SUM RULES AND THE SZEGŐ CONDITION FOR ORTHOGONAL POLYNOMIALS ON THE REAL LINE

BARRY SIMON^{1} AND ANDREJ ZLATOŠ

Abstract. We study the Case sum rules, especially C_{0}, for general Jacobi matrices. We establish situations where the sum rule is valid. Applications include an extension of Shohat’s theorem to cases with an infinite point spectrum and a proof that if lim n(a_{n} − 1) = α and lim nb_{n} = β exist and 2α < |β|, then the Szegő condition fails.

1. Introduction

This paper discusses the relation among three objects well known to be in one-one correspondence: nontrivial (i.e., not supported on a finite set) probability measures, ν, of bounded support in ℝ; orthogonal polynomials associated to geometrically bounded moments; and bounded Jacobi matrices. One goes from measure to polynomials via the Gram-Schmidt procedure, from polynomials to Jacobi matrices by the three-term recurrence relation, and from Jacobi matrices to measures by the spectral theorem.

We will use J to denote the Jacobi matrix (a_{n} > 0)

J = (\begin{matrix}b_{1} & a_{1} & 0 & … \\ a_{1} & b_{2} & a_{2} & … \\ 0 & a_{2} & b_{3} & … \\ … & … & … & … \end{matrix})  (1.1)

ν will normally denote the spectral measure of the vector δ_{1} ∈ ℓ^{2}(ℤ^{+}) and P_{n}(x) the orthonormal polynomials.

We are interested in J’s close to the free Jacobi matrix, J_{0}, with b_{n} = 0, a_{n} = 1, and dν_{0}(E) = (2π)^{−1}χ_{[−2,2]}\sqrt{4 − E^{2}} dE. Most often, we will suppose J − J_{0} is compact. That means σ_{ess}(J) = [−2, 2] and J has only eigenvalues outside [−2, 2], of multiplicity one denoted E_{j}^{±} with E_{1}^{+} > E_{2}^{+} > ⋯ > 2 and E_{1}^{−} < E_{2}^{−} < ⋯ < −2.

Date: March 26, 2003.

2000 Mathematics Subject Classification. Primary: 47B36; Secondary: 42C05.

1 Supported in part by NSF grant DMS-9707661.

One of the main objects of study here is the Szegő integral

Z(J) = \frac{1}{2π} ∫_{−2}^{2} ln(\frac{\sqrt{4 − E^{2}}}{2πdν_{ac}/dE} )\frac{dE}{\sqrt{4 − E^{2}}}  (1.2)

The Szegő integral is often taken in the literature as

(2π)^{−1} ∫_{−2}^{2} ln(\frac{dν_{ac}}{dE} )\frac{dE}{\sqrt{4 − E^{2}}}

which differs from Z(J) by a constant and a critical minus sign (so the common condition that the Szegő integral not be −∞ becomes Z(J) < ∞ in our normalization). There is an enormous literature discussing when Z(J) < ∞ holds (see, e.g., [1, 2, 7, 9, 13, 14, 16, 17, 22, 24]). It can be shown by Jensen’s inequality that Z(J) ≥ −\frac{1}{2} ln(2) so the integral can only diverge to +∞.

We will focus here on various sum rules that are valid. One of our main results is the following:

Theorem 1. Suppose

A_{0}(J) = lim_{N→∞} (− ∑_{n=1}^{N} ln(a_{n}))  (1.3)

exists (although it may be +∞ or −∞). Consider the additional quantities Z(J) given by (1.2) and

E_{0}(J) = ∑_{±} ∑_{j} ln[\frac{1}{2}(|E_{j}^{±}| + \sqrt{(E_{j}^{±})^{2} − 4} )]  (1.4)

If any two of the three quantities A_{0}(J), E_{0}(J), and Z(J) are finite, then all three are, and

Z(J) = A_{0}(J) + E_{0}(J)  (1.5)

Remarks. 1. It is not hard to see that E_{0}(J) < ∞ if and only if

∑_{±} ∑_{j} \sqrt{(E_{j}^{±})^{2} − 4} < ∞  (1.6)

2. The full theorem (Theorem 4.1) does not require the limit (1.3) to exist, but is more complicated to state in that case.

3. If the three quantities are finite, many additional sum rules hold.

4. This is what Killip-Simon [11] call the C_{0} sum rule.

5. Peherstorfer-Yuditskii [17] (see their remark after Lemma 2.1) prove that if Z(J) < ∞, E_{0}(J) = ∞, then the limit in (1.3) is also infinite.

Theorem 1 is an analog for the real line of a seventy-year old theorem for orthogonal polynomials on the unit circle:

\frac{1}{2π} ∫_{0}^{2π} ln(\frac{dν_{ac}}{dθ} ) dθ = ∑_{n=0}^{∞} ln(1 − |α_{j}|^{2})  (1.7)

where {α_{j}}_{j=1}^{∞} are the Verblunsky coefficients (also called reflection, Geronimus, Schur, or Szegő coefficients) of ν. This result was first proven by Verblunsky [27] in 1935, although it is closely related to Szegő’s 1920 paper [24].

For J’s with J −J_{0} finite rank (and perhaps even with ∑_{n=1}^{∞} n(|a_{n} − 1| + |b_{n}|) < ∞), the sum rule (1.5) is due to Case [2]. Recently, Killip-Simon [11] showed how to exploit these sum rules as a spectral tool (motivated in turn by work on Schrödinger operators by Deift-Killip [5] and Denissov [6]). In particular, Killip-Simon emphasized the importance in proving sum rules on as large a class of J’s as possible.

One application we will make of Theorem 1 and related ideas is to prove the following (≡ Theorem 5.2):

Theorem 2. Suppose σ_{ess}(J) ⊂ [−2, 2] and (1.6) holds. Then Z(J) < ∞ if and only if

lim inf_{N}(− ∑_{n=1}^{N} ln(a_{n})) < ∞  (1.8)

Moreover, if these conditions hold, then

(i) The limit A_{0}(J) in (1.3) exists and is finite. (ii) lim_{N→∞} ∑_{n=1}^{N} b_{n} exists and is finite. (iii)

∑_{n=1}^{∞}(a_{n} − 1)^{2} + ∑_{n=1}^{∞} b_{n}^{2} < ∞  (1.9)

Results of this genre when it is assumed that σ(J) = [−2, 2] go back to Shohat [22] with important contributions by Nevai [14]. The precise form is from Killip-Simon [11]. Nikishin [16] showed how to extend this to Jacobi matrices with finitely many eigenvalues. Peherstorfer-Yuditskii [17] proved Z(J) < ∞ implies (i) under the condition E_{0}(J) < ∞, allowing an infinity of eigenvalues for the first time. Our result cannot extend to situations with E_{0}(J) = ∞ since Theorem 1 says if (i) holds and Z(J) < ∞, then E_{0}(J) < ∞.

We will highlight one other result we will prove later (Corollary 6.3).

Theorem 3. Let a_{n}, b_{n} be Jacobi matrix parameters so that

lim_{n→∞} n(a_{n} − 1) = α  lim_{n→∞} nb_{n} = β  (1.10)

exist and are finite. Suppose that

|β| > 2α  (1.11)

Then Z(J) = ∞.

Remark. In particular, if α < 0, (1.11) always holds. (1.11) describes three-quarters of the (2α, β) plane.

In Section 6, we will discuss the background for this result, and describe results of Zlatoš [28] that show if |β| ≤ 2α and one has additional information on the approach to the limit (1.10), then Z(J) < ∞. Thus Theorem 3 captures the precise region where one has (1.10) and one can hope to prove Z(J) = ∞.

Theorem 3 will actually follow from a more general result (see Theorem 4.4, 6.1, and 6.2).

Theorem 4. Suppose (1.9) holds and that either lim sup(− ∑_{j=1}^{n}(a_{j} − 1+ \frac{1}{2}b_{j})) = ∞ or lim sup(− ∑_{j=1}^{n}(a_{j} −1− \frac{1}{2}b_{j})) = ∞. Then Z(J) = ∞.

The main technique in this paper exploits the m-function, the Borel transform of the measure, ν:

m_{ν}(E) = ∫ \frac{dν(x)}{x − E}  (1.12)

Since ν is supported on [−2, 2] plus the set of points {E_{j}^{±}}, we can write

m_{ν}(E) = ∑_{±} ∑_{j} \frac{ν(\{E_{j}^{±}\})}{E_{j}^{±} − E} + ∫_{−2}^{2} \frac{dν(x)}{x − E}  (1.13)

It is useful to transfer everything to the unit circle, using the fact that z ↦ E = z + z^{−1} maps 𝔻 = {z | |z| < 1} onto the cut plane ℂ\[−2, 2]. Thus we can define for |z| < 1

M(z) = −m_{ν}(z + z^{−1})  (1.14)

The minus sign is picked so Im M(z) > 0 if Im z > 0. We use M(z; J) when we want to make the J-dependence explicit. The function M is meromorphic in 𝔻 with poles at (β_{j}^{±})^{−1} such that

E_{j}^{±} = β_{j}^{±} + (β_{j}^{±})^{−1}  (1.15)

with |β_{j}^{±}| > 1. We sometimes drop the explicit ± symbol and count the β_{j}’s in one set.

We define a signed measure dμ^{#} on [0, 2π] by Im M(re^{iθ})dθ → dμ^{#}(θ) weakly as r ↑ 1. Hence μ^{#} is positive on (0, π) and negative on (π, 2π). Actually, M(z) = \overline{M(z̄)} implies dμ^{#}(π + θ) = −dμ^{#}(π − θ), so we let

μ ≡ μ^{#} ↾ [0, π]  (1.16)

By general principles [21],

Im M(e^{iθ}) ≡ lim_{r↑1} Im M(re^{iθ}) = \frac{dμ_{ac}}{dθ} (θ) = π \frac{dν_{ac}}{dx} (2 cos θ) (1.17)

for a.e. θ ∈ (0, π). One actually has that if

dμ̃(θ) ≡ 2 sin θ dμ^{#}(θ) = 2| sin θ| d|μ^{#}|(θ)  (1.18)

then for any interval I ⊂ (0, π) ∪ (π, 2π)

μ̃(I) = πν(2 cos I)  (1.19)

The reason why we exclude 0, π, 2π is possible mass points of ν at ±2. These do not translate to μ^{#} because Im M(±r) = 0 (notice that r + r^{−1} → ±2 as r → ±1 is not a nontangential limit).

By (1.19), μ̃([0, 2π]) ≤ 2π, so μ̃ is a finite measure. This need not be true for μ^{#}, as can be seen from (1.18) and (1.19). Indeed, these formulae show that μ^{#} is finite if and only if

∫_{−2}^{2} χ_{(−2,2)}(x)\frac{dν(x)}{\sqrt{4 − x^{2}}} < ∞  (1.20)

where χ_{(−2,2)} ensures that possible mass points at ±2 do not enter here.

