A CANONICAL FACTORIZATION FOR MEROMORPHIC HERGLOTZ FUNCTIONS ON THE UNIT DISK AND SUM RULES

FOR JACOBI MATRICES

BARRY SIMON

Abstract. We prove a general canonical factorization for meromorphic Herglotz functions on the unit disk whose notable elements are that there is no restriction (other than interlacing) on the zeros and poles for their Blaschke product to converge and there is no singular inner function. We use this result to provide a significant simplification in the proof of Killip-Simon [5] of their result characterizing the spectral measures of Jacobi matrices, J, with J − J_{0} Hilbert-Schmidt. We prove a nonlocal version of Case and step-by-step sum rules.

1. Introduction

The canonical factorization for Nevanlinna functions, f, that is, functions obeying

sup_{0<r<1} ∫_{0}^{2π} log_{+}|f(re^{iθ})| \frac{dθ}{2π} < ∞  (1.1)

says that [8]

f = BOI  (1.2)

Here B is a Blaschke product of the zeros {z_{k}}_{k=1}^{∞} of f, B(z) = ∏_{k=1}^{∞} b(z, z_{k}) with

b(z, z_{k}) = \frac{|z_{k}|}{z_{k}} \frac{z_{k} − z}{1 − z̄_{k}z}  (1.3)

(with |z_{k}|/z_{k} interpreted as −1 if z_{k} = 0); b(z, 0) = z). O is the outer factor, that is,

O(z) = exp(∫_{0}^{2π} \frac{e^{iθ} + z}{e^{iθ} − z} log(|f(e^{iθ})|) \frac{dθ}{2π} )  (1.4)

and I is a singular inner function, that is, I has no zeros and lim_{r↑1}|I(re^{iθ})| = 1 for Lebesgue a.e. θ.

Date: November 12, 2003.

∗ Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125. E-mail: bsimon@caltech.edu. Supported in part by NSF grant DMS-0140592.

One of our goals in this paper is to prove an analog of this result for a rather different class of functions:

Theorem 1.1. Let f be a meromorphic function on 𝔻 = {z | |z| < 1} so that Im f > 0 (resp. < 0) if Im z > 0 (resp. < 0). Then for a.e. θ ∈ [0, 2π),

f(e^{iθ}) ≡ lim_{r↑1} f(re^{iθ})  (1.5)

exists and is a.e. nonzero. Moreover,

∫_{0}^{2π}|log|f(e^{iθ})||^{p} \frac{dθ}{2π} < ∞  (1.6)

for all p < ∞. If p_{1}^{+} < p_{2}^{+} < ⋯ are the poles of f in [0, 1), z_{1}^{+} < z_{2}^{+} < ⋯ the zeros there and p_{1}^{−} > p_{2}^{−} > ⋯ and z_{1}^{−} > z_{2}^{−} > ⋯ the poles and zeros in (−1, 0), then

B(z) = lim_{n→∞} ∏_{j=1}^{n} b(z, z_{j}^{+})b(z, p_{j}^{+})^{−1}b(z, z_{j}^{−})b(z, p_{j}^{−})^{−1}  (1.7)

converges uniformly on compact subsets on 𝔻\{p_{j}^{±}} and

f(z) = ±B(z) exp(∫_{0}^{2π} \frac{e^{iθ} + z}{e^{iθ} − z} log|f(e^{iθ})| \frac{dθ}{2π} )  (1.8)

where the ± sign in front is sgn(f(0)) if f(0) ≠ 0 and + if f(0) = 0.

Three aspects of this theorem should be emphasized. First, unlike the Nevanlinna function case, where one proves ∑(1 − |z_{j}|) < ∞ and needs this to assure convergence of the product defining that B, our f’s can have arbitrary interlaced poles and zeros on (−1, 1) since the interlacing will assure convergence. Second, (1.8) has no singular inner part. This is connected with (1.6).

