitrary numbering
so that (1.8) holds. The result by Schmidt and Spitzer [10], [1, Theorem
11.17], is that
lim inf
spTn(a) = lim sup
spTn(a) = Γ0 (1.9)
where
Γ0 := {λ ∈ C | |zq(λ)| = |zq+1(λ)|}. (1.10)
This result gives a description of the asymptotic location of the eigenvalues.
The eigenvalues accumulate on the set Γ0, which is known to be a disjoint
union of a finite number of (open) analytic arcs and a finite number of ex-
ceptional points [1, Theorem 11.9]. It is also known that Γ0 is connected
EIGENVALUES OF BANDED TOEPLITZ MATRICES 3
[13], [1, Theorem 11.19], and that C \ Γ0 need not be connected [1, Theo-
rem 11.20], [2, Proposition 5.2]. See [1] for many beautiful illustrations of
eigenvalues of banded Toeplitz matrices.
The limiting eigenvalue distribution was determined by Hirschman [5], [1,
Theorem 11.16]. He showed that there exists a Borel probability measure
µ0 on Γ0 such that the normalized eigenvalue counting measure of Tn(a)
converges weakly to µ0, as n → ∞. That is,
λ∈spTn(a)
δλ → µ0, (1.11)
where in the sum each eigenvalue is counted according to its multiplicity.
The measure µ0 is absolutely continuous with respect to the arclength mea-
sure on Γ0 and has an analytic density on each open analytic arc in Γ0,
which can be explicitly represented in terms of the solutions of the algebraic
equation (1.7) as follows. Equip every open analytic arc in Γ0 with an orien-
tation. The orientation induces ±-sides on each arc, where the +-side is on
the left when traversing the arc according to its orientation, and the −-side
is on the left. The limiting measure µ0 is then given by
dµ0(λ) =
zj+(λ)
zj−(λ)
dλ. (1.12)
where dλ is the complex line element on Γ0 (taken according to the orien-
tation), and where zj±(λ), λ ∈ Γ0, is the limiting value of zj(λ
′) as λ′ → λ
from the ± side of the arc. These limiting values exist for every λ ∈ Γ0,
with the possible exception of the finite number of exceptional points.
Note that the right-hand side of (1.12) is a priori a complex measure and
it is not immediately clear that it is in fact a probability measure. In the
original paper [5] and in the book [1, Theorem 11.16], the authors give a
different expression for the limiting density, from which it is clear that the
measure is non-negative. We prefer to work with the complex expression
(1.12), since it allows for a direct generalization which we will need in this
paper.
Note also that if we reverse the orientation on an arc in Γ0, then the ±-
sides are reversed. Since the complex line element dλ changes sign as well,
the expression (1.12) does not depend on the choice of orientation.
The following is a very simple example, which however serves as a moti-
vation for the results in the paper.
Example 1.1. Consider the symbol a(z) = z+1/z. In this case we find that
Γ0 = [−2, 2] and µ0 is absolutely continuous with respect to the Lebesgue
measure and has density
dµ0(λ)
4− λ2
, λ ∈ (−2, 2). (1.13)
4 MAURICE DUITS AND ARNO B.J. KUIJLAARS
This measure is well-known in potential theory and is called the arcsine
measure or the equilibrium measure of Γ0, see e.g. [9]. It has the property
that it minimizes the energy functional I defined by
I(µ) =
|x− y| dµ(x) dµ(y), (1.14)
among all Borel probability measures µ on [−2, 2]. The measure µ0 is also
characterized by the equilibrium condition
log |x− λ| dµ0(λ) = 0, x ∈ [−2, 2], (1.15)
which is the Euler-Lagrange variational condition for the minimization prob-
The fact that µ0 is the equilibrium measure of Γ0 is special for symbols
a with p = q = 1. In that case one may think of the eigenvalues of Tn(a) as
charged particles on Γ0, each eigenvalue having a total charge 1/n, that repel
each other with logarithmic interaction. The particles seek to minimize the
energy functional (1.14). As n → ∞, they distribute themselves according
to µ0 and µ0 is the minimizer of (1.14) among all probability measures
supported on Γ0.
The aim of this paper is to characterize µ0 for general symbols a of the
form (1.3) also in terms of an equilibrium problem from potential theory.
The corresponding equilibrium problem is more complicated since it involves
not only the measure µ0, but a sequence of p+ q − 1 measures
µ−q+1, µ−q+2, . . . , µ−1, µ0, µ1, . . . , µp−2, µp−1
that jointly minimize an energy functional.
2. Statement of results
2.1. The energy functional. To state our results we need to introduce
some notions from potential theory. Main references for potential theory in
the complex plane are [8] and [9].
We will mainly work with finite positive measures on C, but we will also
use ν1 − ν2 where ν1 and ν2 are positive measures. The measures need not
have bounded support. If ν has unbounded support then we assume that
log(1 + |x|) dν(x) < ∞. (2.1)
In that case the logarithmic energy of ν is defined as
I(ν) =
|x− y| dν(x)dν(y) (2.2)
and I(ν) ∈ (−∞,+∞].
EIGENVALUES OF BANDED TOEPLITZ MATRICES 5
Definition 2.1. We define Me as the collection of positive measures ν on
C satisfying (2.1) and having finite energy, i.e., I(ν) < +∞. For c > 0 we
define
Me(c) = {ν ∈ Me | ν(C) = c}. (2.3)
The mutual energy I(ν1, ν2) of two measures ν1 and ν2 is
I(ν1, ν2) =
|x− y| dν1(x)dν2(y). (2.4)
It is well-defined and finite if ν1, ν2 ∈ Me and in that case we have
I(ν1 − ν2) = I(ν1) + I(ν2)− 2I(ν1, ν2). (2.5)
If ν1, ν2 ∈ Me(c) for some c > 0, then
I(ν1 − ν2) ≥ 0, (2.6)
with equality if and only if ν1 = ν2. This is a well-known result if ν1 and ν2
have compact support [9]. For measures in Me(c) with unbounded support,
this is a recent result of Simeonov [11], who obtained this from a very elegant
integral representation for I(ν1 − ν2). It is a consequence of (2.6) that I is
strictly convex on Me(c), since
ν1 + ν2
(I(ν1) + I(ν2))− I
ν1 − ν2
(I(ν1) + I(ν2)) , for ν1, ν2 ∈ Me(c),
with equality if and only if ν1 = ν2.
Before we can state the equilibrium problem we also need to introduce
the sets
Γk := {λ ∈ C | |zq+k(λ)| = |zq+k+1(λ)|}, k = −q + 1, . . . , p− 1, (2.7)
which for k = 0 reduces to the definition (1.10) of Γ0. We will show that
each Γk is the disjoint union of a finite number of open analytic arcs and
a finite number of exceptional points. All Γk are unbounded, except for Γ0
which is compact.
The equilibrium problem will be defined for a vector of measures denoted
by ~ν = (ν−q+1, . . . , νp−1). The component νk is a measure on Γk satisfying
some additional properties that are given in the following definition.
Definition 2.2. We call a vector of measures ~ν = (ν−q+1, . . . , νp−1) admis-
sible if νk ∈ Me, νk is supported on Γk, and
νk(Γk) =
if k ≤ 0,
if k ≥ 0,
(2.8)
for every k = −q + 1, . . . , p − 1.