We would now like to write (1.13) (or rather its imaginary part) in terms of M. The pole terms (including those at ±2, if they are present) translate directly, and so

Im M(z) = Im ∑_{±} \frac{μ(\{±1\})}{z + z^{−1} − 2} + Im ∑_{j} \frac{μ(\{β_{j}^{−1}\})}{z + z^{−1} − (β_{j} + β_{j}^{−1})} + K(z) (1.21)

where

K(z) ≡ Im ∫_{−2}^{2} χ_{(−2,2)}(x)\frac{dν(x)}{z + z^{−1} − x}

and we use μ({β_{j}^{−1}}) for the weights ν({E_{j}}) (and μ({±1}) for ν({±2})). We note that since μ({β_{j}^{−1}}) are point masses of a probability measure, we have

∑_{j} μ({β_{j}^{−1}}) ≤ 1

with the μ({±1}) terms included in the sum as β_{∞}^{±} ≡ ±1.

We will rewrite K(z) in terms of the Poisson kernel. Assume first that (1.20) holds, that is, μ^{#} is a finite measure. Notice that K(re^{iθ}) is a harmonic function in 𝔻. Moreover, since the imaginary parts of the pole terms go to 0 as r ↑ 1,

K(re^{iθ}) − Im M(re^{iθ}) → 0

as r ↑ 1, uniformly for θ in compact subsets of (0, π) ∪ (π, 2π). This means that K(re^{iθ}) dθ → dμ^{#}(θ) weakly as measures on (0, π)∪(π, 2π). Clearly, K(±1) = 0 and μ^{#}({0, π}) = 0, and so K(re^{iθ}) dθ → dμ^{#}(θ) as measures on [0, 2π]. Since K(z) is harmonic in 𝔻, it follows (see, e.g., [21]) that

K(z) = ∫_{0}^{2π} P (z, e^{iφ}) \frac{dμ^{#}(φ)}{2π}  (1.22)

where P (z, w), with |z| < 1, |w| = 1, is the Poisson kernel

P (z, w) ≡ \frac{1 − |z|^{2}}{|z − w|^{2}}  (1.23)

or

P_{r}(θ, φ) ≡ P (re^{iθ}, e^{iφ}) = \frac{1 − r^{2}}{1 + r^{2} − 2r cos(θ − φ)}  (1.24)

Then using the fact that μ^{#} is odd under reflection, we can rewrite (1.21) as

Im M(re^{iθ}) = Im ∑_{j} \frac{μ(\{β_{j}^{−1}\})}{re^{iθ} + r^{−1}e^{−iθ} − (β_{j} + β_{j}^{−1})} + ∫_{0}^{π} D_{r}(θ, φ) \frac{dμ(φ)}{2π} (1.25)

where

D_{r}(θ, φ) = P_{r}(θ, φ) − P_{r}(θ, −φ)  (1.26)

This is because M(z) = \overline{M(z̄)}, so that μ = −μ^{#} ↾ [−π, 0].

As we shall see, it turns out that (1.25) holds even if (1.20) fails, although (1.22) is meaningless in that case. We only need to consider θ, φ ∈ [0, π]. Then obviously, D_{r}(θ, φ) ≥ 0, and (1.24) and (1.26) show

D_{r}(θ, φ) = \frac{(1 − r^{2})2r(cos(θ − φ) − cos(θ + φ))}{(1 + r^{2} − 2r cos(θ − φ))(1 + r^{2} − 2r cos(θ + φ))}

= P_{r}(θ, φ)\frac{4r sin θ sin φ}{1 + r^{2} − 2r cos(θ + φ)}

Notice that if θ, φ ∈ [0, π], then 2r cos(θ + φ) ≤ 2r| cos θ|. Since r < 1, we have

sin^{2}(θ) + 2| cos θ| = 1 − cos^{2}(θ) + 2| cos θ| ≤ 2 ≤ \frac{1 + r^{2}}{r}

which implies

\frac{r sin^{2}(θ)}{1 + r^{2} − 2r| cos θ|} ≤ 1

Hence

0 ≤ \frac{D_{r}(θ, φ)}{sin φ} ≤ \frac{4P_{r}(θ, φ)}{sin θ}  (1.27)

for θ, φ ∈ (0, π) and r < 1.

Using (1.19), the integral in (1.25) can now be estimated as

0 ≤ ∫_{0}^{π} \frac{D_{r}(θ, φ)}{sin φ} \frac{dμ̃(φ)}{4π} ≤ \frac{∥P_{r}∥_{∞}}{sin θ}

and so is finite (notice that if θ = 0, π, then D_{r}(θ, φ) ≡ 0). Moreover, if ν_{n} are probability measures which coincide with ν outside of [−2, −2 + \frac{1}{n}] ∪ [2 − \frac{1}{n}, 2], satisfy (1.20), and ν_{n} → ν weakly, then clearly for any θ ∈ [0, π] and r < 1,

M(re^{iθ}; ν_{n}) → M(re^{iθ}; ν)

and

∫_{0}^{π} D_{r}(θ, φ) dμ_{n}(φ) → ∫_{0}^{π} D_{r}(θ, φ) dμ(φ)

Since the sum in (1.25) is the same for ν_{n} and ν, and (1.25) holds for ν_{n}, it holds for ν.

Section 2, the technical core of the paper, proves some convergence results about integrals of ln[Im M(re^{iθ})]. It is precisely such integrals that arise in Section 3 where, following Killip-Simon [11], we use the well-known

−m(z; J)^{−1} = z − b_{1} + a_{1}^{2}m(z; J^{(1)})

where J^{(1)} is J with the top row and leftmost column removed. We will be able to prove sum rules that compare J and J^{(1)}. In Section 4, we will then list various sum rules, including Theorems 1 and 4. Section 5 proves Theorem 2 and Section 6 discusses Coulomb Jacobi matrices (J − J_{0} decays as n^{−1}) and Theorem 3 in particular.

It is a pleasure to thank Mourad Ismail, Rowan Killip, and Paul Nevai for useful discussions.

2. Continuity of Integrals of ln(Im M)

In this section, we will prove a general continuity result about boundary values of interest for M-functions of the type defined in (1.24). We will consider suitable weight functions, w(φ), on [0, π], of which the examples of most interest are w(φ) = sin^{k}(φ), k = 0 or 2. Our goal is to prove that

lim_{r↑1} ∫ ln[Im M(re^{iφ})] w(φ) dφ = ∫ ln[Im M(e^{iφ})] w(φ) dφ (2.1)

and that the convergence is in L^{1} if the integral on the right is finite. All integrals in this section are from 0 to π if not indicated otherwise.

We define

d(φ) ≡ min(φ, π − φ)  (2.2)

and we suppose that

0 ≤ w(φ) ≤ C_{1} d(φ)^{−1+α}  (2.3)

for some C_{1}, α > 0 and that w is C^{1} with

|w′(φ)w(φ)^{−1}| ≤ C_{2} d(φ)^{−β}  (2.4)

for C_{2}, β > 0. For weights of interest, one can take α = β = 1.

Remarks. 1. For the applications in mind, we are only interested in allowing “singularities” (i.e., w vanishing or going to infinity) at 0 or π, but all results hold with unchanged proofs if d(φ) ≡ min{|φ − φ_{j}|} for any finite set {φ_{j}}. For example, w(φ) = sin^{2}(mφ) as in [12] is fine.

2. Note that by (2.3), ∫_{0}^{π} w(φ) dφ < ∞.

The main technical result we will need is:

Theorem 2.1. Let M be a function with a representation of the form (1.25) and let w be a weight function obeying (2.3) and (2.4). Then (2.1) holds. Moreover, if

∫ ln[Im M(e^{iφ})]w(φ) dφ > −∞  (2.5)

(it is never +∞), then

lim_{r↑1} ∫ |ln[Im M(re^{iφ})] − ln[Im M(e^{iφ})]| w(φ) dφ = 0  (2.6)

Let ln_{±} be defined by

ln_{±}(y) = max(0, ± ln(y))

so

ln(y) = ln_{+}(y) − ln_{−}(y)
|ln(y)| = ln_{+}(y) + ln_{−}(y)

We will prove Theorem 2.1 by proving

Theorem 2.2. For any a > 0 and p < ∞, ln_{+}[Im(M(e^{iφ}))/a] ∈ L^{p}((0, π), w(φ)dφ), and

lim_{r↑1} ∫ |ln_{+} (\frac{Im M(re^{iφ})}{a}) − ln_{+}(\frac{Im M(e^{iφ})}{a})|^{p} w(φ) dφ = 0 (2.7)

Theorem 2.3. For any a > 0, we have

lim_{r↑1} ∫ ln_{−}(\frac{Im M(re^{iφ})}{a})w(φ) dφ = ∫ ln_{−}(\frac{Im M(e^{iφ})}{a})w(φ) dφ

(2.8)

Proof of Theorem 2.1 given Theorems 2.2 and 2.3. By Fatou’s lemma and the fact that for a.e. φ, Im M(re^{iφ}) → Im M(e^{iφ}), we have

lim inf_{r↑1} ∫ ln_{−}[Im M(re^{iφ})] w(φ) dφ ≥ ∫ ln_{−}[Im M(e^{iφ})] w(φ) dφ

(2.9)

Since Theorem 2.2 says that sup_{0<r≤1} ∫ ln_{+}[Im M(re^{iφ})]w(φ) dφ < ∞, it follows that if ∫ ln_{−}[Im M(e^{iφ})]w(φ) dφ = ∞, then (2.1) holds.