Third, Theorem 1.1 is related to but distinct from results on functions, f, which obey Im f > 0 on all of 𝔻 (usually phrased instead for f’s with Re f > 0) — namely, the celebrated result of Kolmogorov [6] that any such f lies in H^{p} for all p < 1, and the less well-known result of Smirnov [11] that such functions have no singular inner part. These results and conformal mapping of half-disks to the disk provide some information on the functions of Theorem 1.1, but say nothing about the Blaschke product (1.7) nor about a singular inner component supported at {±1}. It should, however, be emphasized that Kolmogorov’s result and conformal mapping do imply (1.5) and a stronger result than (1.6), namely

∫_{0}^{2π}(|f(e^{iθ})| + |f(e^{iθ})|^{−1})^{p} \frac{dθ}{2π} < ∞  (1.9)

for p < \frac{1}{2}. The example f(x) = −(1 + z)^{−2} shows that (1.9) is optimal in that the integral is infinite for p = \frac{1}{2}.

Also interesting is that the proof of Theorem 1.1 has fewer technicalities; for example, the product defining B will actually converge on all of ℂ_{+} = {z | Im z > 0}, so |B(e^{iθ})| = 1 will be immediate without the separate argument needed in the Nevanlinna function case (see [8, p. 312]).

While this theorem is interesting for its own sake, we found it in the course of simplifying some recent results of Killip-Simon [5] on Jacobi matrices.

We will consider Jacobi matrix spectral and inverse spectral theory, mainly using the notation of Killip-Simon [5]. A Jacobi matrix, J, is a tridiagonal selfadjoint matrix

J = (\begin{matrix}b_{1} & a_{1} & 0 & 0 & ⋯ \\ a_{1} & b_{2} & a_{2} & 0 & ⋯ \\ 0 & a_{2} & b_{3} & a_{3} & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix})  (1.10)

with a_{n} > 0 and sup_{n}|a_{n}|+|b_{n}| < ∞. J_{0} is the special case with b_{n} ≡ 0, a_{n} ≡ 1.

μ_{J} (often just μ) is the spectral measure for J and vector δ_{1}, that is,

m(E) ≡ ∫ \frac{dμ(x)}{x − E} = ⟨δ_{1}, (J − E)^{−1}δ_{1}⟩  (1.11)

for E ∈ ℂ_{+} = {E | Im E > 0}. The a’s and b’s are then the recursion coefficients for the orthonormal polynomials for μ, that is, if p_{n}(x) is defined recursively by

xp_{n}(x) = a_{n+1}p_{n+1}(x) + b_{n+1}p_{n}(x) + a_{n}p_{n−1}(x)  (1.12)

then p_{n}(x) = γ_{n}x^{n}+ lower order with γ_{n} = (a_{1} … a_{n})^{−1} > 0 and

∫ p_{n}(x)p_{ℓ}(x) dμ(x) = δ_{nℓ}  (1.13)

Note that for J = J_{0}, the dμ is

dμ_{0} = (2π)^{−1}χ_{[−2,2]}(E)\sqrt{4 − E^{2}} dE  (1.14)

Write

dμ(x) = f(x) dx + dμ_{s}(x)  (1.15)

where dμ_{s}(x) is singular with respect to Lebesgue measure. Our main goal here is to present a simple self-contained proof of the main theorem of Killip-Simon [5], but in doing so, we will prove a general nonlocal sum that has other applications:

Theorem 1.2. J − J_{0} is Hilbert-Schmidt, that is,

∑_{n=1}^{∞}[2(a_{n} − 1)^{2} + b_{n}^{2}] < ∞  (1.16)

if and only if dμ obeys the following four conditions:

(i) (Blumenthal-Weyl)

supp[dμ] = [−2, 2] ∪ {E_{j}^{−}}_{j=1}^{N_{−}} ∪ {E_{j}^{+}}_{j=1}^{N_{+}}  (1.17)

where

E_{1}^{−} < E_{2}^{−} < ⋯ < −2 and E_{1}^{+} > E_{2}^{+} > ⋯ > 2  (1.18)

are discrete pure points of dμ, and if N^{+} = ∞ (resp. N^{−} = ∞), then E_{j}^{+} → 2 (resp. E_{j}^{−} → −2).

(ii) (Lieb-Thirring)

∑_{j=1}^{N_{+}}|E_{j}^{+} − 2|^{3/2} + ∑_{j=1}^{N_{−}}|E_{j}^{−} + 2|^{3/2} < ∞  (1.19)

(iii) (Quasi-Szegő)

∫_{−2}^{2} log(f(E))\sqrt{4 − E^{2}} dE > −∞  (1.20)

(iv) (Normalization)

μ(ℝ) = 1  (1.21)

An important thing is that, like Szegő’s theorem [2, 4, 9, 12], the only condition on the singular part of μ_{s} is the one given by (1.21).