Now we are ready to state our first result. The proof is given in section
6 MAURICE DUITS AND ARNO B.J. KUIJLAARS
Theorem 2.3. Let the symbol a satisfy (1.3) and (1.4), and let the curves
Γk be defined as in (2.7). For each k ∈ {−q + 1, . . . , p − 1}, define the
measure µk on Γk by
dµk(λ) =
zj+(λ)
zj−(λ)
dλ, (2.9)
where dλ is the complex line element on each analytic arc of Γk according
to a chosen orientation of Γk (cf. discussion after (1.12)). Then
(a) ~µ = (µ−q+1, . . . , µp−1) is admissible.
(b) There exist constants lk such that
log |λ− x| dµk(x) =
log |λ− x| dµk+1(x) +
log |λ− x| dµk−1(x) + lk,
(2.10)
for k = −q + 1, . . . , p − 1, and λ ∈ Γk. Here we let µ−q and µp be
the zero measures.
(c) ~µ = (µ−q+1, . . . , µp−1) is the unique minimizer of the energy func-
tional J defined by
J(~ν) =
k=−q+1
I(νk)−
k=−q+1
I(νk, νk+1) (2.11)
for admissible vectors of measures ~ν = (ν−q+1, . . . , νp−1).
The relations (2.10) are the Euler-Lagrange variational conditions for the
minimization problem for J among admissible vectors of measures.
It may not be obvious that the energy functional (2.11) is bounded from
below. This can be seen from the alternative representation
J(~ν) =
I(ν0) +
k(k + 1) I
ν−q+k
− ν−q+k+1
k + 1
k(k + 1) I
− νp−k−1
k + 1
. (2.12)
We leave the calculation leading to this identity to the reader. Under the
normalizations (2.8) it follows by (2.6) that each term in the two finite sums
on the right-hand side of (2.12) is non-negative, so that
J(~ν) ≥
I(ν0).
Since ν0 is a Borel probability measure on Γ0 and Γ0 is compact, we indeed
have that the energy functional is bounded from below on admissible vectors
of measures ~ν.
The alternative representation (2.12) will play a role in the proof of The-
orem 2.3.
EIGENVALUES OF BANDED TOEPLITZ MATRICES 7
Yet another representation for J is
J(~ν) =
j,k=−q+1
Ajk I(νj , νk) (2.13)
where the interaction matrix A has entries
Ajk =
1, if j = k,
, if |j − k| = 1,
0, if |j − k| ≥ 2.
(2.14)
The energy functional in the form (2.13) and (2.14) also appears in the
theory of simultaneous rational approximation, where it is the interaction
matrix for a Nikishin system [7, Chapter 5].
It allows for the following physical interpretation: on each of the curves Γk
one puts charged particles with total charge (q+k)/q or (p−k)/p, depending
on whether k ≤ 0 or k ≥ 0. Particles that lie on the same curve repel each
other. The particles on two consecutive curves interact in the sense that
they attract each other but in a way that is half as strong as the repulsion
on a single curve. Particles on different curves that are not consecutive do
not interact with each other in a direct way.
2.2. The measures µk as limiting measures of generalized eigenval-
ues. By (1.12) and Theorem 2.3 we know that the measure µ0 that appears
in the minimizer of the energy functional J is the limiting measure for the
eigenvalues of Tn(a). It is natural to ask about the other measures µk that
appear in the minimizer. In our second result we show that the measures
µk can be obtained as limiting counting measures for certain generalized
eigenvalues.
Let k ∈ {−q+1, . . . , p− 1}. We use Tn(z−k(a−λ) to denote the Toeplitz
matrix with the symbol z 7→ z−k(a(z) − λ). For example, for k = 1, q = 1
and p = 2, we have
−k(a−λ)) =
a1 a0 − λ a−1
a2 a1 a0 − λ a−1
a2 a1 a0 − λ a−1
. . .
. . .
. . .
. . .
a2 a1 a0 − λ a−1
a2 a1 a0 − λ
a2 a1
Definition 2.4. For k ∈ {−q + 1, . . . , p − 1} and n ≥ 1, we define the
polynomial Pk,n by
Pk,n(λ) = detTn(z
−k(a− λ)) (2.15)
and we define the kth generalized spectrum of Tn(a) by
spk Tn(a) = {λ ∈ C | Pk,n(λ) = 0}. (2.16)
8 MAURICE DUITS AND ARNO B.J. KUIJLAARS
Finally, we define µk,n as the normalized zero counting measure of spk Tn(a)
µk,n =
λ∈spk Tn(a)
δλ (2.17)
where in the sum each λ is counted according to its multiplicity as a zero of
Pk,n.
Note that λ ∈ spk Tn(a) is a generalized eigenvalue (in the usual sense) for
the matrix pencil (Tn(z
−ka), Tn(z
−k)), that is, det(A − λB) = 0 with A =
−ka) and B = Tn(z
−k). If k = 0, then B = I and sp0 Tn(a) = spTn(a).
If k 6= 0, then B is not invertible and the generalized eigenvalue problem is
singular, causing that there are less than n generalized eigenvalues. In fact,
since Tn(z
−k(a−λ)) has exactly n−|k| entries a0−λ, we easily get that the
degree of Pk,n is at most n− |k| and so there are at most n− |k| generalized
eigenvalues. Due to the band structure of Tn(z
−k(a−λ)) the actual number
of generalized eigenvalues is substantially smaller.
Proposition 2.5. Let k ∈ {−q+1, . . . , p−1}. Let Pk,n(λ) = γk,nλdk,n + · · ·
have degree dk,n and leading coefficient γk,n 6= 0. Then
dk,n ≤
n, if k < 0,
n, if k > 0.
(2.18)
Equality holds in (2.18) if either k > 0 and n is a multiple of p, or k < 0
and n is a multiple of q, and in those cases we have
γk,n =
(−1)(k+1)na|k|n/q−q , if k < 0 and n ≡ 0 mod q,
(−1)(k+1)nakn/pp , if k > 0 and n ≡ 0 mod p.
(2.19)
We now come to our second main result. It is the analogue of the results
of Schmidt-Spitzer and Hirschman for the generalized eigenvalues.
Theorem 2.6. Let k ∈ {−q + 1, . . . , p− 1}. Then
lim inf
spk Tn(a) = lim sup
spk Tn(a) = Γk, (2.20)
φ(z) dµk,n(z) =
φ(z) dµk(z) (2.21)
holds for every bounded continuous function φ on C.
The key element in the proof of Theorem 2.6 is a beautiful formula of
Widom [14], see [1, Theorem 2.8], for the determinant of a banded Toeplitz
matrix. In the present situation Widom’s formula yields the following. Let
λ ∈ C be such that the solutions zj(λ) of the algebraic equation (1.7) are
mutually distinct. Then
Pk,n(λ) = detTn(z
−k(a− λ)) =
CM (λ) (wM (λ))
, (2.22)
EIGENVALUES OF BANDED TOEPLITZ MATRICES 9
where the sum is over all subsets M ⊂ {1, 2, . . . , p+ q} of cardinality |M | =
p− k and for each such M , we have
wM (λ) := (−1)p−kap
zj(λ), (2.23)
and (with M := {1, 2, . . . , p+ q} \M),
CM (λ) :=
zj(λ)
(zj(λ)− zl(λ))−1. (2.24)
The formula (2.22) shows that for large n, the main contribution comes from
those M for which |wM (λ)| is the largest possible. For λ ∈ C \ Γk there is a
unique such M , namely
M = Mk := {q + k + 1, q + k + 2, . . . , p + q} (2.25)
because of the ordering (1.8).