If (2.5) holds, then

lim_{a↓0} ∫ ln_{−}[\frac{Im M(e^{iφ})}{a}]w(φ) dφ = 0

since ln_{−}(y/a) is monotone decreasing to 0 as a decreases. Given ε, first find a so

∫ ln_{−}[\frac{Im M(e^{iφ})}{a}]w(φ) dφ < \frac{ε}{3}

and then, by (2.8), r_{1} < 1 so for r_{1} < r < 1,

∫ ln_{−}[\frac{Im M(re^{iφ})}{a}]w(φ) dφ < \frac{ε}{3}

By (2.7), find r_{2} < 1, so for r_{2} < r < 1,

∫ |ln_{+}[\frac{Im M(re^{iφ})}{a}] − ln_{+}[\frac{Im M(e^{iφ})}{a}]| w(φ) dφ < \frac{ε}{3}

Writing

|ln(α) − ln(β)| ≤ |ln_{+}(\frac{α}{a}) − ln_{+}(\frac{β}{a} )| + ln_{−}(\frac{α}{a}) + ln_{−}(\frac{β}{a} )

we see that if max(r_{1}, r_{2}) < r < 1, then

∫ |ln[Im M(re^{iφ})] − ln[Im M(e^{iφ})]| w(φ) dφ < ε

so (2.6) holds. □

We will prove Theorem 2.2 by using the dominated convergence theorem and standard maximal function techniques. We let the maximal function of the measure μ̃ defined in (1.18) be

μ̃^{∗}(x) = sup_{0<a<π} \frac{μ̃(x − a, x + a)}{2a}

The Hardy-Littlewood maximal inequality for measures (see Rudin [21]) says that

|\{x | μ̃^{∗}(x) > λ\}| ≤ \frac{3μ̃(0, π)}{λ}  (2.10)

Lemma 2.4. Let M satisfy (1.25), and let α be the sum of the weights of the poles (β_{j}^{±})^{−1}. Then for 0 < r < 1 and 0 ≤ θ ≤ π,

Im M(re^{iθ}) ≤ μ̃^{∗}(θ)[sin θ]^{−1} + αr^{−1}[sin θ]^{−2}  (2.11)

Proof. Since D_{r}(θ, φ) ≤ P_{r}(θ, φ) and P_{r} is a convolution operator with a positive even function of φ decreasing on [0, π] with ∫_{0}^{2π} P_{r}(φ) dφ/2π = 1, we have, by standard calculations, (1.18), (1.19), and (1.27), that

∫_{0}^{π} D_{r}(θ, φ) \frac{dμ(φ)}{2π} ≤ ∫_{0}^{π} \frac{4P_{r}(θ, φ)}{sin θ} \frac{dμ̃(φ)}{4π} ≤ \frac{μ̃^{∗}(θ)}{sin θ}

On the other hand, for |β| ≥ 1,

|\frac{1}{z + z^{−1} − β − β^{−1}} | = |\frac{z}{(z − β)(z − β^{−1})}| ≤ \frac{|z|}{|Im z|^{2}} = \frac{1}{r sin^{2} θ}

if z = re^{iθ}, so summing the pole term shows,

Im ∑_{j} \frac{μ(\{β_{j}^{−1}\})}{z + z^{−1} − β_{j} − β_{j}^{−1}} ≤ \frac{∑_{j} μ(\{β_{j}^{−1}\})}{r sin^{2}(θ)}

□

Proof of Theorem 2.2. Let

f_{1}(θ) = μ̃^{∗}(θ)[sin θ]^{−1} f_{2}(θ) = 2[sin(θ)]^{−2}

For a.e. θ, ln_{+}[(Im M(re^{iθ}))/a] → ln_{+}[(Im M(e^{iθ}))/a]. By (2.11) for all \frac{1}{2} < r < 1, ln_{+}[(Im M(re^{iθ}))/a] ≤ ln_{+}[(f_{1}(θ) + f_{2}(θ))/a]. Thus if we prove that for all p < ∞,

∫ |ln_{+}(\frac{f_{1}(φ) + f_{2}(φ)}{a})|^{p}w(φ) dφ < ∞

we obtain (2.7) by the dominated convergence theorem. Since

|ln_{+}(x)|^{p} ≤ C(p, q)|x|^{q}

for any p < ∞, q > 0, and suitable C(p, q), and

|x + y|^{q} ≤ 2^{q}|x|^{q} + 2^{q}|y|^{q}

it suffices to find some q > 0, so

∫ (|f_{1}(φ)|^{q} + |f_{2}(φ)|^{q}) w(φ) dφ < ∞

Since for v^{−1} + t^{−1} = 1,

∫ |f_{1}(φ)|^{q}w(φ) dφ ≤ (∫ |f_{1}(φ)|^{qv} dφ)^{1/v}(∫ |w(φ)|^{t} dφ)^{1/t} and w(φ) ∈ L^{t} for some t > 1 by (2.3), it suffices to find some s > 0 with

∫ (|f_{1}(φ)|^{s} + |f_{2}(φ)|^{s}) dφ < ∞  (2.12)

By (2.10) and Cauchy-Schwartz,

∫ |f_{1}(φ)|^{s} dφ ≤ ( ∫ |μ̃^{∗}(φ)|^{2s} dφ)^{1/2}( ∫ | sin φ|^{−2s} dφ)^{1/2} < ∞

and ∫ |f_{2}(φ)|^{s} dφ < ∞ whenever s < \frac{1}{2}. □

As a preliminary to the proof of Theorem 2.3, we need

Lemma 2.5. Let w obey (2.4). Let 0 < φ_{0} < π and let φ_{1}, φ_{2} ∈ [0, π] obey

(a)  d(φ_{1}) ≥ d(φ_{0}), d(φ_{2}) ≥ d(φ_{0})  (2.13)
(b)  |φ_{1} − φ_{2}| ≤ d(φ_{0})^{β}  (2.14)

Then for C_{3} = C_{2} exp(C_{2}),

|\frac{w(φ_{1})}{w(φ_{2})} − 1| ≤ C_{3}|φ_{1} − φ_{2}| d(φ_{0})^{−β}  (2.15)

Proof.

|\frac{w(φ_{1})}{w(φ_{2})} − 1| = |exp( ∫_{φ_{1}}^{φ_{2}} \frac{w′(η)}{w(η)} dη) − 1|
≤ |exp(C_{2}|φ_{2} − φ_{1}| d(φ_{0})^{−β}) − 1|  (2.16)

by (2.4) and (2.13). But |e^{x} − 1| ≤ e^{|x|}|x|, so by (2.14),

|\frac{w(φ_{1})}{w(φ_{2})} − 1| ≤ C_{2} exp(C_{2})|φ_{1} − φ_{2}| d(φ_{0})^{−β}

which is (2.15). □

We will also need the following pair of lemmas:

Lemma 2.6. Let 0 < η < θ < π − η and

N_{r}(θ, η) = ∫_{θ−η}^{θ+η} D_{r}(θ, φ) \frac{dφ}{2π}

Then

0 ≤ [1 − N_{r}(θ, η)] ≤ \frac{4(1 − r)}{r sin^{2}(η)}  (2.17)

Proof. We have

1 = ∫_{0}^{2π} P_{r}(θ, φ) \frac{dφ}{2π}

so since D_{r} ≤ P_{r}, N_{r} ≤ 1 and

1 − N_{r}(θ, η) ≤ \frac{2}{2π} ∫\begin{matrix}φ∈[0,2π] \\ |θ−φ|≥η \end{matrix} P_{r}(θ, φ) dφ

If |θ − φ| ≥ η, then

P_{r}(θ, φ) = \frac{1 − r^{2}}{(1 − r)^{2} + 4r sin^{2}[\frac{1}{2}(θ − φ)]}

≤ \frac{2(1 − r)}{4r sin^{2}(\frac{1}{2}η)}

≤ \frac{2(1 − r)}{r sin^{2}(η)}

and (2.17) is immediate. □

Lemma 2.7. If ∫_{0}^{π} Im M(e^{iθ}) dθ ≠ 0, then for θ ∈ [0, π], r ∈ (\frac{1}{2}, 1),

Im M(re^{iθ}) ≥ c(r^{−1} − r) sin θ  (2.18)

Proof. In terms of the real line m function, for E_{2} > 0, E_{1} real,

Im[−m(E_{1} − iE_{2})] ≥ \frac{E_{2}}{π} ∫_{−2}^{2} \frac{Im m(E) dE}{(E_{1} − E)^{2} + E_{2}^{2}}  (2.19)

since we have dropped the positive contributions of ν_{sing} to Im(−m). Now if z = re^{iθ},

M(z) = −m(E_{1} − iE_{2})

where z + z^{−1} = E_{1} − iE_{2} or E_{1} = (r + r^{−1}) cos θ, E_{2} = (r^{−1} − r) sin θ. If r > \frac{1}{2}, then |E_{1}| ≤ \frac{5}{2}, |E_{2}| ≤ \frac{3}{2}, and in (2.19), |E| ≤ 2. Thus

Im M(z) ≥ cE_{2}(z)

which is (2.18). □

Proof of Theorem 2.3. Since ln_{−} is a decreasing function, to get upper bounds on ln_{−}[Im M(re^{iθ})/a], we can use a lower bound on Im M. The elementary bound

ln_{−}(ab) ≤ ln_{−}(a) + ln_{−}(b)  (2.20)

will be useful.

As already noted, Fatou’s lemma implies the lim inf of the left side of (2.8) is bounded from below by the right side, so it suffices to prove that

lim sup_{r↑1} ∫_{0}^{π} ln_{−}(\frac{Im M(re^{iφ})}{a})w(φ) dφ ≤ ∫_{0}^{π} ln_{−}(\frac{Im M(e^{iφ})}{a})w(φ) dφ

(2.21)

Pick γ and κ so 0 < max(β, 1)γ < κ < \frac{1}{2} and let θ_{0}(r) = (1 − r)^{γ}, η(r) = (1 − r)^{κ}. We will bound Im M(re^{iθ}) from below for d(θ) ≤ θ_{0}(r) using (2.18), and for d(θ) ≥ θ_{0}(r), we will use the Poisson integral for the region |φ − θ| ≤ η(r).

By (2.18) and (2.3),

∫_{d(φ)≤θ_{0}(r)} ln_{−}(\frac{Im M(re^{iφ})}{a})w(φ) dφ ≤ C_{a}θ_{0}^{α}[ln_{−}(r^{−1} − r) + ln_{−} θ_{0}]

which goes to zero as r ↑ 1 for any a. So suppose d(θ) > θ_{0}. Write

Im M(re^{iθ}) ≥ ∫_{θ−η(r)}^{θ+η(r)} D_{r}(θ, φ) Im M(e^{iφ}) \frac{dφ}{2π}

= N_{r}(θ, η) ∫_{θ−η(r)}^{θ+η(r)} \frac{D_{r}(θ, φ)}{2πN_{r}(θ, η)} Im M(e^{iφ}) dφ (2.22)

For later purposes, note that for d(θ) > θ_{0}, (2.17) implies

0 ≤ 1 − N_{r}(θ, η) ≤ C(1 − r)^{1−2κ}  (2.23)

which goes to zero since κ < \frac{1}{2}. Using (2.22) and (2.20), we bound ln_{−}[Im M(re^{iθ})/a] as two ln_{−}’s. Since ln_{−} is convex and D_{r}(θ, φ)/2πN_{r}(θ, η) χ_{(θ−η,θ+η)}(φ) dφ is a probability measure, we can use Jensen’s inequality to see that

w(θ) ln_{−}[\frac{Im M(re^{iθ})}{a}]

≤ w(θ) ln_{−}[N_{r}(θ, η)] + ∫_{θ−η(r)}^{θ+η(r)} \frac{w(θ)}{w(φ)} \frac{D_{r}(θ, φ)}{N_{r}(θ, η)} w(φ) ln_{−}[\frac{Im M(e^{iφ})}{a}] \frac{dφ}{2π}

(2.24)

In the first term for the θ’s with d(θ) ≥ θ_{0}(r), N_{r} obeys (2.23) so

∫_{d(θ)≥θ_{0}(r)} w(θ) ln_{−}[N_{r}(θ, η)] dθ = O((1 − r)^{1−2κ}) → 0  (2.25)

In the second term, note that for the θ’s in question, N_{r}(θ, η)^{−1} − 1 = O((1 − r)^{1−2κ}) and by (2.15), w(θ)/w(φ) − 1 = O((1 − r)^{κ−βγ}). Since D_{r}(θ, φ) ≤ P_{r}(θ, φ), we thus have

∫_{d(θ)≥θ_{0}} ln_{−}[\frac{Im M(re^{iθ})}{a}]w(θ) dθ

≤ O((1 − r)^{1−2κ}) + [1 + O((1 − r)^{1−2κ})][1 + O((1 − r)^{κ−βγ})]

∫\begin{matrix}d(θ)≥θ_{0} \\ |φ−θ|≤η \end{matrix} P_{r}(θ, φ)w(φ) ln_{−}[\frac{Im M(e^{iφ})}{a}] dφ \frac{dθ}{2π} (2.26)

Since the integrand is positive, we can extend it to {(θ, φ) | θ ∈ [0, 2π], φ ∈ [0, π]} and do the θ integration using ∫ P_{r}(θ, φ)dθ/2π = 1. The result is (2.21). □

This concludes the proof of Theorem 2.1. By going through the proof, one easily sees that

Theorem 2.8. Theorem 2.1 remains true if in (2.1) and (2.6), ln[Im M(re^{iφ})] is replaced by ln[g(r) sin φ + Im M(re^{iφ})] where g(r) ≥ 0 and g(r) → 0 as r ↑ 1.

