MULTIPLIERS OF DE BRANGES SPACES

B. Mark Davis and John E. McCarthy

0. Introduction

Hilbert spaces of functions on the unit circle 𝕋 that can be extended to analytic functions on the disk 𝔻 lie at the heart of harmonic analysis, from the solution of the prediction theory problems of Szegö [24], Kolmogorov [14], Helson-Szegö [13] and Helson-Sarason [12], to the success of the theory of de Branges and Rovnyak on spaces contractively contained in H^{2} in [4]. In this paper we consider certain of these latter spaces.

Let b be a fixed function in the unit ball of H^{∞}, the bounded analytic functions on 𝔻; let D_{b} = (1 − T_{b}Tb̄)^{\frac{1}{2}} , where T_{m} : H^{2} → H^{2} denotes the Toeplitz operator of multiplication by the function m followed by projection onto H^{2}, the space of squareintegrable functions on the circle whose negative Fourier coefficients vanish. Then the de Branges space H(b) is defined to be the range of D_{b}, with an inner product that makes D_{b} into a partial isometry. H(b) is a (not necessarily closed) subspace of H^{2}, on which the evaluation functionals at points in the disk are continuous.

The spaces H(b) were introduced by de Branges and Rovnyak in [5] and [6] and have been studied and utilised in various contexts, ranging from model theory [1] and kernels of Toeplitz operators [10], to exposed points in H^{1} [23] and complex function theory [22]. The structure of the spaces, however, is still not well understood; a natural question to ask is, given a specific b, what are the multipliers of H(b), i.e. what functions (necessarily in H^{∞}) multiply H(b) into itself? This question has been addressed in [16],[17],[20] and [21]; we shed some further light on it below, but the general solution is still unknown.

For any (positive) measure μ on the circle, H^{2}(μ) will denote the closure of the (analytic) polynomials in L^{2}(μ). In Section 1 we introduce the measure μ_{b} associated with b; a

Typeset by AMS-TEX 2 B. MARK DAVIS AND JOHN E. MCCARTHY

function m is a multiplier of H(b) if and only if the Toeplitz operator Tm̄, defined a priori on polynomials, is bounded on H^{2}(μ_{b}).

Let us call a measure μ Szegö if it is absolutely continuous with respect to Lebesgue measure σ on the circle, and if evaluation at any point of 𝔻 is a continuous linear functional on H^{2}(μ); any Szegö measure can be written as |h|^{2}σ, where h is some H^{2} function ([9,p.144]). If, in addition, the truncation operator Π that takes a formal Fourier series ∑_{n=−∞}^{∞} a_{n}z^{n} to ∑_{n=0}^{∞} a_{n}z^{n} is bounded on L^{2}(μ), then μ is called Helson-Szegö. In Section 2 we show that if μ = |h|^{2}σ is a Szegö measure then a necessary condition for Tm̄ to be bounded on H^{2}(μ) is that m be in the range of Th̄; in the special case that μ = |p|^{2}ρ, where p is a polynomial and ρ is Helson-Szegö, we show that this condition is also sufficient (Theorem 2.3).

In section 3 we consider the effect of adding a singular measure to an absolutely continuous one. Let T (μ) = {m ∈ H^{∞} : Tm̄ is bounded on H^{2}(μ)}, and let μ = μ_{a} + μ_{s} be the decomposition of μ into absolutely continuous and singular parts (with respect to σ); then we can classify T (μ) in terms of T (μ_{a}) when μ_{s} consists of a finite number of atoms:

Theorem 3.8. Let μ = |h|^{2}σ + ∑_{n=1}^{N} r_{n}δ_{λ_{n}}, where h is an outer function in H^{2}, and each r_{n} is positive. Then the following are equivalent:

(i) m is in T (μ). (ii) For each n, m is in T (|h|^{2}σ + δ_{λ_{n}}). (iii) (a) m is in T (|h|^{2}σ)

(b) For each n, the Fourier series of m at λ_{n} converges, to some number m(λ_{n}) say.

(c) For each n, m is in Π(λ̄_{n} − z̄)h̄H^{2}.

Moreover, condition (c) is equivalent to:

(c′) For each n,

∫_{𝕋} |Π\frac{1}{h̄(z)} \frac{m(z) − m(λ_{n})}{z − λ_{n}}|^{2}dσ(z) < ∞.

The absolutely continuous part of μ_{b} being Szegö is equivalent to the function b not being an extreme point of the unit ball of H^{∞}; whether or not this is the case introduces a dichotomy into the study of multipliers of H(b). For example, in [20] Sarason showed

MULTIPLIERS OF DE BRANGES SPACES 3

that a (non-constant) polynomial is a multiplier of H(b) if and only if b is not extreme. In section 4 we show that the polynomials are not the only functions that are always multipliers in the non-extreme case; in particular, we show:

Theorem 4.2. The H^{∞} function m is a multiplier of every H(b) space, when b is not an extreme point of the ball of H^{∞}, if and only if there is some constant c > 0 such that m̂(n) = O(exp(−c\sqrt{n})).