2.3. Overview of the rest of the paper. In section 3 we will state some
preliminary results about analyticity properties of the solutions zj of the al-
gebraic equation (1.7). These results will be needed in the proof of Theorem
2.3 which is given in section 4. In section 5 we will prove Proposition 2.5
and Theorem 2.6. Finally, we conclude the paper by giving some examples
in section 6.
3. Preliminaries
In this section we collect a number of properties of the curves Γk and the
solutions z1(λ), . . . , zp+q(λ) of the algebraic equation (1.7). For convenience
we define throughout the rest of the paper
Γ−q = Γp = ∅, and µ−q = µp = 0. (the zero-measure).
Occasionally we also use
z0(λ) = 0, zp+q+1(λ) = +∞.
3.1. The structure of the curves Γk. We start with a definition, cf. [1,
§11.2].
Definition 3.1. A point λ0 ∈ C is called a branch point if a(z) − λ0 = 0
has a multiple root. A point λ0 ∈ Γk is an exceptional point of Γk if λ0 is a
branch point, or if there is no open neighborhood U of λ such that Γk ∩U is
an analytic arc starting and terminating on ∂U .
If λ0 is a branch point, then there is a z0 such that a(z0) = λ0 and
a′(z0) = 0. Then we may assume that z0 = zq+k(λ0) = zq+k+1(λ0) for some
k and λ0 ∈ Γk. For a symbol a of the form (1.3), the derivative a′ has exactly
p+q zeros (counted with multiplicity), so that there are exactly p+q branch
points counted with multiplicity.
10 MAURICE DUITS AND ARNO B.J. KUIJLAARS
The solutions zk(λ) also have branching at infinity (unless p = 1 or q = 1).
There are p solutions of (1.7) that tend to infinity as λ → ∞, and q solutions
that tend to 0. Indeed, we have
zk(λ) =
−1/q(1 +O(λ−1/q)), for k = 1, . . . , q,
1/p(1 +O(λ−1/p)), for k = q + 1, . . . , p + q,
(3.1)
as λ → ∞. Here c1, . . . , cq are the q distinct solutions of cq = a−q (taken in
some order depending on λ), and cq+1, . . . , cp+q are the p distinct solutions
of cp = a−1p (again taken in some order depending on λ).
The following proposition gives the structure of Γk at infinity.
Proposition 3.2. Let k ∈ {−q+1, . . . , p−1}\{0}. Then there is an R > 0
such that Γk ∩ {λ ∈ C | |λ| > R} is a finite disjoint union of analytic arcs,
each extending from |λ| = R to infinity.
Proof. The proof is similar to the proof of [1, Proposition 11.8] where a
similar structure theorem was proved for finite branch points. We omit the
details. �
It follows from Proposition 3.2 that the exceptional points for Γk are in a
bounded set. Since the set of exceptional point is discrete we conclude that
there are only finitely many exceptional points. Then we have the following
result about the structure of Γk.
Proposition 3.3. For every k ∈ {−q + 1, . . . , p − 1}, the set Γk is the
disjoint union of a finite number of open analytic arcs and a finite number
of exceptional points. The set Γk has no isolated points.
Proof. This was proved for k = 0 in [10] and [1, Theorem 11.9]. For general
k, there are only finitely many exceptional points and the proof follows in a
similar way. �
3.2. The Riemann surface. From Proposition 3.3 it follows that the curves
Γk can be taken as cuts for the p + q-sheeted Riemann surface of the alge-
braic equation (1.7). We number the sheets from 1 to p+ q, where the kth
sheet of the Riemann surface is
Rk = {λ ∈ C | |zk−1(λ)| < |zk(λ)| < |zk+1(λ)|} = C \ (Γ−q+k−1 ∪ Γ−q+k).
(3.2)
Thus zk is well-defined and analytic on Rk.
The easiest case to visualize is the case where consecutive cuts are disjoint,
that is, Γ−q+k−1 ∩ Γ−q+k = ∅ for every k = 2, . . . , p+ q− 2. In that case we
have that Rk is connected to Rk+1 via Γ−q+k in the usual crosswise manner,
and zk+1 is the analytic continuation of zk across Γ−q+k.
The general case is described in the following proposition.
Proposition 3.4. Suppose A is an open analytic arc such that A ⊂ Γ−q+k,
for k = k1, . . . , k2, and A ∩ (Γ−q+k1−1 ∪ Γ−q+k2+1) = ∅. Then for k =
EIGENVALUES OF BANDED TOEPLITZ MATRICES 11
k1, . . . , k2+1, we have that the analytic continuation of zk across A is equal
to zk1+k2−k+1. Thus across A, we have that Rk is connected to Rk1+k2−k+1.
Proof. We have that
|zk1(λ)| = |zk1+1(λ)| = · · · = |zk2(λ)| = |zk2+1(λ)|
for λ ∈ A, with strict inequalities (<) for λ on either side of A. Choose an
orientation for A. Then there is a permutation π of {k1, . . . , k2 + 1} such
that zπ(k) is the analytic continuation of zk from the +-side of A to the
−-side of A.
Assume that there are k, k′ ∈ {k1, . . . , k2 + 1} such that k < k′ and
π(k) < π(k′). Take a regular λ0 ∈ A and a small neighborhood U of λ0 such
that A∩U = Γ−q+k ∩U = Γ−q+k′ ∩U and A∩U is an analytic arc starting
and terminating on ∂U . Then we have a disjoint union U = U+∪U−∪(A∩U)
where U+ (U−) is the part of U on the +-side (−-side) of A. The function
φ defined by
φ(λ) =
zk(λ)
zk′ (λ)
, for λ ∈ U+,
zπ(k)(λ)
zπ(k′)(λ)
, for λ ∈ U−,
has an analytic continuation to U , and satisfies |φ(λ)| < 1 for λ ∈ U+ ∪ U−
and |φ(λ)| = 1 for λ ∈ A ∩ U . This contradicts the maximum principle for
analytic functions. Therefore π(k) > π(k′) for every k, k′ ∈ {k1, . . . , k2 + 1}
with k < k′, and this implies that π(k) = k1 + k2 − k + 1 for every k =
k1, . . . , k2 + 1, and the proposition follows. �
3.3. The functions wk(λ). A major role is played by the functions wk,
which for k ∈ {−q + 1, . . . , p − 1}, are defined by
wk(λ) =
zj(λ), for λ ∈ C \ Γk. (3.3)
Note that wk = (−1)p−ka−1p w{1,...,k} in the notation of (2.23).
Proposition 3.5. The function wk is analytic in C \ Γk.
Proof. Since zj is analytic on Rj = C \ (Γ−q+j−1 ∪ Γ−q+j), see (3.2), we
obtain from its definition that wk is analytic in C \
j=1 Γ−q+j. Let A be
an analytic arc in Γ−q+j \ Γk for some j < k + q. Choose an orientation on
A. Since the arc is disjoint from Γk, we have that zj+(λ) = zπ(j)−(λ), for
λ ∈ A and j = 1, . . . , q+k, where π is a permutation of {1, . . . , q+k}. Since
wk is symmetric in the zj ’s for j = 1, . . . , q + k, it then follows that
wk+(λ) = wk−(λ), for λ ∈ A,
which shows the analyticity in C \Γk with the possible exception of isolated
singularities at the exceptional points of Γ−q+1, Γ−q+2, . . . , Γk−1. However,
each zj , and therefore also wk, is bounded near such an exceptional point,
so that any isolated singularity is removable. �
12 MAURICE DUITS AND ARNO B.J. KUIJLAARS
In the rest of the paper we make frequently use of the logarithmic de-
rivative w′k/wk of wk. By the fact that wk does not vanish on C \ Γk and
Proposition 3.5, it follows that w′k/wk is analytic in C \ Γk. By Proposition
3.4 it moreover has an analytic continuation across every open analytic arc
A ⊂ Γk. Near the exceptional points that are no branch points w′k/wk re-
mains bounded. At the branch points it can however have singularities of a
certain order.