Proof. In the ln_{+} bounds, we get an extra [sup_{\frac{1}{2} <r<1} g(r)] sin θ in f_{2}(θ). Since we still have pointwise convergence, we easily get the analog of Theorem 2.2. In the proof of Theorem 2.3, Fatou is unchanged since g(r) → 0, and since

ln_{−}(g(r) sin φ + Im M(re^{iφ})) ≤ ln_{−}(Im M(re^{iφ}))

the lim sup bound has an unchanged proof. □

3. The Step-by-Step Sum Rules

We will call J a BW matrix (for Blumenthal-Weyl) if J is a bounded Jacobi matrix with σ_{ess}(J) = [−2, 2], for example, if J − J_{0} is compact. Let J^{(n)} be the matrix resulting from removing the first n rows and columns. Let {E_{j}^{±}(J)}_{j=1}^{∞} be the eigenvalues of J above/below ±2, ordered by ±E_{1}^{±} ≥ ±E_{2}^{±} ≥ ⋯ with E_{j}^{±}(J) defined to be ±2 if there are only finitely many eigenvalues k < j above/below ±2. Then by the min-max principle,

±E_{j+n}^{±}(J) ≤ ±E_{j}^{±}(J^{(n)}) ≤ ±E_{j}^{±}(J)  (3.1)

We have lim_{j→∞} E_{j}^{±}(J) = ±2 if J is a BW matrix.

It follows by the convergence of sums of alternating series that if f is even or odd and monotone on [2, ∞) with f(2) = 0, then

lim_{N→∞} ∑_{±} ∑_{j=1}^{N} [f(E_{j}^{±}(J)) − f(E_{j}^{±}(J^{(n)}))] ≡ δf_{n}(J)  (3.2)

exists and is finite. If β_{j}^{±} is defined by E_{j}^{±} = β_{j}^{±} +(β_{j}^{±})^{−1} with |β_{j}| > 1, we define X_{ℓ}^{(n)}(J) as δf_{n}(J) for

f(E) = \begin{cases}ln|β| & ℓ = 0 \\ −\frac{1}{ℓ}[β^{ℓ} − β^{−ℓ}] & ℓ = 1, 2 … \end{cases}  (3.3)

In addition, we will need

ζ_{ℓ}^{(n)}(J) = \begin{cases}− ∑_{j=1}^{n} ln(a_{j}) & ℓ = 0 \\ \frac{2}{ℓ} lim_{m→∞}[Tr(T_{ℓ}(\frac{1}{2}J_{m;F})) − Tr(T_{ℓ}(\frac{1}{2}J_{m−n;F}^{(n)} ))] & ℓ = 1, 2, … \end{cases}

(3.4)

where J_{m;F} is the finite matrix formed from the first m rows and columns of J and T_{ℓ} is the ℓ-th Chebyshev plynomial (of the first kind). As noted in [11, Proposition 4.3], the limit in (3.4) exists since the expression is independent of m once m > ℓ + n.

Note that

ζ_{1}^{(n)}(J) = ∑_{j=1}^{n} b_{j}  (3.5)

ζ_{2}^{(n)}(J) = ∑_{j=1}^{n} \frac{1}{2}b_{j}^{2} + (a_{j}^{2} − 1)  (3.6)

as computed in [11].

Note that by construction (with J^{(0)} ≡ J),

X_{ℓ}^{(n)}(J) = ∑_{j=0}^{n−1} X_{ℓ}^{(1)}(J^{(j)})  (3.7)

and

ζ_{ℓ}^{(n)}(J) = ∑_{j=0}^{n−1} ζ_{ℓ}^{(1)}(J^{(j)})  (3.8)

As final objects we need

Z(J) = \frac{1}{4π} ∫_{0}^{2π} ln(\frac{sin θ}{Im M(e^{iθ}, J)}) dθ  (3.9)

and for ℓ ≥ 1,

Z_{ℓ}^{±}(J) = \frac{1}{4π} ∫_{0}^{2π} ln(\frac{sin θ}{Im M(e^{iθ}, J)}) (1 ± cos(ℓθ)) dθ  (3.10)
Y_{ℓ}(J) = −\frac{1}{2π} ∫_{0}^{2π} ln(\frac{sin θ}{Im M(e^{iθ}, J)}) cos(ℓθ) dθ  (3.11)

We include “sin θ” inside ln(…) so that Z(J_{0}) = Z_{ℓ}^{±}(J_{0}) = Y_{ℓ}(J_{0}) = 0 because M(z, J_{0}) = z. Notice that (3.9) is the same as (1.2). Indeed,

Im M(e^{iθ}, J) = sgn(π − θ) π \frac{dν_{ac}}{dE} (2 cos θ)

for a.e. θ ∈ (0, 2π), and the factor (4π)^{−1} replaces (2π)^{−1} because under z ↦ z + z^{−1} the unit circle covers (−2, 2) twice.

Of course,

Z_{ℓ}^{±}(J) = Z(J) ∓ \frac{1}{2} Y_{ℓ}(J)  (3.12)

when all integrals converge. By Theorem 2.2, the ln_{−} piece of the integrals in (3.9)–(3.11) always converges. Since 1 ± cos(ℓθ) ≥ 0, the integrals defining Z(J), Z_{ℓ}^{±}(J) either converge or diverge to +∞. We therefore always define Z(J) and Z_{ℓ}^{±}(J) although they may take the value +∞. Since [1 ± cos(ℓθ)] ≤ 2, Z(J) < ∞ implies Z_{ℓ}^{±}(J) < ∞, so we define Y_{ℓ}(J) by (3.12) if and only if Z(J) < ∞.

If Z(J) < ∞, we say J obeys the Szegő condition or J is Szegő. If Z_{1}^{±}(J) < ∞, we say J is Szegő at ±2 since, for example, if Z_{1}^{+}(J) < ∞, the integral in (3.9) converges near θ = 0 (E = 2 cos(θ) near +2) and if Z_{1}^{−}(J) < ∞, the integral converges near θ = π (i.e., E = −2). Note that while Z_{1}^{+}(J) < ∞ only implies convergence of (3.9) at θ = 0, it also implies that at θ = π the integral with a sin^{2} θ inserted converges (quasi-Szegő condition).

Our main goal in this section is to prove the next three theorems

Theorem 3.1 (Step-by-Step Sum Rules). Let J be a BW matrix. Z(J) < ∞ if and only if Z(J^{(1)}) < ∞, and if Z(J) < ∞, we have

Z(J) = − ln(a_{1}) + X_{0}^{(1)}(J) + Z(J^{(1)})  (3.13)
Y_{ℓ}(J) = ζ_{ℓ}^{(1)}(J) + X_{ℓ}^{(1)}(J) + Y_{ℓ}(J^{(1)});  ℓ = 1, 2, 3, …  (3.14)

Remarks. 1. By iteration and (3.7)/(3.8), we obtain if Z(J) < ∞, then Z(J^{(n)}) < ∞ and

Z(J) = − ∑_{j=1}^{n} ln(a_{j}) + X_{0}^{(n)}(J) + Z(J^{(n)})  (3.15)
Y_{ℓ}(J) = ζ_{ℓ}^{(n)}(J) + X_{ℓ}^{(n)}(J) + Y_{ℓ}(J^{(n)});  ℓ = 1, 2, 3, … (3.16)

2. We call (3.13)/(3.14) the step-by-step Case sum rules.

Theorem 3.2 (One-Sided Step-by-Step Sum Rules). Let J be a BW matrix. Z_{1}^{±}(J) < ∞ if and only if Z_{1}^{±}(J^{(1)}) < ∞, and if Z_{1}^{±}(J) < ∞, then we have for ℓ = 1, 3, 5, …,

Z_{ℓ}^{±}(J) = − ln(a_{1}) ∓ \frac{1}{2} ζ_{ℓ}^{(1)}(J) + X_{0}^{(1)}(J) ∓ \frac{1}{2} X_{ℓ}^{(1)}(J) + Z_{ℓ}^{±}(J^{(1)}) (3.17)

Remark. Theorem 3.2 is intended to be two statements: one with all the upper signs used and one with all the lower signs used.

Theorem 3.3 (Quasi-Step-by-Step Sum Rules). Let J be a BW matrix. Z_{2}^{−}(J) < ∞ if and only if Z_{2}^{−}(J^{(1)}) < ∞, and if Z_{2}^{−}(J) < ∞, then for ℓ = 2, 4, …, we have

Z_{ℓ}^{−}(J) = − ln(a_{1}) + \frac{1}{2} ζ_{ℓ}^{(1)}(J) + X_{0}^{(1)}(J) + \frac{1}{2} X_{ℓ}^{(1)}(J) + Z_{ℓ}^{−}(J^{(1)}) (3.18)

Remarks. 1. The name comes from the fact that since 1 − cos 2θ = 2 sin^{2} θ, Z_{2}^{−}(J) is what Killip-Simon [11] called the quasi-Szegő integral

Z_{2}^{−}(J) = \frac{1}{2π} ∫_{0}^{2π} ln(\frac{sin θ}{Im M(e^{iθ}, J)}) sin^{2} θ dθ  (3.19)

2. Since Z(J) < ∞ implies Z_{1}^{+}(J) and Z_{1}^{−}(J) < ∞, and Z_{1}^{+}(J) or Z_{1}^{−}(J) < ∞ imply Z_{2}^{−}(J) < ∞, we have additional sum rules in various cases.