As in [5], this theorem follows immediately from a sufficiently general form of a sum rule.

Given two measures μ and ν on a compact Hausdorff space, define their relative entropy by

S(μ | ν) = \begin{cases}−∞ & if μ is not ν a.c. \\ − ∫ log(\frac{dμ}{dν}) dμ & if μ is ν a.c. \end{cases}

Let μ_{0} be the measure given by (1.14) and let

Q(μ) = −S(μ_{0} | μ)

We will need the fact ([5, Corollary 5.3]) that

μ_{n} ⟶^{w} μ ⇒ Q(μ) ≤ lim inf Q(μ_{n})  (1.22)

(semicontinuity of the entropy).

Also define two functions: G on (0, ∞) by

G(a) = a^{2} − 1 − 2 log(a)  (1.23)

and F on (2, ∞) by

F (E) = \frac{1}{4} (β^{2} + β^{−2} − log β^{4})  (1.24)
E = β + β^{−1};  β > 1  (1.25)

The P_{2} sum rule of Killip-Simon [5] says

Q(μ) + ∑_{n,±} F (|E_{n}^{±}|) = \frac{1}{4} ∑_{n=1}^{∞} b_{n}^{2} + \frac{1}{2} ∑_{n=1}^{∞} G(a_{n})  (1.26)

and the point of [5] is that it always holds in the sense that all terms are nonnegative and either both sides are finite and equal or both are infinite. Since the right side of (1.26) is finite if and only if b_{n} and a_{n}−1 are ℓ_{2} and the left side is finite if and only if (i)–(iv) of Theorem 1.2 hold, (1.26) immediately implies Theorem 1.2.

As in [5], one proves (1.26) by showing LHS ≤ RHS and RHS ≤ LHS. The proof in [5] of the first inequality is conceptually and technically simple — essentially, one proves (1.26) when only finitely many b_{n} and a_{n} − 1 are nonzero and uses (1.22). The proof in the other direction in [5] is involved and opaque; Simon-Zlatoš [10] improved this part, but their argument still has several pages of calculations. Instead, our key technical input will be Theorem 1.1.

As in [5] and related to ideas going back at least as far as Szegő [12], we will map ℂ ∪ {∞}\[−2, 2] to the open unit disk 𝔻 by the inverse of the map z ↦ (z + z^{−1}). This allows us to define a natural function M on 𝔻 associated to the m of (1.11) by

M(z) = −m(z + z^{−1})  (1.27)

M is meromorphic in 𝔻 and has poles at the points (β_{j}^{±})^{−1} defined by (1.25). Since m(E) ∼ (−E)^{−1} for E ∼ ∞,

M(z) = z + O(z^{2})  (1.28)

near z = 0. Since Im M > 0 if Im z > 0, between any two poles of M is a zero and the zeros (other than z = 0) are easily seen to be related to the eigenvalues of the Jacobi matrix J^{(1)} obtained by removing the top row and leftmost column in J. Let M_{1} be the M-function for J^{(1)}. Then our main technical result is the following representation for M:

Theorem 1.3. For any Jacobi matrix, Im M and Im M_{1} have boundary values a.e. on ∂𝔻 and up to Lebesgue measure zero sets:

S ≡ {θ | Im M(θ) ≠ 0} = {θ | Im M_{1}(θ) ≠ 0}  (1.29)

Moreover, for any p < ∞,

∫_{S} | log(\frac{Im M}{Im M_{1}} )|^{p} \frac{dθ}{2π} < ∞  (1.30)

Most importantly,

a_{1}M = zB^{+}(z)B^{−}(z) exp[\frac{1}{4π} ∫_{0}^{2π} log(\frac{Im M(θ)}{Im M_{1}(θ)})[\frac{e^{iθ} + z}{e^{iθ} − z} ] dθ]

(1.31)

where B^{+} and B^{−} are suitable convergent products of the zeros and poles.

This theorem not only yields Theorem 1.2 but has other applications [1].