Proposition 3.6. Let λ0 ∈ Γk be a branch point of Γk. Then there exists
an m ∈ N such that
w′k(λ)
wk(λ)
(λ− λ0)−m/(m+1)
, (3.4)
as λ → λ0 with λ ∈ C \ Γk.
Proof. Let 1 ≤ j ≤ q+k. We investigate the behavior of zj(λ) when λ → λ0
such that λ remains in a connected component of C \ (Γj−1 ∪ Γj). Then
zj(λ) → z0 for some z0 ∈ C with a(z0) = λ0. Let m0 + 1 be the multiplicity
of z0 as a solution of a(z) = λ0. Then
a(z) = λ0 + c0(z − z0)m0+1(1 +O(z − z0)), z → z0, (3.5)
for some nonzero constant c0. Therefore,
zj(λ) = z0 +O((λ− λ0)1/(m0+1)), (3.6)
z′j(λ) = O((λ− λ0)−m0/(m0+1)), (3.7)
for λ → λ0 such that λ remains in the same connected component of C \
(Γj−1 ∪ Γj). Let m be the maximum of all the multiplicities of the roots of
a(z) = λ0. Then it follows from (3.6) and (3.7) that
z′j(λ)
zj(λ)
= O((λ− λ0)−m/(m+1))
as λ → λ0 with λ ∈ C \ Γk. Then we obtain (3.4) in view of (3.3). �
We end this section by giving the asymptotics of w′k/wk for λ → ∞.
Proposition 3.7. As λ → ∞ with λ ∈ C \ Γk, we have
w′k(λ)
wk(λ)
− q+k
λ−1 +O
λ−1−1/q
, for k = −q + 1, . . . ,−1,
−λ−1 +O(λ−2), for k = 0,
λ−1 +O
λ−1−1/p
, for k = 1, . . . , p − 1.
(3.8)
Proof. This follows directly from (3.1) and (3.3). �
EIGENVALUES OF BANDED TOEPLITZ MATRICES 13
4. Proof of Theorem 2.3
We use the function wk introduced in (3.3). We define µk by the formula
(2.9) and we note that
dµk(λ) =
w′k+(λ)
wk+(λ)
w′k−(λ)
wk−(λ)
dλ. (4.1)
Proposition 4.1. For each k = −q + 1, . . . , p − 1, we have that µk is a
measure on Γk with total mass µk(Γk) = (q + k)/q if k ≥ 0, and µk(Γk) =
(p− k)/p if k ≥ 0.
Proof. We first show that µk is a measure, i.e., that it is non-negative on
each analytic arc of Γk. Let A be an analytic arc in Γk consisting only of
regular points. Let t 7→ λ(t) be a parametrization of A in the direction of
the orientation of Γk. Then
dµk(λ) =
w′k+(λ(t))
wk+(λ(t))
w′k−(λ(t))
wk−(λ(t))
λ′(t)dt
wk+(λ(t))
wk−(λ(t))
To conclude that µk is non-negative on A, it is thus enough to show that
Re log
wk+(λ)
wk−(λ)
= 0, for λ ∈ A, (4.2)
Im log
wk+(λ)
wk−(λ)
increases along A. (4.3)
Since |wk+(λ)| = |wk−(λ)| for λ ∈ A, we have (4.2) so that it only remains
to prove (4.3).
There is a neighborhood U of A such that U \ Γk has two components,
denoted U+ and U−, where U+ is on the +-side of Γk and U− on the −-side.
It follows from Proposition 3.4 that wk has an analytic continuation from
U− to U , which we denote by ŵk, and that |wk(λ)| < |ŵk(λ)| for λ ∈ U+,
and equality |wk+(λ)| = |ŵk(λ)| holds for λ ∈ A. Thus it follows that
Re log
wk(λ)
ŵk(λ)
≤ 0, for λ ∈ A,
where ∂
denotes the normal derivative to A in the direction of U+. Then by
the Cauchy-Riemann equations we have that Im log
wk+(λ)
ŵk+(λ)
is increasing
along A. Since ŵk+(λ) = wk−(λ) for λ ∈ A, we obtain (4.3). Thus µk is a
measure.
Next we show that µk is a finite measure, which means that we have to
show that
w′k+(λ)
wk+(λ)
w′k−(λ)
wk−(λ)
(4.4)
14 MAURICE DUITS AND ARNO B.J. KUIJLAARS
Figure 1. Illustration for the proofs of Propositions 4.1 and
4.2. The solid line is a sketch of a possible contour Γk. The
dashed line is the contour Γ̃k,R and the dotted line is the
boundary of a disk of radius R around 0.
is integrable near infinity on Γk and near every branch point on Γk. This
follows from Propositions 3.7 and 3.6. Indeed, from Proposition 3.7 it follows
w′k+(λ)
wk+(λ)
w′k−(λ)
wk−(λ)
λ−1−δ
as λ → ∞, λ ∈ Γk. (4.5)
where δ = 1/q if k < 0 and δ = 1/p if k > 0. Since δ > 0 we see that
(4.4) is integrable near infinity. For a branch point λ0 of Γk, we have from
Proposition 3.6 that there exist an m ≥ 1 such that
w′k+(λ)
wk+(λ)
w′k−(λ)
wk−(λ)
(λ− λ0)−m/(m+1)
as λ → λ0, λ ∈ Γk. (4.6)
This shows that (4.4) is integrable near every branch point. Thus µk is a
finite measure.
Finally we compute the total mass of µk. LetD(0, R) = {z ∈ C | |z| < R}.
Then for R large enough, so that D(0, R) contains all exceptional points of
Γk and all connected components of C \ Γk (if any),
µk(Γk ∩D(0, R)) =
Γk∩D(0,R)
w′k+(λ)
wk+(λ)
Γk∩D(0,R)
w′k−(λ)
wk−(λ)
(4.7)
where we have used the behavior (4.6) near the branch points in order to
be able to split the integrals. Again using (4.6) we can then turn the two
integrals into a contour integral over a contour Γ̃k,R as in Figure 1. The
contour Γ̃k,R passes along the ±-sides of Γk ∩D(0, R) and if we choose the
orientation that is also shown in Figure 1 (and which is independent of the
EIGENVALUES OF BANDED TOEPLITZ MATRICES 15
choice of orientation for Γk), then
µk(Γk ∩D(0, R)) =
Γ̃k,R
w′k(λ)
wk(λ)
dλ. (4.8)
The parts of Γ̃k,R that belong to bounded components of C \Γk form closed
contours along the boundary of each bounded component. By Cauchy’s
theorem their contribution to the integral (4.8) vanishes. The parts of Γ̃k,R
that belong to the unbounded components of C \Γk can be deformed to the
circle ∂D(0, R) with the clockwise orientation. Thus if we use the positive
orientation on ∂D(0, R) as in Figure 1, then we obtain from (4.8)
µk(Γk ∩D(0, R)) = −
∂D(0,R)
w′k(λ)
wk(λ)
Letting R → ∞ and using Proposition 3.7, we then find that µk is a measure
on Γk with total mass µk(Γk) = (q + k)/q if k ≤ 0, and µk(Γk) = (p− k)/p
if k ≥ 0. �
The following proposition is the next step in showing that the measures
µk from (2.9) satisfy the equations (2.10).