3. In [12], Laptev et al. prove sum rules for Z_{ℓ}^{−}(J) where ℓ = 4, 6, 8, . . .. One can develop step-by-step sum rules in this case and use it to streamline the proof of their rules as we streamline the proof of the Killip-Simon P_{2} rule (our Z_{2}^{−} sum rule) in the next section.

The step-by-step sum rules were introduced in Killip-Simon, who first take r < 1 (in our language below), then take n → ∞, and then r ↑ 1 with some technical hurdles to take r ↑ 1. By first letting r ↑ 1 with n < ∞, and then n → ∞ as in the next section, we can both simplify their proof and obtain additional results. The idea of using the imaginary part of

−M(z; J)^{−1} = −(z + z^{−1}) + b_{1} + a_{1}^{2}M(z; J^{(1)})  (3.20)

is taken from Killip-Simon [11].

Proof of Theorem 3.1. Taking imaginary parts of both sides of (3.20) with z = re^{iθ} and r < 1, we obtain

[Im M(re^{iθ}; J)] |M(re^{iθ}; J)|^{−2} = (r^{−1} − r) sin θ + a_{1}^{2} Im M(re^{iθ}; J^{(1)})

(3.21)

Taking ln’s of both sides, we obtain

ln(\frac{sin θ}{Im M(re^{iθ}; J)}) = t_{1} + t_{2} + t_{3}  (3.22)

where

t_{1} = −2 ln|M(re^{iθ}; J)|  (3.23)
t_{2} = −2 ln a_{1}  (3.24)

t_{3} = ln(\frac{sin θ}{g(r) sin θ + Im M(re^{iθ}; J^{(1)})})  (3.25)

where

g(r) = a_{1}^{−2}(r^{−1} − r)  (3.26)

Let

f(z) = \frac{M(rz; J)}{rz},

so f(0) = 1 (see (3.20)). In the unit disk, f(z) is meromorphic and has poles at {(rβ_{j}^{±}(J))^{−1} | j so that |β_{j}^{±}(J)| > r^{−1}} and zeros at {(rβ_{j}^{±}(J^{(1)}))^{−1} | j so that |β_{j}^{±}(J^{(1)})| > r^{−1}}. Thus, by Jensen’s formula for f:

\frac{1}{4π} ∫_{0}^{2π} t_{1} dθ = − ln r+ ∑_{|β_{j}^{±}(J)|>r^{−1}} ln|rβ_{j}^{±}(J)|− ∑_{|β_{j}^{±}(J^{(1)})|>r^{−1}} ln|rβ_{j}^{±}(J^{(1)})|

By (3.1), the number of terms in the sums differs by at most 2, so that the ln(r)’s cancel up to at most 2 ln(r) → 0 as r ↑ 1. Thus as r ↑ 1,

\frac{1}{4π} ∫_{0}^{2π}(t_{1} + t_{2}) dθ → − ln(a_{1}) + X_{0}^{(1)}(J)  (3.27)

It follows by (3.22) and Theorems 2.1 and 2.8 (with w(φ) = 1) that Z(J) < ∞ if and only if Z(J^{(1)}) < ∞, and if they are finite, (3.13) holds.

It also follows that if Z(J) < ∞, we have L^{1} convergence of the ln’s to their r = 1 values. That implies convergence of the integrals with cos(ℓθ) inside. Higher Jensen’s formula as in [11] then implies (3.14). In place of ln|βr^{−1}|, we have (rβ)^{ℓ} −(rβ)^{−ℓ}, but the sums still converge to the r = 1 limit since we can separate the β^{ℓ} and β^{−ℓ} terms, and then the r’s factor out. □

Proofs of Theorems 3.2 and 3.3. These are the same as the above proof, but now the weight w is either 1±cos(θ) or 1−cos(2θ) and that weight obeys (2.3) and (2.4). □

Corollary 3.4. Let J be a BW matrix. If J and J̃ differ by a finite rank perturbation, then J is Szegő (resp. Szegő at ±2) if and only if J̃ is.

Proof. For some n, J^{(n)} = J̃^{(n)}, so this is immediate from Theorems 3.1 and 3.2. □

Conjecture 3.5. Let J be a BW matrix. If J and J̃ differ by a trace class perturbation, then J is Szegő (resp. Szegő at ±2) if and only if J̃ is. It is possible this conjecture is only generally true if J − J_{0} is only assumed compact or is only assumed Hilbert-Schmidt.

This conjecture for J = J_{0} is Nevai’s conjecture recently proven by Killip-Simon. Their method of proof and the ideas here would prove this conjecture if one can prove a result of the following form. Let J, J̃ differ by a finite rank operator so that by the discussion before (3.2),

lim_{N→∞} ∑_{±} ∑_{j=1}^{N}(\sqrt{E_{j}^{±}(J)^{2} − 4} − \sqrt{E_{j}^{±}(J̃)^{2} − 4} ) ≡ δ(J, J̃)

exists and is finite. The conjecture would be provable by the method of [11] and this paper (by using the step-by-step sum rule to remove the first m pieces of J and then replacing them with the first m pieces of J̃) if one had a bound of the form

|δ(J, J̃)| ≤ (const.)Tr(|J − J̃|)  (3.28)

(3.28) with J = J_{0} is the estimate of Hundertmark-Simon [10]. We have counterexamples that show (3.28) does not hold for a universal constant c. However, in these examples, ∥J∥ → ∞ as c → ∞. Thus it could be that (3.28) holds with c only depending on J for some class of J’s. If it held with a bound depending only on ∥J∥, the conjecture would hold in general. If J was required in J_{0}+ Hilbert-Schmidt, we would get the conjecture for such J’s.

4. The Z_{0}, Z_{1}^{±}, and Z_{2}^{−} Sum Rules

Our goal here is to prove that sum rules of Case type hold under certain hypotheses. Of interest on their own, these considerations also somewhat simplify the proof of the P_{2} sum rule in Section 8 of [11], and considerably simplify the proof of the C_{0} sum rule for trace class J −J_{0} from Section 9 of [11]. Throughout, J will be a BW matrix. There are two main tools. As in [11], lower semicontinuity of the Z’s in J (in the topology of pointwise convergence of matrix elements) gets inequalities in one direction. We use step-by-step sum rules and boundedness from below of Z for the other direction.

We first introduce some quantities involving a fixed Jacobi matrix:

Ā_{0}(J) = lim sup_{n→∞}(− ∑_{j=1}^{n} ln(a_{j}))  (4.1)

\underline{A}_{0}(J) = lim inf_{n→∞}(− ∑_{j=1}^{n} ln(a_{j}))

Ā_{1}^{±}(J) = lim sup_{n→∞}(− ∑_{j=1}^{n}(a_{j} − 1 ± \frac{1}{2}b_{j}))  (4.2)

\underline{A}_{1}^{±}(J) = lim inf_{n→∞}(− ∑_{j=1}^{n}(a_{j} − 1 ± \frac{1}{2}b_{j}))

A_{2}(J) = ∑_{j=1}^{∞} [\frac{1}{4}b_{j}^{2} + \frac{1}{2}G(a_{j})]  (4.3)

where

G(a) = a^{2} − 1 − ln(a^{2})

Since G(a) ≥ 0, the finite sums have a limit (which may be +∞).

We note that for a near 1, G(a) ∼ 2(a − 1)^{2}. Thus A_{2}(J) is finite if and only if J − J_{0} is Hilbert-Schmidt. In (4.2), we can use a_{j} − 1 in place of ln(a_{j}) because if {a_{j} − 1} ∈ ℓ^{2} (e.g., if J − J_{0} is Hilbert-Schmidt), then ∑|ln(a_{j}) − (a_{j} − 1)| < ∞. Notice also that in the case of a discrete Schrödinger operator (i.e., a_{n} ≡ 1), Ā_{0}(J) = \underline{A}_{0}(J) = 0.

Next, we introduce some functions of the eigenvalues:

E_{0}(J) = ∑_{j,±} ln|β_{j}^{±}|  (4.4)

E_{1}^{±}(J) = ∑_{j} \sqrt{(E_{j}^{±})^{2} − 4}  (4.5)
E_{2}(J) = ∑_{j,±} F (E_{j}^{±})  (4.6)

where F (E) = \frac{1}{4}[β^{2} − β^{−2} − ln(β^{4})] with E = β + β^{−1} and |β| > 1. For |E| ∼ 2, F (E) is O((|E| − 2)^{3/2}). In (4.4) and (4.6), we sum over + and −. In (4.5), we define E_{1}^{+} and E_{1}^{−} with only the + or only the − terms.

We need the following basis-dependent notion: Definition. Let B be a bounded operator on ℓ^{2}(ℤ^{+}). We say B has a conditional trace if

lim_{ℓ→∞} ∑_{j=1}^{ℓ}⟨δ_{j}, Bδ_{j}⟩ ≡ c-Tr(B)  (4.7)

exists and is finite.

If B is not trace class, this object is not unitarily invariant.

Our goal in this section is to prove the following theorems whose proof is deferred until after all the statements.

Theorem 4.1. Let J be a BW matrix. Consider the four statements:

(i) Ā_{0}(J) > −∞ (ii) \underline{A}_{0}(J) < ∞ (iii) Z(J) < ∞ (iv) E_{0}(J) < ∞ Then

(a) (ii) + (iv) ⇒ (iii) + (i)

(b) (i) + (iii) ⇒ (iv) + (ii)

(c) (iii) ⇒ Ā_{0}(J) < ∞

(d) (iv) ⇒ \underline{A}_{0}(J) > −∞ Thus (iii) + (iv) ⇒ (i) + (ii). In particular, if \underline{A}_{0}(J) = Ā_{0}(J), that is, the limit exists, then the finiteness of any two of Z(J), E_{0}(J), and Ā_{0}(J) implies the finiteness of the third.

If all four conditions hold and J − J_{0} is compact, then (e)

lim_{n→∞} (− ∑_{j=1}^{n} ln(a_{j})) ≡ A_{0}(J)  (4.8)

exists and is finite, and

Z(J) = A_{0}(J) + E_{0}(J)  (4.9)

(f) For each ℓ = 1, 2, …,

− ∑_{j,±} ℓ^{−1}[β_{j}^{±}(J)^{ℓ} − β_{j}^{±}(J)^{−ℓ}] ≡ X_{ℓ}^{(∞)}(J)  (4.10)

converges absolutely and equals lim_{n→∞} X_{ℓ}^{(n)}(J).

(g) For each ℓ = 1, 2, …,

B_{ℓ}(J) = \frac{2}{ℓ}\{T_{ℓ}(\frac{J}{2} ) − T_{ℓ}(\frac{J_{0}}{2} )\}  (4.11)

has a conditional trace and

c-Tr(B_{ℓ}(J)) = lim_{n→∞} ζ_{ℓ}^{(n)}(J)  (4.12)

for example, if ℓ = 1, ∑_{j=1}^{n} b_{j} converges to a finite limit.