For the case of interest, |S| = 2π and then (1.31) is intended literally. If |S| < 2π, we interpret Im M(θ)/ Im M_{1}(θ) for θ ∉ S by

\frac{Im M(θ)}{Im M_{1}(θ)} ≡ lim_{r↑1} \frac{Im M(re^{iθ})}{Im M_{1}(re^{iθ}) + a_{1}^{−2}(r^{−1} − r) sin θ}  (1.32)

Part of the theorem is that with this extended definition, log(Im M(θ)/ Im M_{1}(θ)) ∈ L^{p}(dθ/2π) for all p < ∞.

The connection between Theorem 1.1 and Theorem 1.3 is evident; essentially, the later is implied by the former and the formula already used in [5]:

\frac{Im M(θ)}{Im M_{1}(θ)} = a_{1}^{2}|M(e^{iθ})|^{2}  (1.33)

In Section 2, we will discuss Blaschke products of zeros and poles, and in Section 3, we will prove Theorems 1.1 and 1.3. In Section 4, we will see that the Taylor coefficients of the logs of the two sides of (1.31) are the step-by-step sum rules of [5] and [10], and use them following [5] and [10] to prove (1.26). Section 5 has a few closing remarks, including a brief indication of the analog of M for orthogonal polynomials on the unit circle.

I would like to thank Leonid Golinskii for a comment that got me thinking in the right direction, and Rowan Killip for useful comments. Our joint work extending [5] to the continuum case (in preparation) was an important guide to my thinking.

2. Alternating Blaschke Products

Given real numbers 0 < p_{1} < z_{1} < ⋯ < p_{n} < z_{n} < ⋯ with lim p_{n} = lim z_{n} = 1, we define a sequence B_{j}(z) of alternating Blaschke factors as follows:

c_{2j}(z) = \frac{z_{j} − z}{1 − z_{j}z}  j = 1, 2, …  (2.1)
c_{2j−1}(z) = \frac{1 − p_{j}z}{p_{j} − z}  j = 1, 2, …  (2.2)

B_{j}(z) = ∏_{k=1}^{j} c_{k}(z)  (2.3)

so B_{j} has poles inside 𝔻 at p_{1}, p_{2}, …, p_{ℓ}, and zeros at z_{1}, …, z_{m} where ℓ = m = j/2 if j is even and ℓ = (j + 1)/2, m = (j − 1)/2 if j is odd. Here is the convergence theorem we will prove in this section:

Theorem 2.1. For all z ∈ ℂ\{p_{j}} ∪ {z_{j}^{−1}} ∪ {1} ≡ S, B_{∞}(z) = lim_{j→∞} B_{j}(z) exists and the convergence is uniform on compact subsets of S. B_{∞} is nonvanishing on S\{z_{j}} ∪ {p_{j}^{−1}}. For |z| = 1 and z ≠ 1, |B_{∞}(z)| = 1. If log B_{∞} is defined on ℂ_{+} to be continuous with lim_{y↓0} B_{∞}(iy) > 0, then

|Im(log B_{∞}(z))| ≤ π  (2.4)

Remarks. 1. This should be viewed as a souped-up version of the fact that if a_{n} → 0, |a_{n}| ≥ |a_{n+1}|, and (−1)^{n}a_{n} ≥ 0, then ∑_{n=1}^{N} a_{n} converges.

2. There is no regular Blaschke condition that ∑_{j=1}^{∞}(1 − |z_{j}|) < ∞ because we can instead use

∑_{j=1}^{∞} |z_{j} − p_{j}| < ∞  (2.5)

Killip has remarked that this is sufficient for all parts of the theorem to be true, save (2.4).

3. (2.4) holds because there is a cancellation of phases from the zeros and poles.

4. This theorem is a close relative to the one in Gesztesy-Simon [3] that if p_{1} < z_{1} < ⋯ with p_{n} → ∞, then lim_{n→∞} ∏_{j=1}^{n}(1 − z/z_{j})/(1 − z/p_{j}) converges and defines a Herglotz function. See also Levin [7].