In Section 5 we consider the extreme point case. We prove

Theorem 5.1. If b is an extreme point of the ball of H^{∞}, and m is a non-constant multiplier of H(b), then m is a cyclic vector of S^{∗}.

In Section 6, we ask some questions, and make some conjectures, on what the multipliers of H(b) are in general.

We would like to thank Derek Westwood for valuable discussions, and Donald Sarason for many useful remarks on an earlier draft of this paper.

1. The measure μ_{b}

Let b be a fixed non-constant function in the unit ball of H^{∞}. The function \frac{1+b}{1−b} then has positive real part, so can be represented as a Herglotz integral

\frac{1 + b(z)}{1 − b(z)} = ∫_{𝕋} \frac{e^{iθ} + z}{e^{iθ} − z} dμ_{b}(e^{iθ}) + ic,

where c is a real constant and μ_{b} is a unique positive Borel measure.

For each function f in H^{2}(μ_{b}), define a function V_{b}f on 𝔻 by

(V_{b}f)(z) = (∫ \frac{f(e^{iθ})}{1 − ze^{−iθ}} dμ_{b}(e^{iθ}))(1 − b(z)).

Sarason proved in [23] that V_{b} is an isometry from H^{2}(μ_{b}) onto H(b).

For any measure μ on the circle, H^{2}(μ) decomposes into the direct sum of L^{2}(μ_{s}) and H^{2}(μ_{a}); if log(\frac{dμ_{a}}{dσ}) is integrable (with respect to σ), μ_{a} is Szegö; if log(\frac{dμ_{a}}{dσ}) is not integrable, then μ_{a} is not Szegö, and H^{2}(μ_{a}) = L^{2}(μ_{a}). For a discussion, see e.g. [2] or Considering the measure μ_{b}, notice that the absolutely continuous part of μ_{b} is

\frac{1 − |b|^{2}}{|1 − b|^{2}} σ,

(look at radial limits of ℜ\frac{1+b}{1−b}). By the above remarks, the absolutely continuous part of μ_{b} is therefore Szegö if and only if log(1 − |b|^{2}) is integrable; this integrability in turn is equivalent to b not being an extreme point of the unit ball of H^{∞} [7]. So if b is an extreme point of the ball, H^{2}(μ_{b}) = L^{2}(μ_{b}), and H^{2}(μ_{b}) has no analytic structure; if b is not an extreme point, then evaluation at a point of the disk is a continuous functional on H^{2}(μ_{b}), so this space can be thought of as having analytic extensions to the disk. This disparity is reflected by the very different behaviour of the multipliers of H(b) in the two cases.

For future reference, let us also note that μ_{b} is purely singular if and only if |b| = 1 σ-almost everywhere, i.e. if and only if b is inner.

By looking at what V_{b}^{−1} does to multiplication by m on H(b), Lotto showed in [15] that a function m in H^{∞} is a multiplier of H(b) if and only if the operator Tm̄, defined on polynomials by Tm̄(p) = Πm̄p, has a continuous extension to all of H^{2}(μ_{b}). Thus the problem of determining multipliers of H(b) is reduced to the more tractable one of finding which co-analytic Toeplitz operators are bounded on H^{2}(μ_{b}), i.e. finding T (μ_{b}).

Let <, > denote the formal inner product of two Fourier series, i.e. if f(z) ∼ ∑_{n=−∞}^{∞} a_{n}z^{n}, and g(z) ∼ ∑_{n=−∞}^{∞} b_{n}z^{n}, then < f, g >= ∑_{n=−∞}^{∞} a_{n}b̄_{n}. We shall often write < f, g > as ∫ fḡdσ, where, if necessary to avoid convergence problems, we think of this integral as lim_{r↑1} ∫ f(re^{iθ})\overline{g(re^{iθ})}dσ(θ).

Let S be the unilateral shift, which we think of as multiplication by the independent variable. Then the action of Tm̄ on a polynomial p of degree d is given by

(Tm̄p)(z) =< p, m > + < p, Sm > z + ...+ < p, S^{d}m > z^{d}.  (1.1)

2. Szegö measures

Suppose μ is a Szegö measure. Then it can be written as |h|^{2}σ for some outer function h in H^{2} (see e.g. [9]). Moreover, μ being Szegö means that the functional that assigns to

MULTIPLIERS OF DE BRANGES SPACES 5

a polynomial its zeroth coefficient is continuous on H^{2}(μ). So if Tm̄ is bounded on H^{2}(μ), the functional p ↦ (Tm̄p)(0) is bounded, which from (1.1) means there is a constant C such that

| < p, m > | ≤ C√ ∫_{𝕋} |p|^{2}dμ  (2.1)

for all polynomials p. The set of analytic functions m that satisfy (2.1) can be identified with the dual of H^{2}(μ), and we shall denote this set by H^{2}(μ)^{∗}.