(h) The Case sum rule holds:

Y_{ℓ}(J) = c-Tr(B_{ℓ}(J)) + X_{ℓ}^{(∞)}(J)  (4.13)

where Y_{ℓ} is given by (3.11), X_{ℓ}^{(∞)} by (4.10), and c-Tr(B_{ℓ}(J)) by (4.7), (4.11), and (4.12).

Remarks. 1. In one sense, this is the main result of this paper.

2. We will give examples later where Ā_{0}(J) = \underline{A}_{0}(J) and one of the conditions (i)/(ii), (iii), (iv) holds and the other two fail.

3. For ℓ odd, T_{ℓ}(J_{0}/2) vanishes on-diagonal. By Proposition 2.2 of [11] and the fact that the diagonal matrix elements of J_{0}^{k} are eventually constant, it follows that for ℓ even, T_{ℓ}(J_{0}/2) eventually vanishes ondiagonal and c-Tr(T_{ℓ}(J_{0}/2)) = −\frac{1}{2}. Thus (g) says c-Tr(T_{ℓ}(J/2)) exists and the sum rule (4.13) can replace c-Tr(B_{ℓ}(J)) by \frac{2}{ℓ}c-Tr(T_{ℓ}(J/2)) plus a constant (zero if ℓ is odd and 1/ℓ if ℓ is even). For ℓ even, c-Tr(T_{ℓ}(J_{0}/2)) = −\frac{1}{2} while Tr(T_{ℓ}(J_{0,n;F} /2)) = −1 for n large because T_{ℓ}(J_{0,n;F} /2) has two ends.

Corollary 4.2. Let J−J_{0} be compact. If Z(J) < ∞, then − ∑_{j=1}^{n} ln(a_{j}) either converges or diverges to −∞.

Remarks. 1. We will give an example later where Z(J) < ∞, and lim_{n→∞}(− ∑_{j=1}^{n} ln(a_{j})) = −∞.

2. In other words, if J − J_{0} is compact and Ā_{0}(J) ≠ \underline{A}_{0}(J), then Z(J) = ∞.

3. Similarly, if J − J_{0} is compact and E_{0}(J) < ∞, then the limit exists and is finite or is +∞.

Proof. If Z(J) < ∞ and Ā_{0} > −∞, then by (b) of the theorem, all four conditions hold, and so by (e), the limit exists. On the other hand, if Ā_{0} = −∞, then Ā_{0} = \underline{A}_{0} = −∞. □

Corollary 4.3. If J − J_{0} is trace class, then Z(J) < ∞, E_{0}(J) < ∞, and the sum rules (4.9) and (4.13) hold.

Remark. This is a result of Killip-Simon [11]. Our proof that Z(J) < ∞ is essentially the same as theirs, but our proof of the sum rules is much easier.

Proof. Since J − J_{0} is trace class, it is compact. Clearly, Ā_{0} = \underline{A}_{0}, and is neither ∞ nor −∞ since a_{j} > 0 and ∑|a_{j} − 1| < ∞ imply ∑|ln(a_{j})| < ∞. By the bound of Hundertmark-Simon [10], E_{0}(J) < ∞. The sum rules then hold by (a), (e), and (h) of Theorem 4.1. □

Theorem 4.4. Suppose J − J_{0} is Hilbert-Schmidt. Then

(i) \underline{A}_{1}^{±} < ∞ and E_{1}^{±} < ∞ implies Z_{1}^{±} < ∞. (ii) Z_{1}^{±} < ∞ implies Ā_{1}^{±} < ∞. (iii) Z_{1}^{±} < ∞ and Ā_{1}^{±} > −∞ implies E_{1}^{±} < ∞. (iv) E_{1}^{±} < ∞ implies \underline{A}_{1}^{±} > −∞.

Remarks. 1. Each of (i)–(iv) is intended as two statements.

2. In Section 6, we will explore (ii), which is the most striking of these results since its contrapositive gives very general conditions under which the Szegő condition fails.

3. The Hilbert-Schmidt condition in (i) and (iv) can be replaced by the somewhat weaker condition that

∑_{j,±} (|E_{j}^{±}| − 2)^{3/2} < ∞  (4.14)

That is true for (ii) and (iii) also, but by the Z_{2}^{−} sum rule, (4.14) plus Z_{1}^{±} < ∞ implies J − J_{0} is Hilbert-Schmidt.

Theorem 4.5. Let J be a BW matrix. Then

Z_{2}^{−}(J) + E_{2}(J) = A_{2}(J)  (4.15)

Remarks. 1. This is, of course, the P_{2} sum rule of Killip-Simon [11]. Our proof that Z_{2}^{−}(J) + E_{2}(J) ≤ A_{2}(J) is identical to that in [11], but our proof of the other half is somewhat streamlined.

2. As in [11], the values +∞ are allowed in (4.15).

Proof of Theorem 4.1. As in [11], let J_{n} be the infinite Jacobi matrix obtained from J by replacing a_{ℓ} by 1 if ℓ ≥ n and b_{ℓ} by 0 if ℓ ≥ n + 1. Then (3.15) (noting J_{n}^{(n)} = J_{0} and Z(J_{0}) = 0) reads

Z(J_{n}) = − ∑_{j=1}^{n} ln(a_{j}) + ∑_{j,±} ln|β_{j}^{±}(J_{n})|  (4.16)

[11, Section 6] implies the eigenvalue sum converges to E_{0}(J) if J −J_{0} is compact, and in any event, is bounded above by E_{0}(J) + c_{0} where c_{0} = 0 if J − J_{0} is compact and otherwise,

c_{0} = ln|β_{1}^{+}(J)| + ln|β_{1}^{−}(J)|  (4.17)

Moreover, by semicontinuity of the entropy [11, Section 5], Z(J) ≤ lim inf Z(J_{n}). Thus we have

Z(J) ≤ \underline{A}_{0}(J) + E_{0}(J) + c_{0}  (4.18)

Thus far, the proof is directly from [11]. On the other hand, by (3.15), we have

Z(J) ≥ Ā_{0}(J) + lim inf X_{0}^{(n)}(J) + lim inf Z(J^{(n)})  (4.19)

By the lemma below, lim_{n→∞} X_{0}^{(n)}(J) = E_{0}(J). Moreover, by Theorem 5.5 (eqn. (5.26)) of Killip-Simon [11], Z(J^{(n)}) ≥ −\frac{1}{2} ln(2), and if J^{(n)} → J_{0} in norm, that is, J − J_{0} is compact, then by semicontinuity of Z, 0 = Z(J_{0}) ≤ lim inf Z(J^{(n)}). Therefore, (4.19) implies that

Z(J) ≥ Ā_{0}(J) + E_{0}(J) − c  (4.20)

where

c = 0 if J − J_{0} is compact;  c = \frac{1}{2} ln(2) in general  (4.21)

24 B. SIMON AND A. ZLATOŠ

With these preliminaries out of the way, \underline{Proof of (d).} (iv) and (4.18) imply that

Ā_{0}(J) ≥ \underline{A}_{0}(J) ≥ Z(J) − E_{0}(J) − c_{0} > −∞  (4.22)

\underline{Proof of (a).} (4.18) shows Z(J) < ∞, and (d) shows that (i) holds. \underline{Proof of (c).} By (4.20) and E_{0}(J) ≥ 0,

Z(J) ≥ Ā_{0}(J) − c

so Z(J) < ∞ implies Ā_{0}(J) < ∞. \underline{Proof of (b).} Since Ā_{0}(J) > −∞ and c < ∞, (4.20) plus Z(J) < ∞ implies E_{0}(J) < ∞. (c) shows that (ii) holds.

Note that (iii), (iv), and (4.20) imply that

\underline{A}_{0}(J) ≤ Ā_{0}(J) ≤ Z(J) − E_{0}(J) + \frac{1}{2} ln(2) < ∞  (4.23)

Thus we have shown more than merely (iii) + (iv) ⇒ (i) + (ii), namely, (iii) + (iv) imply by (4.22) and (4.23)

−∞ < Ā_{0}(J) ≤ \underline{A}_{0}(J) + \frac{1}{2} ln(2) + c_{0} < ∞  (4.24)

We can say more if J − J_{0} is compact. \underline{Proof of (e).} (4.23) is now replaced by

\underline{A}_{0}(J) ≤ Ā_{0}(J) ≤ Z(J) − E_{0}(J)

since we can take c = 0 in (4.20). This plus (4.22) with c_{0} = 0 implies Ā_{0}(J) = \underline{A}_{0}(J) and (4.9). \underline{Proof of (f), (g), (h).} We have the sum rules (3.15), (3.16). Z(J) ± \frac{1}{2}Y_{ℓ}(J) is an entropy up to a constant, and so, lower semicontinous. Since ∥J^{(n)} − J_{0}∥ → 0, we have

lim inf(Z(J^{(n)}) ± \frac{1}{2}Y_{ℓ}(J^{(n)})) ≥ 0  (4.25)

On the other hand, since Z(J^{(n)}) < ∞ and E_{0}(J^{(n)}) ≤ E_{0}(J) < ∞, J^{(n)} obeys the sum rule (4.9). Since − ∑_{j=1}^{n} ln(a_{j}) converges conditionally

lim_{n→∞} lim_{m→∞}(− ∑_{j=n}^{m+n} ln(a_{j})) = 0

Moreover, E_{0}(J^{(n)}) → 0 by Lemma 4.6 below and we conclude that lim Z(J^{(n)}) = 0. Thus (4.25) becomes

lim inf_{n} Y_{ℓ}(J^{(n)}) ≥ 0,  lim sup_{n} Y_{ℓ}(J^{(n)}) ≤ 0

or

lim_{n} Y_{ℓ}(J^{(n)}) = 0  (4.26)

By the lemma below, lim_{n} X_{ℓ}^{(n)}(J) = X_{ℓ}^{(∞)}(J) exists and is finite. Since E_{0}(J) < ∞, we have that the sum defining X_{ℓ}^{(∞)}(J) is absolutely convergent. This proves (f).

By this fact, (3.16), and (4.26), lim_{n→∞} ζ_{ℓ}^{(n)}(J) exists, is finite, and obeys the sum rule

Y_{ℓ}(J) = lim_{n→∞} ζ_{ℓ}^{(n)}(J) + X_{ℓ}^{(∞)}(J)

By Propositions 2.2 and 4.3 of Killip-Simon [11], the existence of lim_{n→∞} ζ_{ℓ}^{(n)}(J) is precisely the existence of the conditional trace. □

Lemma 4.6. Let J be a BW matrix. Let f be a monotone increasing continuous function on [2, ∞) with f(2) = 0. Then

lim_{n→∞} ∑_{j=1}^{∞} [f(E_{j}^{+}(J)) − f(E_{j}^{+}(J^{(n)}))] = ∑_{j=1}^{∞} f(E_{j}^{+}(J))  (4.27)

Remarks. 1. The right side of (4.27) may be finite or infinite.