Proof. First, let −1 < x < 0. All numerators and denominators in (2.1) and (2.2) are positive so we can unambiguously take logs. Since c_{k}(z) → 1 as k → ∞ and

\frac{∂}{∂a}(\frac{a − x}{1 − ax}) = \frac{1 − x^{2}}{(1 − ax)^{2}} > 0

if a ∈ (0, 1), we have

|log c_{j}(x)| ≥ |log c_{j+1}(x)| j = 1, 2
log c_{j}(x) → 0  as j → ∞
(−1)^{j+1} log c_{j}(x) > 0

so ∑ log c_{j}(x) is a convergent alternating series and thus, for x ∈ (−1, 0), lim_{j→∞} B_{j}(x) exists.

Noting that

|\frac{z_{j} − z}{p_{j} − z} − 1| = \frac{|z_{j} − p_{j}|}{|p_{j} − z|}  |\frac{1 − p_{j}z}{1 − z_{j}z}^{−1}| = \frac{|z_{j} − p_{j}| |z|}{|1 − z_{j}z|}

one sees that

|B_{2j}(z)| ≤ exp(S(z) ∑_{k=1}^{j} |z_{j} − p_{j}|)

where

S(z) = max_{α∈(p_{1},1)} (\frac{1}{|α − z|} , \frac{|z|}{|1 − αz|})

Since

|c_{2j−1}(z)| ≤ max_{α∈(p_{1},1)} \frac{|1 − αz|}{|z − α|}

we see that on ℂ\[p_{1}, p_{1}^{−1}], sup_{j}|B_{j}(z)| < ∞ and so we get the uniform convergence. A small modification handles the points in [p_{1}, p_{1}^{−1}] other than {p_{j}} ∪ {z_{j}^{−1}}. Since |b_{ℓ}(z)| = 1 if |z| = 1 and B_{j} → B_{∞} uniformly, |B_{∞}(z)| = 1 on ∂𝔻\{−1}.

That proves everything but (2.4). To prove that, note that

∏_{j=1}^{k} \frac{z_{j} − z}{p_{j} − z} \frac{1}{p_{k+1} − z} = ∑_{ℓ=1}^{k} \frac{α_{ℓ}}{p_{ℓ} − z}

where each α_{ℓ} is positive, and so it defines a Herglotz function. Thus B_{2j−1} is a ratio of Herglotz functions and so |Im[log(B_{2j−1})]| < π. Taking limits, we get (2.4). □

3. Factorization of Meromorphic Herglotz Functions

Here we will prove Theorems 1.1 and 1.3.

Proof of Theorem 1.1. Since an analytic function g(z) has points with Im g < 0 in any neighborhood of a polar singularity or zero, f is analytic and nonvanishing in 𝔻_{+} ≡ 𝔻 ∩ ℂ_{+} and 𝔻_{−} = 𝔻 ∩ {z | Im z < 0}. Similarly, since each half-neighborhood of a zero or pole of degree larger than 1 has points with Im g < 0, we conclude that f has only simple poles and zeros on (−1, 1).

Since Im f is 0 on (−1, 1), we have \frac{∂ Im f}{∂y} ≥ 0 so, by the Cauchy-Riemann equations, \frac{∂ Re f}{∂x} = f′(x + i0) ≥ 0. That implies the zeros and poles interlace. Using slightly different labelling of zeros and poles from Section 1, let z_{0} be the smallest zero in [0, 1) and p_{1}^{+} < z_{1}^{+} < ⋯ and p_{1}^{−} > z_{1}^{−} > ⋯ the poles and zeros in (z_{0}, 1) and (−1, z_{0}). Define

B(z) = b(z, z_{0})B_{+}(z)B_{−}(z)  (3.1)

where B_{+} (resp. B_{−}) are the product defined by Theorem 2.1 for z_{j}^{+}, p_{j}^{+} (resp. B_{−} is a similar product for z_{j}^{−}, p_{j}^{−}). It is easy to see that this agrees with slightly reordered product in (1.7).

Define

G(z) = f(z)B(z)^{−1}  (3.2)

Then G(z) is

(i) analytic and nonvanishing on 𝔻 since we have explicitly removed all the zeros and poles.