Proposition 4.2. For k = −q + 1, . . . , p− 1, we have that
dµk(x)
x− λ =
w′k(λ)
wk(λ)
, for λ ∈ C \ Γk, (4.9)
log |λ− x| dµk(x) = − log |wk(λ)|+ αk, for λ ∈ C, (4.10)
where αk is the constant
log |a−q|+ kq log |a−q|, if k ≤ 0,
log |a−q| − kp log |ap|, if k ≥ 0.
(4.11)
Proof. To prove (4.9), we follow the same arguments as in the calculation
of µk(Γk) in the end of the proof of Proposition 4.1. Let λ ∈ C \ Γk, and
choose R > 0 as in the proof of Proposition 4.1. We may assume R > |λ|.
Then similar to (4.7) and (4.8) we can write
Γk∩D(0,R)
dµk(x)
x− λ =
Γ̃k,R
w′k(x)
wk(x)(x− λ)
where Γ̃k,R has the same meaning as in the proof of Proposition 4.1, see also
Figure 1. As in the proof of Proposition 4.1 we deform to an integral over
∂D(0, R), but now we have to take into account that the integrand has a
pole at x = λ with residue w′k(λ)/wk(λ). Therefore, by Cauchy’s theorem
Γk∩D(0,R)
dµk(x)
x− λ =
w′k(λ)
wk(λ)
∂D(0,R)
w′k(x)
wk(x)(x− λ)
dx. (4.12)
Letting R → ∞ and using Proposition 3.7 gives (4.9).
16 MAURICE DUITS AND ARNO B.J. KUIJLAARS
Next we integrate (4.9) over a Jordan curve J in C \ Γk from λ1 to λ2.
x− λ dµk(x) dλ = −
∫ ∫ λ2
x− λ dλ dµk(x)
(log |λ1 − x| − log |λ2 − x|+ i∆J [arg(λ− x)]) dµk(x), (4.13)
where ∆J [arg(λ − x)] denotes the change in argument of λ − x as when λ
varies over J from λ1 to λ2. By (4.9) the integral (4.13) is equal to
w′k(λ)
wk(λ)
dλ = log |wk(λ2)| − log |wk(λ1)|+ i∆J [argwk(λ)]. (4.14)
Equating the real parts of (4.13) and (4.14) we get
(log |λ1 − x| − log |λ2 − x|) dµk(x) = − log |wk(λ1)|+ log |wk(λ2)|.
(4.15)
Since λ1 and λ2 can be taken arbitrarily in a connected component of C\Γk,
we find that there exists a constant αk ∈ R (which a priori could depend on
the connected component) such that
log |λ− x| dµk(x) = − log |wk(λ)|+ αk, (4.16)
for all λ in a connected component of C \ Γk. By continuity the equation
(4.16) extends to the closure of the connected component, which shows that
the same constant αk is valid for all connected components. Thus (4.16)
holds for all λ ∈ C.
The exact value of αk can then be determined by expanding (4.16) for
large λ. Suppose for example that k < 0. Then by (3.1) and (3.3)
|wk(λ)| =
|zj(λ)| = |a−q|(q+k)/q|λ|−(q+k)/q
1 +O(λ−1/q)
as λ → ∞. Thus
− log |wk(λ)| =
q + k
log |λ| − q + k
log |a−q|+O(λ−1/q). (4.17)
Since
log |λ− x| dµk(x) = log |λ|µk(Γk) + o(1) =
q + k
log |λ|+ o(1), (4.18)
as λ → ∞, the value (4.11) for αk follows from (4.16), (4.17), and (4.18). The
argument for k > 0 is similar. This completes the proof of the proposition.
To prove part (c) of Theorem 2.3 we also need the following lemma.
EIGENVALUES OF BANDED TOEPLITZ MATRICES 17
Lemma 4.3. Let ~ν1 = (ν1,−q+1 . . . , ν1,p−1) and ~ν2 = (ν2,−q+1 . . . , ν2,p−1) be
two admissible vectors of measures. Then J(~ν1 − ~ν2) is well defined and
J(~ν1 − ~ν2) ≥ 0, (4.19)
with equality if and only if ~ν1 = ~ν2.
Proof. Since both ~ν1 and ~ν2 have finite energy, we find that J(~ν1 − ~ν2) is
well defined. According to the alternative representation (2.12), we have
J(~ν1 − ~ν2) =
I(ν1,0 − ν2,0)
k(k + 1)I
ν1,−q+k
− ν2,−q+k
− ν1,−q+k+1
k + 1
ν2,−q+k+1
k + 1
k(k + 1)I
ν1,p−k
ν2,p−k
ν1,p−k−1
k + 1
ν2,p−k−1
k + 1
(4.20)
Using (2.6) and (2.8), we see that all terms in (4.20) are non-negative and
therefore (4.19) holds.
Suppose now that J(~ν1 − ~ν2) = 0. Then all terms in the right-hand side
of (4.20) are zero, so that
ν1,0 = ν2,0, (4.21)
ν1,−q+k
ν2,−q+k+1
k + 1
ν1,−q+k+1
k + 1
ν2,−q+k
, for k = 1, . . . , q − 1,
(4.22)
ν1,p−k
ν2,p−k−1
k + 1
ν1,p−k−1
k + 1
ν2,p−k
, for k = 1, . . . , p− 1.
(4.23)
Using (4.21) in (4.22) with k = q − 1, we find ν1,−1 = ν2,−1. Proceeding
inductively we then obtain from (4.22) that ν1,k = ν2,k for all k = −q +
1, . . . , 0. Similarly, from (4.21) and (4.23) it follows that ν1,k = ν2,k for
k = 0, . . . , p− 1, so that ~ν1 = ~ν2 as claimed. �
Now we are ready for the proof of Theorem 2.3.
Proof of Theorem 2.3. (a) In view of Proposition 4.1 it only remains to
show that µk ∈ Me for every k = −q + 1, . . . , p − 1. The decay estimate
(4.5) implies that
log(1 + |λ|) dµk(λ) < ∞.
The fact that I(µk) < +∞ follows from (4.10). Indeed,
I(µk) = −
log |λ− x|dµk(x)dµk(λ) =
(log |wk(λ)| − αk)dµk(λ)
18 MAURICE DUITS AND ARNO B.J. KUIJLAARS
and this is finite since µk is a finite measure on Γk with a density that decays
as in (4.5) and log |wk(λ)| is continuous on Γk and grows only as a constant
times log |λ| as λ → ∞. Thus ~µ is admissible and part (a) is proved.
(b) According to (4.10) we have
log |λ− x| dµk(x)−
log |λ− x| dµk+1(λ)−
log |λ− x| dµk−1(λ)
= −2 log |wk(λ)|+ 2αk + log |wk+1(λ)| − αk+1 + log |wk−1(λ)| − αk−1
= log
wk+1(λ)wk−1(λ)
wk(λ)
+ 2αk − αk+1 − αk−1
= log
zq+k+1(λ)
zq+k(λ)
+ 2αk − αk+1 − αk−1. (4.24)
Since |zq+k(λ)| = |zq+k+1(λ)| for λ ∈ Γk, we see from (4.24) that (2.10)
holds with constant
lk = 2αk − αk−1 + αk+1. (4.25)
Note that for k = −q + 1 and k = p − 1, we are using the convention that
µ−q = µp = 0, and we also have put α−q = αp = 0. This proves part (b).