2. The sum on the left is interpreted as the limit of the sum from 1 to n as n → ∞, which exists and is finite by the arguments at the start of Section 3.

3. A similar result holds for E_{j}^{−} and f monotone decreasing on (−∞, −2].

Proof. Call the sum on the left of (4.27) (δf)(J, n). Since E_{j}^{+}(J^{(n)}) ≤ E_{j}^{+}(J), we have

(δf)(J, n) ≥ ∑_{j=1}^{m} [f(E_{j}^{+}(J)) − f(E_{j}^{+}(J^{(n)}))]  (4.28)

so, if we show for each fixed j as n → ∞,

E_{j}^{+}(J^{(n)}) → 2  (4.29)

we have, by taking n → ∞ and then m → ∞, that

lim inf(δf)(J, n) ≥ ∑_{j=1}^{∞} f(E_{j}^{+}(J))  (4.30)

On the other hand, since f ≥ 0, for each m,

∑_{j=1}^{m} [f(E_{j}^{+}(J)) − f(E_{j}^{+}(J^{(n)}))] ≤ ∑_{j=1}^{m} f(E_{j}^{+}(J))

so taking m to infinity and then n → ∞,

lim sup(δf)(J, n) ≤ ∑_{j=1}^{∞} f(E_{j}^{+}(J))  (4.31)

Thus (4.29) implies the result, so we need only prove that.

Fix ε > 0 and look at the solution of the orthogonal polynomial sequence u_{n} = P_{n}(2 + ε) as a function of n. By Sturm oscillation theory [8], the number of sign changes of u_{n} (i.e., number of zeros of the piecewise linear interpolation of u_{n}) is the number of j with E_{j}^{+}(J) > 2 + ε. Since J is a BW matrix, this is finite, so there exist N_{0} with u_{n} of definite sign if n ≥ N_{0} − 1. It follows by Sturm oscillation theory again that for all j,

E_{j}^{+}(J^{(n)}) ≤ 2 + ε

if n ≥ N_{0}. This implies (4.29). □

The combination of this Sturm oscillation argument and Theorem 3.1 gives one tools to handle finitely many bound states as an alternate to Nikishin [16]. For the oscillation argument says that if J has finitely many eigenvalues outside [−2, 2], there is a J^{(n)} with no eigenvalues. On the other hand, by Theorem 3.1, Z(J) < ∞ if and only if Z(J^{(n)}) < ∞.

Proof of Theorem 4.5. Z_{2}^{−}(J) is an entropy and not merely an entropy up to a constant (see [11]). Thus Z_{2}^{−}(J^{(n)}) ≥ 0 for all J^{(n)}. Moreover, since the terms in A_{2} are positive, the limit exists. Thus, following the proofs of (4.18) and (4.20) but using (3.18) in place of (3.15),

Z_{2}^{−}(J) + E_{2}(J) ≤ A_{2}(J)

and

Z_{2}^{−}(J) + E_{2}(J) ≥ A_{2}(J)

which yields the P_{2} sum rule. In the above, we use the fact that in place of Z(J) ≥ −\frac{1}{2} ln(2), one has Z_{2}^{−}(J) ≥ 0, and the fact that A_{2}(J) < ∞ implies that J − J_{0} is compact. □

Proof of Theorem 4.4. Let g(β) = ln β− \frac{1}{2}(β−β^{−1}) in the region β > 0. Then

g′(β) = β^{−1} − \frac{1}{2} − \frac{1}{2}β^{−2} = −\frac{1}{2}β^{−2}(β − 1)^{2}

so g is analytic near β = 1 and g(1) = g′(1) = g′′(1) = 0, that is, g(β) ∼ c(β − 1)^{3}. On the other hand, h(β) = ln β + \frac{1}{2}(β − β^{−1}) is g(β) + (β − β^{−1}) = β − β^{−1} + O((β − 1)^{3}). Since β + β^{−1} = E means β − β^{−1} = \sqrt{E^{2} − 4} and β − 1 = O(\sqrt{E − 2} ), we conclude that

E > 2 ⇒ ln(β) − \frac{1}{2}(β − β^{−1}) = O(|E − 2|^{3/2})

ln(β) + \frac{1}{2}(β − β^{−1}) = \sqrt{E^{2} − 4} + O(|E − 2|^{3/2})

while

E < −2 ⇒ ln(|β|) − \frac{1}{2}(β − β^{−1}) = \sqrt{E^{2} − 4} + O(|E + 2|^{3/2})
ln(|β|) + \frac{1}{2}(β − β^{−1}) = O(|E + 2|^{3/2})

It follows, using Lemma 4.6, that

lim_{n→∞} X_{0}^{(n)}(J) ∓ \frac{1}{2}X_{1}^{(n)}(J) = E_{1}^{±} + bdd

since Theorem 4.5 implies ∑_{j,±}(\sqrt{E_{j}^{±2} − 4} )^{3} < ∞ (or, by results of [10]). Thus for a constant c_{1} dependng only on ∥J − J_{0}∥_{2}, we have

Z_{1}^{±}(J) ≤ c_{1} + \underline{A}_{1}^{±} + E_{1}^{±}  (4.32)

by writing the finite rank sum rule, taking limits and using the argument between (4.16) and (4.17). Since Z_{1}^{±}(J) are entropies up to a constant, we have Z_{1}^{±}(J^{(n)}) ≥ −c_{2} and so by (3.17),

Z_{1}^{±}(J) ≥ −c_{2} + Ā_{1}^{±} + E_{1}^{±} − c∥J − J_{0}∥_{2}^{2}  (4.33)

With these preliminaries, we have \underline{Proof of (i), (iv).} Immediate from (4.32).

\underline{Proof of (ii).} Since E_{1}^{±} ≥ 0, (4.33) implies

Z_{1}^{±}(J) ≥ −c_{2} + Ā_{1}^{±}

so (ii) holds. \underline{Proof of (iii).} Immediate from (4.33). □

Remark. (i)–(iv) of Theorem 4.4 are exactly (a)–(d) of Theorem 4.1 for the Z_{1}^{±} sum rules. One therefore expects a version of (e) of that theorem to hold as well. Indeed, a modification of the above proof yields for J − J_{0} Hilbert-Schmidt that if E_{1}^{+}, Z_{1}^{+}, Ā_{1}^{+} are finite, then

Z_{1}^{+}(J) = − ∑_{n=1}^{∞} [ln(a_{n}) + \frac{1}{2}b_{n}] + ∑_{j,±} [ln|β_{j}^{±}| + \frac{1}{2}(β_{j}^{±} − (β_{j}^{±})^{−1})]

and if E_{1}^{−}, Z_{1}^{−}, Ā_{1}^{−} are finite, then

Z_{1}^{−}(J) = − ∑_{n=1}^{∞} [ln(a_{n}) − \frac{1}{2}b_{n}] + ∑_{j,±} [ln|β_{j}^{±}| − \frac{1}{2}(β_{j}^{±} − (β_{j}^{±})^{−1})]

5. Shohat’s Theorem with an Eigenvalue Estimate

Shohat [22] translated Szegő’s theory from the unit circle to the real line and was able to identify all Jacobi matrices which lead to measures with no mass points outside [−2, 2] and have Z(J) < ∞. The strongest result we know of this type is the following (Theorem 4′) from Killip-Simon [11] (the methods of Nevai [14] can prove the same result): Theorem 5.1. Let σ(J) ⊂ [−2, 2]. Consider

(i) \underline{A}_{0}(J) < ∞ where \underline{A}_{0} is given by (4.1). (ii) Z(J) < ∞ (iii) ∑_{n=1}^{∞}(a_{n} − 1)^{2} + ∑_{n=1}^{∞} b_{n}^{2} < ∞ (iv) \underline{A}_{0} = Ā_{0} and is finite.

(v) lim_{N→∞} ∑_{n=1}^{N} b_{n} exists and is finite. Then (under σ(J) ⊂ [−2, 2]), we have

(i) ⟺ (ii)

and either one implies (iii), (iv), and (v).

We can prove the following extension of this result: Theorem 5.2. Theorem 5.1 remains true if σ(J) ⊂ [−2, 2] is replaced by σ_{ess}(J) ⊂ [−2, 2] and (1.6).

Remarks. 1. Gončar [9], Nevai [14], and Nikishin [16] extended Shohat-type theorems to allow finitely many bound states outside [−2, 2].

2. Peherstorfer-Yuditskii [17] recently proved that E_{0}(J) < ∞ and (ii) implies (iv) and additional results on polynomial asymptotics.

Proof. Let us suppose first σ_{ess}(J) = [−2, 2], so J is a BW matrix. By Theorem 4.1(a), (i) of this theorem plus E_{0}(J) < ∞ implies (ii) of this theorem. By Theorem 4.1(c), (ii) of this theorem implies (i) of this theorem.

If either holds, then (iv) follows from (e) of Theorem 4.1, (v) from the ℓ = 1 case of (g) of Theorem 4.1. (iii) follows from Theorem 4.5 if we note that E_{0} < ∞ implies E_{2} < ∞, that Z(J) < ∞ implies Z_{2}^{−}(J) < ∞ and that G(a) = O((a − 1)^{2}).

If we only have a priori that σ_{ess}(J) ⊂ [−2, 2], we proceed as follows. If Z(J) < ∞, σ_{ac}(J) ⊃ [−2, 2] so, in fact, σ_{ess}(J) = [−2, 2]. If \underline{A}_{0} < ∞, we look closely at the proof of Theorem 4.1(a). (4.18) does not require σ_{ess}(J) = [−2, 2], but only that σ_{ess}(J) ⊂ [−2, 2]. Thus, \underline{A}_{0} < ∞ implies Z(J) < ∞ if E_{0}(J) < ∞. □

There is an interesting way of rephrasing this. Let the normalized orthogonal polynomial obey

P_{n}(x) = γ_{n}x^{n} + O(x^{n−1})  (5.1)

As is well known (see, e.g. [23]),

γ_{n} = (a_{1}a_{2} … a_{n})^{−1}  (5.2)

Thus

\underline{A}_{0} = lim inf ln(γ_{n})  (5.3)

and

Ā_{0} = lim sup ln(γ_{n})  (5.4)

Corollary 5.3. Suppose σ_{ess}(J) ⊂ [−2, 2] and E_{0}(J) < ∞. Then Z(J) < ∞ (i.e., the Szegő condition holds) if and only if γ_{n} is bounded from above (and in that case, it is also bounded away from 0; indeed, lim γ_{n} exists and is in (0, ∞)).

Remark. Actually, lim sup γ_{n} < ∞ is not needed; lim inf γ_{n} < ∞ is enough.