(ii) has

lim_{r↑1}|\frac{G(re^{iθ})}{f(re^{iθ})}| = 1  (3.3)

for θ ≠ 0, π since |B(e^{iθ})| = 1. (iii) We have that in 𝔻

|Im log(G(z))| ≤ 4π  (3.4)

For this holds in 𝔻 ∩ ℂ_{+} since f and ±b are Herglotz functions (for which Im log g(z) ∈ (0, π)) and B_{±} obey (2.4).

G(z) as a nonvanishing function has an analytic logarithm. By (3.4) and the boundedness of the conjugate harmonic function map (see Rudin [8]), we have that

log G(z) ∈ H^{p} all p < ∞  (3.5)

so that

log G(z) = \frac{1}{2π} ∫_{0}^{2π} log|G(e^{iθ})| \frac{e^{iθ} + z}{e^{iθ} − z} dθ  (3.6)

Since (3.3) holds, we have that for all p < ∞,

\frac{1}{2π} ∫_{0}^{2π}|log[|f(e^{iθ})|]|^{p} dθ < ∞

and, by (3.1) and (3.6), (1.8) holds. □

Proof of Theorem 1.3. We use the continued fraction expansion for M (see [5]):

M(z) = (z + z^{−1} − b_{1} − a_{1}^{2}M_{1}(z))^{−1}  (3.7)

Since Im(−M(z)^{−1}) = Im M(z)/|M(z)|^{2}, we see for z ∈ 𝔻,

\frac{Im M(z)}{|M|^{2}} = a_{1}^{2} Im M_{1}(z) − Im(z + z^{−1})  (3.8)

so, since Im M(z) and Im M_{1}(z) are nonzeros on 𝔻 ∩ ℂ_{+}, we have there that

a_{1}^{2}|M(re^{iθ})|^{2} = \frac{Im M(re^{iθ})}{[Im M_{1}(re^{iθ}) + a_{1}^{−2}(r^{−1} − r) sin(θ)]}  (3.9)

Taking boundary values and using the fact that M is Herglotz on 𝔻 so that lim|M(e^{iθ})| is finite and nonzero for a.e. θ (by Theorem 1.1), we see that (1.29) holds and that on S,

a_{1}^{2}|M(e^{iθ})|^{2} = \frac{Im M(e^{iθ})}{Im M_{1}(e^{iθ})}  (3.10)

Off S, this formula holds if we interpret the right side by (1.32). We have proven (1.31) by combining (1.8) and (3.10) □

We will see in the next section that Taylor coefficients of (1.31) are a step-by-step sum rule. We can think of (1.31) as a kind of global (i.e., not just Taylor coefficients at z = 0) step-by-step sum rule. It will be used in [1].

4. The P_{2} Sum Rule

Our goal in this section is to prove (1.26) in the strong sense indicated in Section 1, which then, as explained there, implies Theorem 1.2. The arguments follow those in [5] and [10] and are presented here because our subsidiary results are stronger here and because, by specializing to P_{2}, some details are easier. The initial step is a general step-by-step sum rule:

Theorem 4.1. Let J be an arbitrary Jacobi matrix which obeys condition (i) of Theorem 1.2 (e.g., J −J_{0} is compact). Let J^{(1)} be the matrix with one row and column removed. Then

(i) If E_{n}^{±} and E_{n}^{(1)±} are the eigenvalues for J and J^{(1)}, then with F given by (1.24)/ (1.25),

∑ |F (E_{n}^{±}) − F (E_{n}^{(1)±})| < ∞  (4.1)

where F (E_{n}^{±}) is interpreted as 0 if n > N_{±}, and similarly for F (E_{n}^{(1)±}).

(ii) log(Im M_{1}(θ)/ Im M(θ)) (interpreted as (1.32) if Im M(θ) = 0) lies in all L^{p} for p < ∞.

(iii)

\frac{1}{4π} ∫_{0}^{2π} log(\frac{Im M_{1}(θ)}{Im M(θ)}) sin^{2} θ dθ+∑_{n,±} F (E_{n}^{+})−F (E_{n}^{±(1)}) = \frac{1}{4} b_{1}^{2}+\frac{1}{2} G(a_{1})

(4.2)

with G given by (1.23).

Remark. This proof is essentially the same as Theorem 4.2 of [5] given our Theorem 1.3.