(c) Let ~ν = (ν−q+1, . . . , νp−1) be any admissible vector of measures. From
the representation (2.13) we get
J(~ν) = J(~µ+ ~ν − ~µ)
= J(~µ) + J(~ν − ~µ) + 2
j,k=−q+1
AjkI(µj, νk − µk). (4.26)
Using (2.14), we find from (4.26)
J(~ν) = J(~µ) + J(~ν − ~µ) +
k=−q+1
I(2µk − µk−1 − µk+1, νk − µk) (4.27)
For each k = −q + 1, . . . , p − 1, we have
I(2µk − µk−1 − µk+1, νk − µk)
log |λ− x| d(2µk − µk−1 − µk+1)(x)
d(νk − µk)(λ) (4.28)
By (2.10) the inner integral in the right-hand side of (4.28) is constant for
λ ∈ Γk. Since νk and µk are finite measures on Γk with νk(Γk) = µk(Γk),
we find from (4.28) that
I(2µk − µk−1 − µk+1, νk − µk) = 0, for k = −q + 1, . . . , p− 1.
Then (4.27) shows that J(~ν) = J(~µ)+J(~ν−~µ), which by Lemma 4.3 implies
that J(~ν) ≥ J(~µ) and equality holds if and only if ~ν = ~µ. This completes
the proof of Theorem 2.3. �
EIGENVALUES OF BANDED TOEPLITZ MATRICES 19
5. Proofs of Proposition 2.5 and Theorem 2.6
5.1. Proof of Proposition 2.5. We will now prove Proposition 2.5, which
follows by a combinatorial argument.
Proof of Proposition 2.5. We prove (2.18) and (2.19) for k > 0. The case
k < 0 is similar. Let us first expand the determinant in the definition of
Pk,n(λ) = detTn(z
−k(a− λ)) =
(a− λ)j−π(j)+k. (5.1)
Here Sn denotes the set of all permutation on {1, . . . , n}. By the band struc-
ture of Tn(z
−k(a − λ)) it follows that we only have non-zero contributions
from permutations π that satisfy
k − p ≤ π(j) − j ≤ q + k, for all j = 1, . . . , n. (5.2)
Define for π ∈ Sn,
Nπ = {j | π(j) = j + k}. (5.3)
and denote the number of elements of Nπ by |Nπ|. For each π ∈ Sn we have
j=1(a−λ)j−π(j)+k is a polynomial in λ of degree at most |Nπ|. So by
(5.1)
dk,n = degPk,n ≤ max
|Nπ| (5.4)
where we maximize over permutations π ∈ Sn satisfying (5.2).
Let π ∈ Sn satisfying (5.2). We prove (2.18) by giving an upper bound
for |Nπ|. Since
j=1(π(j) − j) = 0 we obtain
(π(j) − j)+ =
(j − π(j))+, (5.5)
where (·)+ is defined as (a)+ = max(0, a) for a ∈ R. Each j ∈ Nπ gives a
contribution k to the left-hand side of (5.5). Therefore the left-hand side is
at least k|Nπ|. By (5.2) we have that each term in the right hand side is
at most p− k. Moreover, there are at most n− |Nπ| non-zero terms in this
sum. Combining this with (5.5) leads to
k|Nπ| ≤
(π(j) − j)+ =
(j − π(j))+ ≤ (n− |Nπ|)(p − k). (5.6)
Hence, if π is a permutation satisfying (5.2)
|Nπ| ≤
n(p− k)
. (5.7)
Now (2.18) follows by combining (5.7) and (5.4).
To prove (2.19), we assume that n ≡ 0 mod p. We claim that there exists
a unique π such that equality holds in (5.7). Then equality holds in both
20 MAURICE DUITS AND ARNO B.J. KUIJLAARS
inequalities of (5.6) and the above arguments show that this can only happen
π(j) = j + k, or π(j) = j − p+ k, (5.8)
for every j = 1, . . . , n. We claim that there exists a unique such permutation,
namely
π(j) =
j + k, if j ≡ 1, . . . , (p − k) mod p,
j − p+ k, if j ≡ (p− k + 1), . . . , p mod p.
(5.9)
To see this let π be a permutation satisfying (5.8). The numbers 1, . . . , p−
k can not satisfy π(j) = j−p+k and thus satisfy π(j) = j+k. On the other
hand, the numbers 1, . . . , k can not be the image of numbers j satisfying
π(j) = j + k, and thus π(j) = j − p + k for j = p − k + 1, . . . , p. So (5.9)
holds for j = 1, . . . , p. This means in particular that the restriction of π to
{p + 1, . . . , n} is again a permutation, but now on {p + 1, . . . , n}. By the
same arguments we then find that (5.9) holds for j = p + 1, . . . , 2p, and so
on. The result is that (5.9) is indeed the only permutation that satisfies
(5.8).
Finally, a straightforward calculation shows that the coefficient of λ(p−k)n/p
j=1(a − λ)j−π(j)+k with π as in (5.9) is nonzero and given by (2.19).
This proves the proposition. �
5.2. Proof of Theorem 2.6. Before we start with the proof of Theorem
2.6 we first prove the following proposition concerning the asymptotics for
Pk,n for n → ∞.
Proposition 5.1. Let Mk = {q + k + 1, . . . , p+ q}. We have that
Pk,n(λ) = (wMk(λ))
nCMk(λ) (1 +O(exp(−cKn)) , n → ∞, (5.10)
uniformly on compact subsets K of C \ Γk. Here cK is a positive constant
depending on K.
Proof. First rewrite (2.22) as
Pk,n(λ) = (wMk(λ))
nCMk(λ) (1 +Rk,n(λ)) . (5.11)
with Rk,n defined by
Rk,n(λ) =
M 6=Mk
(wM (λ))
nCM (λ)
(wMk(λ))
nCMk(λ)
. (5.12)
Let K be a compact subset of C \ Γk. If K does not contain branch points
then there exists A,B > 0 such that
A < |CM (λ)| < B (5.13)
for all λ ∈ K and M . Moreover, we have
wM (λ)
wMk(λ)
zq+k(λ)
zq+k+1(λ)
≤ sup
zq+k(λ)
zq+k+1(λ)
< 1, (5.14)
EIGENVALUES OF BANDED TOEPLITZ MATRICES 21
for all λ ∈ K and M 6= Mk. Therefore one readily verifies from (5.11)
that there exist cK such that |Rk,n(λ)| ≤ exp(−cKn) for all λ ∈ K and n
large enough. This proves the statement in case K does not contain branch
points.
Suppose that K does contain branch points. Without loss of generality
we can assume that all branch points lie in the interior of K (otherwise we
replace K by a bigger compact set). The boundary ∂K of K is a com-
pact set with no branch points and therefore (5.10) holds for ∂K by the
above arguments. Since wMk and CMk are analytic in K, we find by (5.11)
that Rk,n is analytic in K. The maximum modulus principle for analytic
functions states that supz∈K |Rk,n(z)| = supz∈∂K |Rk,n(z)| and thereby we
obtain that (5.10) also holds for K with the same constant cK = c∂K . �
We now state two particular consequences of (5.10).
Corollary 5.2. Let k ∈ {−q + 1, . . . , p − 1}. For every compact set K ⊂
C \ Γk we have that µk,n(K) = 0 for n large enough.
Proof. Let K be a compact subset of C \ Γk. By (5.10) it follows that Pk,n
has no zeros in K for large n. Since nµk,n(K) equals the number of zeros of
Pk,n in K the corollary follows. �
Corollary 5.3. Let k ∈ {−q + 1, . . . , p− 1}. We have that
dµk,n(x)
x− λ =
dµk(x)
x− λ , (5.15)
uniformly on compact subsets of C \ Γk.