Proof. By (5.3), γ_{n} bounded implies \underline{A}_{0} < ∞, and thus Z(J) < ∞. Conversely, Z(J) < ∞ implies −∞ < \underline{A}_{0} = Ā_{0} < ∞. So by (5.2), it implies γ_{n} is bounded above and below. □

In the case of orthogonal polynomials on the circle, Szegő’s theorem says Z < ∞ if and only if κ_{j} is bounded if and only if ∑_{j=1}^{∞}|α_{j}|^{2} < ∞ where κ_{j} is the leading coefficient of the normalized polynomials, and α_{j} are the Verblunsky (aka Geronimus, aka reflection) coefficients. In the real line case, if one drops the a priori requirement that E_{0}(J) < ∞, it can happen that γ_{n} is bounded but Z(J) = ∞. For example, if a_{n} ≡ 1 but b_{n} = n^{−1}, then Z(J) cannot be finite. For J − J_{0} ∈ ℓ_{2}, so Theorem 4.4(ii) is applicable and thus, Ā_{1}^{−} = ∞ implies Z(J) = ∞.

But the other direction always holds:

Theorem 5.4. Let J be a BW matrix with Z(J) < ∞ (i.e., the Szegő condition holds). Then γ_{n} is bounded. Moreover, if J − J_{0} is compact, then lim_{n→∞} γ_{n} exists.

Remarks. 1. The examples of the next section show Z(J) < ∞ is consistent with lim γ_{n} = 0.

2. This result — even without a compactness hypothesis — is known. For γ_{n} is monotone increasing in the measure (see, e.g., Nevai [15]) and so one can reduce to the case where Shohat’s theorem applies.

Proof. By Theorem 4.1(c), Z(J) < ∞ implies Ā_{0} < ∞ which, by (5.4), implies γ_{n} is bounded. If J − J_{0} is compact, then Corollary 4.2 implies that lim γ_{n} = exp(lim − ∑_{j=1}^{n} ln(a_{j})) exists but can be zero. □

Here is another interesting application of Theorem 5.2.

Theorem 5.5. Suppose b_{n} ≥ 0 and

∑_{n=1}^{∞} |a_{n} − 1| < ∞  (5.5)

Then E_{0}(J) < ∞ if and only if ∑_{n=1}^{∞} b_{n} < ∞.

Proof. If ∑_{n=1}^{∞} b_{n} < ∞, E_{0}(J) < ∞ by (5.5) and the bounds of Hundertmark-Simon [10]. On the other hand, if E_{0}(J) < ∞, (5.5) implies \underline{A}_{0} < ∞, so by Theorem 5.2, ∑_{n=1}^{N} b_{n} is convergent. Since b_{n} ≥ 0, ∑_{n=1}^{∞} b_{n} < ∞. □

6. O(n^{−1}) Perturbations

In this section, we will discuss examples where

a_{n} = 1 + αn^{−1} + E_{a}(n)  (6.1)
b_{n} = βn^{−1} + E_{b}(n)  (6.2)

where E_{·}(n) is small compared to \frac{1}{n} in some sense. Our main result will involve the very weak requirement on the errors that n(|E_{a}(n)| + |E_{b}(n)|) → 0. (In fact, we only need the weaker condition that ∑_{j=1}^{n}(|E_{a}(j)|+ |E_{b}(j)|) is o(ln n).) In discussing the historical context, we will consider stronger assumptions like

E_{·}(n) = \frac{γ}{n^{2}} + o(\frac{1}{n^{2}} )  (6.3)

We will also mention examples where the leading n^{−1} terms are replaced by (−1)^{n}n^{−1}.

These examples are natural because they are just at the borderline beyond J − J_{0} trace class or \underline{A}_{0}(J) < ∞ or Ā_{0}(J) > −∞.

Here is the general picture for these examples. The (α, β) plane is divided into four regions:

(a) |β| < −2α. Szegő fails at both −2 and 2.

(b) |β| ≤ 2α. Szegő holds.

(c) β > 2|α| or β = −2α with β > 0. Szegő holds at +2 but fails at −2.

(d) β < −2|α| or β = 2α with β < 0. Szegő holds at −2 but fails at +2.

Remarks. 1. These are only guidelines and the actual result that we can prove requires estimates on the errors.

2. Put more succinctly, Szegő holds at ±2 if and only if 2α ± β ≥ 0.

3. We need strong hypotheses at the edges of our regions where |β| = 2|α|. For example, “generally” Szegő should hold if β = 2α > 0, but if a_{n} = 1+αn^{−1} −(n ln(n))^{−1} and b_{n} = 2αn^{−1}, the Szegő condition fails (at −2), as follows from Theorem 6.1 below.

Here is the history of these kinds of problems:

(1) Pollaczek [18, 19, 20] found an explicit class of orthogonal polynomials in the region (in our language) |β| < −2α, one example for each such (α, β) with further study by Szegő [24, 26] (but note formula (1.7) in the appendix to Szegő’s book [26] is wrong — he uses in that formula the Bateman project normalization of the parameters he calls a, b, not the normalization he uses elsewhere). They found that for these polynomials, the Szegő condition fails.

(2) In [13], Nevai reported a conjecture of Askey that (with O(n^{−2}) errors) Szegő fails for all (α, β) ≠ (0, 0).

(3) In [1], Askey-Ismail found some explicit examples with b_{n} ≡ 0 and α > 0, and noted that the Szegő condition holds (!), so they concluded the conjecture needed to be modified.

(4) In [7], Dombrowski-Nevai proved a general result that Szegő holds when b_{n} ≡ 0 and α > 0 with errors of the form (6.3).

(5) In [3], Charris-Ismail computed the weights for Pollaczek-type examples in the entire (α, β) plane to the left of the line α = 1, and considered a class depending on an additional parameter, λ. While they did not note the consequence for the Szegő condition, their example is consistent with our picture above.

In addition, we note that in [13], Nevai proved that the Szegő condition holds if a_{n} = 1+(−1)^{n}α/n+O(n^{−2}) and b_{n} = (−1)^{n}β/n+O(n^{−2}); see also [4].

With regard to this class, here is our result in this paper:

Theorem 6.1. Suppose

∑_{n=1}^{∞}(a_{n} − 1)^{2} + b_{n}^{2} < ∞  (6.4)

lim sup_{N}(− ∑_{n=1}^{N}(a_{n} − 1 ± \frac{1}{2}b_{n})) = ∞  (6.5)

for either plus or minus. Then the Szegő condition fails at ±2.

Proof. (6.5) implies that Ā_{1}^{±}(J) = ∞ so by Theorem 4.4(ii), Z_{1}^{±}(J) = ∞. □

Remark. The same kind of argument lets us also prove the failure of the Szegő condition without assuming (6.4), and with (6.5) replaced by the slightly stronger condition that

lim sup_{N}(− ∑_{n=1}^{N}(ln(a_{n}) ± p b_{n})) = ∞  (6.6)

for some 0 ≤ p < \frac{1}{2}. For one can use the step-by-step sum rule for the weight 1±2p cos θ. (6.4) is not needed to control errors in E-sums since they have a definite sign near both +2 and −2, and it is not needed to replace ln(a) by a − 1 since (6.6) has ln(a_{n}).

These considerations yield another interesting result. One can prove Theorem 4.1 for the weight w(θ) = 1 ± 2p cos θ just as we did it for the weight 1. Since w(θ) is bounded away from zero, the corresponding Z^{±} term is finite if only if Z is. Since p < \frac{1}{2}, the corresponding eigenvalue term is finite if and only if E_{0} is. Using Theorem 4.1(a)–(d) for this w(θ), we obtain

Theorem 6.2. Let |p| < \frac{1}{2} and |q| < \frac{1}{2}.

(i) If

lim sup_{N}(− ∑_{n=1}^{N}(ln(a_{n}) + p b_{n})) > −∞

and

lim inf_{N}(− ∑_{n=1}^{N}(ln(a_{n}) + q b_{n})) = −∞

then Z(J) = ∞. (ii) If

lim inf_{N}(− ∑_{n=1}^{N}(ln(a_{n}) + p b_{n})) < ∞

and

lim sup_{N}(− ∑_{n=1}^{N}(ln(a_{n}) + q b_{n})) = ∞

then E_{0}(J) = ∞.

In particular, if a_{n} = 1, b_{n} ≥ 0, and ∑_{n=1}^{∞} b_{n} = ∞, we have Z(J) = ∞ and E_{0}(J) = ∞. On the other hand, if instead ∑_{n=1}^{∞} b_{n} < ∞, then Z(J) < ∞ and E_{0}(J) < ∞ (see [11, 10]).

Corollary 6.3. If a_{n}, b_{n} are given by (6.1), (6.2) with

lim_{n→∞} n[|E_{a}(n)| + |E_{b}(n)|] = 0  (6.7)

and 2α ± β < 0, then the Szegő condition fails at ±2.

Remarks. 1. This is intended as separate results for + and for −.

2. All we need is

lim_{n→∞} (ln N)^{−1} ∑_{n=1}^{N}(|E_{a}(n)| + |E_{b}(n)|) = 0

instead of (6.7). In particular, trace class errors can be accommodated.

Proof. If (6.7) holds,

∑_{n=1}^{N}(a_{n} − 1) ± \frac{1}{2}b_{n} = (α ± \frac{1}{2}β) ln N + o(ln N)

so (6.5) holds if 2α ± β < 0. □

As for the complementary region |β| ≤ 2α, one of us has proven (see Zlatoš [28]) the following:

Theorem 6.4 (Zlatoš [28]). Suppose |β| ≤ 2α and

a_{n} = 1 + αn^{−1} + O(n^{−1−ε})
b_{n} = βn^{−1} + O(n^{−1−ε})

for some ε > 0. Then the Szegő condition holds.

Remarks. 1. This is a corollary of a more general result (see [28]).

2. In these cases, − ∑_{n=1}^{N} ln(a_{n}) diverges to −∞. This is only consistent with (4.18) because E_{0}(J) = ∞, that is, the eigenvalue sum diverges and the two infinities cancel.

We can use these examples to illustrate the limits of Theorem 4.1:

(1) If a_{n} = 1 and b_{n} = \frac{1}{n}, then Z(J) = ∞ (by Corollary 6.3) while Ā_{0}(J) = \underline{A}_{0}(J) < ∞. Thus E_{0}(J) = ∞.

(2) If a_{n} = 1 − \frac{1}{n}, b_{n} = 0, then Z(J) = ∞ (by Corollary 6.3) Ā_{0}(J) = \underline{A}_{0}(J) = ∞, but E_{0}(J) < ∞ since J has no spectrum outside [−2, 2].

(3) If a_{n} = 1 + \frac{1}{n}, b_{n} = 0, then Z(J) < ∞ (by Theorem 6.4), but Ā_{0}(J) = \underline{A}_{0}(J) = −∞ and so E_{0}(J) = ∞.

Finally, we note that Nevai’s [13] (−1)^{n}/n theorem shows that we can have Z(J) < ∞, E_{0}(J) < ∞, and have the sums ∑ a_{n} and/or ∑ b_{n} be only conditionally and not absolutely convergent.

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Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125

E-mail address: bsimon@caltech.edu; andrej@caltech.edu