Proof. By (1.28) and (3.7), one finds

log(\frac{M(z)}{z}) = 1 + b_{1}z + (\frac{1}{2} b_{1}^{2} + a_{1}^{2} − 1)z^{2} + O(z^{3})  (4.3)

Computing the Taylor series for B_{∞}(z) up to order 2, one finds by taking logs of (1.31) and looking at the zeroth and second Taylor coefficient that

\frac{1}{4π} ∫_{0}^{2π} log(\frac{Im M_{1}(e^{iθ})}{Im M(e^{iθ})})dθ−∑_{n,±} [log|β_{n}^{±}(J)|−log|β_{n}^{±}(J^{(1)})|] = − log(a_{1})

(4.4)

and

−\frac{1}{2π} ∫_{0}^{2π} log(\frac{Im M_{1}(e^{iθ})}{Im M(e^{iθ})}) cos 2θ dθ

+ \frac{1}{2} ∑_{n,±}\{β_{n}^{±}(J)^{2} − β^{±}(J_{n})^{2} − (β_{n}^{±}(J)^{−2} + (β^{±}(J_{n}))^{−2}\} = \frac{1}{2} b_{1}^{2} + (a_{1}^{2} − 1)

(4.5)

In these formulae, the ∑_{n,±} converge because |E_{n}^{±}| ≤ |E_{n}^{(1)±}| ≤ |E_{n+1}^{±}| so the sums are alternating sums since log|β| and β^{2} − β^{−2} are monotone. (4.3) is obtained by adding \frac{1}{2}× (4.5) to (4.4). □

Remark. For cases where M has a meromorphic continuation past ∂𝔻, (4.2) was proven by Killip-Simon [5], and if ∫ − log[Im M(e^{iθ})] sin^{2} θ\frac{dθ}{4π} < ∞, it was proven by Simon-Zlatoš [10]. In this generality, it is new.

Proof of (1.26). We can iterate (4.2) to get

∫_{0}^{2π} \frac{1}{4π} log(\frac{Im M(e^{iθ}, J^{(ℓ)})}{Im M(e^{iθ})}) sin^{2} θ dθ

+ ∑_{n,±} F (E_{n}^{±}) − F (E_{n}^{±(ℓ)}) = ∑_{j=1}^{ℓ}[\frac{1}{4} b_{j}^{2} + \frac{1}{2} G(a_{j})]

(4.6) with J^{(ℓ)} the matrix with the top ℓ rows and leftmost ℓ columns removed. In particular, if J^{(ℓ)} = J_{0} for some ℓ, we have (1.26) in that case.

Given a general J, let J_{ℓ} be defined by having a values (a_{1}, a_{2}, …, a_{ℓ−1}, 1, 1, …) and b values (b_{1}, …, b_{ℓ}, 0, 0 …) so (J_{ℓ})^{(ℓ)} = J_{0}. Thus (1.26) holds for J_{ℓ}. The limit of the right side of (1.26) converges as ℓ → ∞. By (1.22), the left sides obey an inequality. The result is

RHS of (1.26) ≤ LHS of (1.26)  (4.7)

Suppose J is such that RHS of (1.26) < ∞. Then each term is finite, so we can separate the logs and infinite sums, and (4.6) becomes

RHS of (1.26) for J = [ ∑_{j=1}^{ℓ} \frac{1}{4} b_{j}^{2} + \frac{1}{2} G(a_{j})] + RHS of (1.26) for J^{(ℓ)}

≥ ∑_{j=0}^{ℓ} \frac{1}{4} b_{j}^{2} + \frac{1}{2} G(a_{j})

since the right-hand side of (1.26) is always nonnegative. Taking ℓ → ∞, we find

RHS of (1.26) ≥ LHS of (1.26)

□

5. Remarks

The ideas of this paper also provide a proof of Theorem 9.14 of [5] that when J − J_{0} is trace class, we have that the Jost function u(z; J) has no singular inner component. This proof avoids Lemma 9.13 and its several pages of argument (plus the need to prove (2.50) of [5]).