Proof. Let K be a compact subset of C \ Γk. Note that
dµk,n(x)
x− λ =
λi∈spk Tn(a)
λi − λ
P ′k,n(λ)
nPk,n(λ)
, (5.16)
for all λ ∈ K. With Mk and cK as in Proposition 5.1 we obtain from (5.10)
P ′k,n(λ)
nPk,n(λ)
w′Mk(λ)
wMk(λ)
+O(1/n), n → ∞, (5.17)
uniformly on K. Let us rewrite the right-hand side of (5.17). By expanding
both sides of zq(a(z)− λ) = ap
j=1(z − zj(λ)) and collecting the constant
terms we obtain
(−zj(λ)) =
. (5.18)
Since λ /∈ Γk, we can split this product in two parts, take the logarithmic
derivative and use (3.3) and (2.23) to obtain
z′j(λ)
zj(λ)
j=q+k+1
z′j(λ)
zj(λ)
w′k(λ)
wk(λ)
w′Mk(λ)
wMk(λ)
. (5.19)
22 MAURICE DUITS AND ARNO B.J. KUIJLAARS
Combining (5.16), (5.17) and (5.19), we obtain
dµk,n(x)
x− λ =
w′k(λ)
wk(λ)
(5.20)
uniformly on K. Then (5.15) follows from (5.20) and (4.9). �
Now we are ready for the proof of Theorem 2.6.
Proof of Theorem 2.6.
First we prove (2.21). By Proposition 2.5 and the fact that ~µ is admissible,
we get (see (2.8))
µk,n(C) =
degPk,n ≤ µk(C), (5.21)
for every n ∈ N.
Let C0(C) be the Banach space of continuous functions on C that vanish
at infinity. The dual space C0(C)
∗ of C0(C) is the space of regular complex
Borel measures on C. By (5.21) the sequence (µk,n)n∈N belongs to the ball in
C0(C)
∗ centered at the origin with radius µk(C), which is weak
∗ compact by
the Banach-Alaoglu theorem. Let µk,∞ be the limit of a weak
∗ convergent
subsequence of (µk,n)n∈N.
By weak∗ convergence and Corollary 5.2 we obtain that µk,∞ is supported
on Γk. Combining this with (5.15) and the weak
∗ convergence leads to
dµk(x)
x− λ =
dµk,∞(x)
x− λ , (5.22)
for every λ ∈ C \ Γk. The integrals in (5.22) are known in the literature as
the Cauchy transforms of the measures µk and µk,∞. The Cauchy transform
on Γk is an injective map that maps measures on Γk to functions that are
analytic in C \ Γk (one can find explicit inversion formulae, see for example
the arguments in [9, Theorem II.1.4] or the Stieltjes-Perron inversion formula
in the special case Γk ⊂ R). Thus it follows from (5.22) that µk,∞ = µk.
Therefore
µk,n = µk (5.23)
in the sense of weak∗ convergence in C0(C)
∗. Thus (2.21) holds if φ is a
continuous function that vanishes at infinity.
From (5.21) and (5.23) it also follows that
µk,n(C) = µk(C), (5.24)
Then the sequence (µk,n)n∈N is tight. That is, for every ε > 0 there exists a
compact K such that µk,n(C \K) < ε for every n ∈ N. By a standard ap-
proximation argument one can now show that (2.21) holds for every bounded
continuous function φ on C.
Having (2.21) and Proposition 5.1, we can prove (2.20) as in [1, Theo-
rem 11.17]. Indeed, the sets lim infn→∞ spk Tn(a) and lim supn→∞ spk Tn(a)
equal the support of µk, which is Γk. �
EIGENVALUES OF BANDED TOEPLITZ MATRICES 23
–2 –1.5 –1 –0.5 0.5 1 1.5
lambda
–2 –1.5 –1 –0.5 0.5 1 1.5
lambda
Figure 2. Illustration for Example 1: The densities of the
measures µ0 (left) and µ1 (right) for a =
4(z+1)3
6. Examples
6.1. Example 1. As a first example consider the symbol a defined by
a(z) =
4(z + 1)3
. (6.1)
In this case we have p = 2 and q = 1. So we obtain two contours Γ0 and
Γ1 with two associated measures µ0 and µ1. This example appeared in [3],
in which the authors gave explicit expressions for Γ0 and µ0. The following
proposition also contains expressions for Γ1 and µ1. In what follows we take
the principal branches for all fractional powers.
Proposition 6.1. With a as in (6.1), we have that Γ0 = [0, 1] and
dµ0(λ) =
dλ. (6.2)
Moreover, Γ1 = (−∞, 0] and
dµ1(λ) =
)1/3 −
1− λ− 1
(−λ)2/3
dλ. (6.3)
Proof. A straightforward calculation shows that λ = 0 and λ = 1 are the
branch points.
Let λ ∈ Γ0 ∪ Γ1 and assume that λ is not a branch point. There exist
y1, y2 ∈ C such that y1 6= y2, |y1| = |y2| and a(y1) = a(y2) = λ. Then it
follows from (6.1) that |y1+1| = |y2+1|. Therefore y1 and y2 are intersection
points of a circle centered at −1 and a circle centered at the origin. Since
y1 6= y2, this means that y1 = y2 and therefore λ = a(y1) = a(y2) = a(y1) =
λ, so that λ ∈ R. A further investigation shows that a(z)−λ has 3 different
real zeros if λ > 1. If λ < 1 and λ 6= 0 then a(z) − λ has precisely 1 real
zero and 2 conjugate complex zeros. Therefore, Γ0 ∪ Γ1 = (−∞, 1].
24 MAURICE DUITS AND ARNO B.J. KUIJLAARS
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
k = 0
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
k = 1
Figure 3. Illustration for Example 1: The spectrum
spT50(a) (top) and the generalized spectrum sp1T50(a) (bot-
tom), for the symbol a =
4(z+1)3
Now we will show that Γ0 = [0, 1] and Γ1 = (−∞, 0]. By Cardano’s
formula the solutions of the algebraic equation a(z) = λ are given by
zj(λ) = −1−
3λ1/3
1 + (1− λ)1/2
+ ω−j
1− (1− λ)1/2
(6.4)
for λ ∈ [0, 1] and
zj(λ) = −1+
3(−λ)1/3
1 + (1− λ)1/2
− ω−j−2
(1− λ)1/2 − 1
(6.5)
for λ ∈ (−∞, 0]. Here ω = e2πi/3. One can check that |z1(λ)| = |z2(λ)| <
|z3(λ)| for λ ∈ (0, 1] and |z1(λ)| < |z2(λ)| = |z3(λ)| for λ ∈ (−∞, 0). More-
over, for λ = 0 we have z1(0) = z2(0) = z3(0) = −1. Therefore Γ0 = [0, 1]
and Γ1 = (−∞, 0].
The density (6.2) was already given in [3] and (6.3) follows in a similar
way. �
In Figure 2 we plotted the densities of µ0 and µ1. Note that, due to the
interaction between µ0 and µ1 in the energy functional, there is more mass
of µ0 near 0 than near 1. We also see that the singularities of the densities
for µ0 and µ1 are of order O(|λ|−2/3) for λ → 0, whereas the typical nature
of a singularity in each of the measures is a square root singularity. The
stronger singularity is due to the fact that a(z) − λ has a triple root for
λ = 0.
EIGENVALUES OF BANDED TOEPLITZ MATRICES 25
–4 –2 2 4
lambda
–4 –2 2 4
lambda
Figure 4. Illustration for Example 2: The densities of the
measures µ0 (left) and µ1 = µ−1 (right) for a(z) = z
2 + z +
z−1 + z−2.
In Figure 3 we plotted the eigenvalues and generalized eigenvalues for
n = 50. It is known that the eigenvalues are simple and positive [3, §2.3],
which we also see in Figure 3.
6.2. Example 2. For the symbol a defined by
a(z) = z2 + z + z−1 + z−2. (6.6)
we have p = q = 2. From the symmetry a(1/z) = a(z) it follows that
Γ−1 = Γ1 and µ−1 = µ1.
The interesting feature of this example is that the contours Γ0 and Γ±1
overlap. To be precise, the interval (−9/4, 0) is contained in all three con-
tours Γ−1,Γ0 and Γ1. This can be most easily seen by investigating the
image of the unit circle under a. Consider
a(eit) = 2 cos 2t+ 2cos t, for t ∈ [0, 2π). (6.7)
A straightforward analysis shows that for every λ ∈ (−9/4, 0), the equation
a(eit) = λ has four different solutions for t in [0, 2π). This means that the
four solutions of the equation a(z) = λ are on the unit circle, and so in
particular have the same absolute value.
The equation a(z) − λ = 0 can be explicitly solved by introducing the
variable y = z + 1/z. In exactly the same way as in the previous example
one can obtain the limiting measures. We will not give the explicit formulas,
but only plot the densities in Figure 4. The branch points are λ = −9/4,
λ = 0 and λ = 4. The contours are given by
Γ0 = [−9/4, 4], Γ−1 = Γ1 = (−∞, 0]. (6.8)
The densities have singularities at the branch points in the interior of their
supports. The singularities are only felt at one side of the branch points.
Consider first µ0, whose density has a singularity at 0. However the limiting
value when 0 is approached from the positive real axis is finite. The change
in behavior of µ0 has to do with the fact that z1 is analytic on (0, 4) but not
26 MAURICE DUITS AND ARNO B.J. KUIJLAARS
on (−9/4, 0). Therefore we find by (1.12) that
dµ0(λ) =
z1+(λ)
z2+(λ)
z1−(λ)
z2−(λ)
dλ (6.9)
on (−9/4, 0), and
dµ0(λ) =
z2+(λ)
z2−(λ)
dλ (6.10)
on (0, 4).
For µ−1 = µ1 a similar phenomenon happens at λ = −9/4. This is a
consequence of the fact that z1 has an analytic continuation into z2 when
we cross (−∞,−9/4), but it has an analytic continuation into z4 when we
cross (−9/4, 0).
6.3. Example 3. As a final example, consider the symbol
a(z) = zp + z−q, (6.11)
with p, q ≥ 1 and gcd(p, q) = 1. This example appeared in [10], where the
authors mentioned that Γ0 is given by the star
Γ0 = {rωj | j = 1, . . . , p+ q, 0 ≤ r ≤ R} (6.12)
with ω = e2πi/(p+q) and R = (p + q)p−p/(p+q)q−q/(p+q). The other contours
also have a star shape, namely
Γk = {(−1)krωj | j = 1, . . . , p+ q, 0 ≤ r < ∞} (6.13)
for k 6= 0. Note that the star Γk for k 6= 0 is unbounded.
In Figure 5 we plotted the eigenvalues and the generalized eigenvalues
for p = 2, q = 3 and n = 50. All the (generalized) eigenvalues appear
to lie exactly on the contours. In the special case p = 1 it is known that
the eigenvalues of Tn(a) lie indeed precisely on the star (6.12) and are all
simple (possibly except for 0) [4, Theorem 3.2], see also [6] for a connection
to Chebyshev-type quadrature.
6.4. Numerical stability. In Figure 3 and Figure 5 the eigenvalues and
the generalized eigenvalues of T50(a) were computed numerically. To control
the stability of the numerical computation of the eigenvalues one needs to
analyze the pseudo-spectrum. For banded Toeplitz matrices the pseudo-
spectrum is well understood [12, Th. 7.2]. To this date, a similar analysis
of the pseudo-spectrum for the matrix pencil (Tn(z
−ka), Tn(z
−k)) has not
been carried out. See [12, §X.45] for some remarks on the pseudo-spectrum
for the generalized eigenvalue problem.
EIGENVALUES OF BANDED TOEPLITZ MATRICES 27
−5 0 5
k = −2
−5 0 5
k = −1
−5 0 5
k = 0
−5 0 5
k = 1
Figure 5. Illustration for Example 3: The contours Γk and
the eigenvalues and generalized eigenvalues for T50(a) for the
symbol a = z2 + z−3.
References
1. A. Böttcher and S. M. Grudsky, Spectral Properties of Banded Toeplitz Matrices,
SIAM, Philadelphia, PA, 2005.
2. A. Böttcher and S. M. Grudsky, Can spectral values sets of Toeplitz band matrices
jump?, Linear Algebra Appl., 351-352 (2002), pp. 99-116.
3. E. Coussement, J. Coussement and W. Van Assche, Asymptotic zero distribution for
a class of multiple orthogonal polynomials, Trans. Amer. Math. Soc., (to appear)
4. M. Eiermann and R. Varga, Zeros and local extreme points of Faber polynomials
associated with hypocycloidal domains, Electron. Trans. Numer. Anal., 1 (1993), pp.
49-71.
5. I. I. Hirschman, Jr., The spectra of certain Toeplitz matrices, Illinois J. Math., 11
(1967), pp. 145-159.
6. A. Kuijlaars, Chebyshev quadrature for measures with a strong singularity, J. Comput.
Appl. Math., 65 (1995), pp. 207-214.
7. E. Nikishin and V. Sorokin, Rational Approximations and Orthogonality, Translations
of Mathematical Monographs 92, American Mathematical Society, Providence, RI,
(1991).
8. T. Ransford, Potential Theory in the Complex Plane, London Mathematical Society
Student Texts 28, Cambridge University Press, Cambridge, 1995.
9. E.B. Saff and V. Totik, Logartihmic Potentials with External Fields, Grundlehren der
Mathematischen Wissenschaften 316, Springer-Verlag, Berlin, 1997.
10. P. Schmidt and F. Spitzer, The Toeplitz matrices of an arbitrary Laurent polynomial,
Math. Scand., 8 (1960), pp. 15-38.
28 MAURICE DUITS AND ARNO B.J. KUIJLAARS
11. P. Simeonov, A weigthed energy problem for a class of admissible weights, Houston
J. Math., 31 (2005), pp. 1245-1260.
12. L.N. Trefethen and M. Embree, Spectra and Pseudospectra, Princeton University
Press, Princeton, NJ, 2005.
13. J.L. Ullman, A problem of Schmidt and Spitzer, Bull. Amer. Math. Soc., 73 (1967),
pp. 883-885.
14. H. Widom, On the eigenvalues of certain Hermitean operators, Trans. Amer. Math.
Soc., 88 (1958), pp. 491-522.
	1. Introduction
	2. Statement of results
	2.1. The energy functional
	2.2. The measures k as limiting measures of generalized eigenvalues
	2.3. Overview of the rest of the paper
	3. Preliminaries
	3.1. The structure of the curves k
	3.2. The Riemann surface
	3.3. The functions wk()
	4. Proof of Theorem 2.3
	5. Proofs of Proposition 2.5 and Theorem 2.6
	5.1. Proof of Proposition 2.5
	5.2. Proof of Theorem 2.6
	6. Examples
	6.1. Example 1
	6.2. Example 2
	6.3. Example 3
	6.4. Numerical stability
	References