For, as in [5], u(z; J) is a Nevanlinna function, so it has a factorization (1.26)

u = B_{0}O_{0}S_{0}  (5.1)

Similarly, u_{1} ≡ u(z; J^{(1)}) has a factorization

u_{1} = B_{1}O_{1}S_{1}  (5.2)

But the m-function obeys

a_{1}M(z) = \frac{zu_{1}(z; J)}{u_{0}(z; J)}

Recognizing that B_{1}/B_{0} is exactly the product zB_{+}B_{−} for the canonical product for M and that this product has no singular part, we see that

S_{0} = S_{1}  (5.3)

Let

u_{n} = B_{n}O_{n}S_{n}  (5.4)

be the Nevanlinna factorization for u(z; J^{(n)}). Since J is trace class, B_{n} → 1, and using estimates from Section 2 of [5], u_{n} → 1, O_{n} → 1. It follows that S_{n} → 1. But S_{0} = S_{1} = ⋯ = S_{n} so S_{0} ≡ 1. This theme will be further pursued in [1].

The second topic concerns analogs of Theorem 1.3 in the case of orthogonal polynomials on the unit circle [2, 4, 9, 12]. If dμ = \frac{w(θ)}{2π}dθ+dμ_{s} is a measure on ∂𝔻, w(θ) is the analog of Im M(e^{iθ}). Just as dμ in the real line case is associated to a set of Jacobi parameters, a_{n} > 0 and b_{n} ∈ ℝ, dμ in the circle case is associated to a sequence {α_{0}, α_{1}, α_{2}, … } of complex Verblunsky coefficients (also called Schur parameters or reflection coefficients). There is a measure, dμ_{1}, associated with Verblunsky coefficients {α_{1}, α_{2}, … } analogous to μ_{J^{(1)}} in the real line case and a weight w_{1}. There is a natural analog of the M-function, viz., the Carathéodory function

F (z) = ∫ \frac{e^{iθ} + z}{e^{iθ} − z} dμ(e^{iθ})

and w(θ) = lim_{r↑1} Re F (re^{iθ}).

Alas, there is nothing as simple as (3.9) and no sense in which boundary values of |F (re^{iθ})| are related to w(θ)/w_{1}(θ). F is simply not the proper analog of M for single-step sum rules. Instead, the function that replaces a_{1}M in the step-by-step sum rules for this case is

(δ_{0}D)(z) = \frac{1 − α_{0}f}{ρ_{0}} \frac{1 − zf_{1}}{1 − zf}  (5.5)

where f is the Schur function for dμ, that is, F = (1 + zf)/(1 − zf), f_{1} is the Schur function of dμ_{1}, and ρ_{0} = (1 − |α_{0}|^{2})^{1/2}. This δ_{0}D is an outer function and that produces suitable step-by-step sum rules. The details will appear in [9].

References

[1] D. Damanik and B. Simon, Jost functions and Jost solutions for Jacobi matrices, in preparation.

[2] Ya. L. Geronimus, Polynomials Orthogonal on a Circle and Their Applications, Amer. Math. Soc. Translation 1954 (1954), no. 104, 79 pp.

[3] F. Gesztesy and B. Simon, On the determination of a potential from three spectra, Amer. Math. Soc. Transl. (2) 189 (1999), 85–92.

[4] U. Grenander and G. Szegő, Toeplitz Forms and Their Applications, 2nd edition, Chelsea, New York, 1984; 1st edition, University of California Press, Berkeley-Los Angeles, 1958.

[5] R. Killip and B. Simon, Sum rules for Jacobi matrices and their applications to spectral theory, Ann. of Math. 158 (2003), 253–321.

[6] A. Kolmogorov, Sur les fonctions harmoniques conjuguées et les séries de Fourier, Fund. Math. 7 (1925), 24–29.

[7] B. Ja. Levin, Distribution of Zeros of Entire Functions, revised edition, American Mathematical Society, Providence, RI, 1980.

[8] W. Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill, New York, 1987.

[9] B. Simon, Orthogonal Polynomials on the Unit Circle, AMS Book Series, expected 2004.

[10] B. Simon and A. Zlatoš, Sum rules and the Szegő condition for orthogonal polynomials on the real line, to appear in Comm. Math. Phys.

[11] V.I. Smirnov, Sur les valeurs limites des fonctions régulières á l’intérieur d’un cercle, J. Société Phys.-Math. de Léningrade 2 (1929), 22–37.

[12] G. Szegő, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., Vol. 23, American Mathematical Society, Providence, RI, 1939; 3rd edition, 1967.