In [11] it is observed that the space H^{2}(|h|^{2}σ) is the same as the space of quotients H^{2}/h, and consequently that m is in H^{2}(μ)^{∗} if and only if it is in the range of Th̄, i.e. if and only if there exists some f in H^{2} such that m = Πh̄f. Thus we get:

Proposition 2.2. A necessary condition for a function m in H^{∞} to be in T (|h|^{2}σ) is that m be in the range of Th̄.

It follows from [21] that this condition is not, in general, sufficient; it does suffice, however, if the distant future and past are at a positive angle in L^{2}(μ). Let F_{d} be the set of analytic polynomials, whose first d coefficients are zero; and let P_{d} be the set of coanalytic polynomials whose first d coefficients vanish. In this terminology, a Helson-Szegö measure is one for which P_{−1} and F_{0} are at a positive angle. In [12] Helson and Sarason showed that P_{−1} and F_{d} are at a positive angle in L^{2}(μ) if and only if μ = |p|^{2}ρ, where p is a polynomial of degree less than or equal to d, and ρ is a Helson-Szegö measure. For a measure of this form, we can prove the converse to Proposition 2.2:

Theorem 2.3. Let μ = |h|^{2}σ = |p|^{2}ρ, where h is outer, p is a polynomial and ρ is a Helson-Szegö measure. Then a function m is in T (μ) if and only if m is in the range of Th̄ intersected with L^{∞}.

Proof. Necessity follows from Proposition 2.2.

For sufficiency, let d be the degree of p. Because m is in H^{2}(μ)^{∗}, so are Sm, S^{2}m, .., and S^{d}m. Therefore there is a constant C_{1} such that, for any polynomial q,

| < q, m > | + ... + | < q, S^{d}m > | ≤ C_{1}√ ∫_{𝕋} |q|^{2}dμ.

6 B. MARK DAVIS AND JOHN E. MCCARTHY

Now, starting with a polynomial q, multiply it by m̄ : this increases the norm (in L^{2}(μ)) by at most a factor of ∥m∥_{∞}. Subtract off the terms in 1, z, ... up to z^{d} : the norm is now increased by at most a factor of ∥m∥_{∞} + C_{1}. Now apply Π. Π will annihilate the negative terms in the Fourier series of m̄q; whilst this in general will be an infinite series, and so not in P_{−1}, it is the (complex) conjugate of a function in H_{0}^{∞}, and so lies in the L^{2}(μ) closure of P_{−1} (because H^{∞} is contained in H^{2}(μ)). Therefore, applying Π can increase the norm by at most a factor of twice the cosecant of the angle between P_{−1} and F_{d}, a finite number, C_{2} say. Now add back the terms in 1, z, ..., z^{d}. The result is Tm̄q, and the norm of q has been increased by at most a factor of (C_{2}(∥m∥_{∞} + C_{1}) + C_{1}). Therefore Tm̄ is bounded on polynomials, as required. □

It follows from the proof that if μ is actually Helson-Szegö, then all of H^{∞} is in T (μ). The converse follows from the closed graph theorem:

Corollary 2.4. T (μ) = H^{∞} if and only if μ is Helson-Szegö.

Proof. If μ is Helson-Szegö, then Tm̄ is just multiplication by the bounded function m̄ followed by the bounded projection Π, so the composition is bounded.

Conversely, if all of H^{∞} is in T (μ), it is easy to check, using the closed graph theorem, that the inclusion map from H^{∞} into the bounded operators on H^{2}(μ), that sends m to Tm̄, is continuous, so there is some constant C_{3} such that ∥Tm̄∥ ≤ C_{3}∥m∥_{∞}. In particular, ∥Tz̄^{n}| is less than or equal to C_{3} for all n.

But if μ is not Helson-Szegö, then there exists some polynomial p of norm less than 1/2C_{3} for which ∥Tz̄^{n}p∥ = 1. Therefore μ must be Helson-Szegö. □

A measure |h|^{2}σ satisfies the Helson-Szegö condition if and only if |h|^{2} satisfies Muckenhoupt’s A_{2} condition, viz. the infimum over all intervals I of

(\frac{1}{|I|} ∫_{I} |h|^{2}dσ)(\frac{1}{|I|} ∫_{I} \frac{1}{|h|^{2}} dσ)

must be positive (see [9]). One can construct bounded functions h that satisfy this condition for which the essential infimum of |h| is zero; in other words the Toeplitz operator Th̄

MULTIPLIERS OF DE BRANGES SPACES 7

need not be invertible; however, its range must always contain all of H^{∞}, by combining Corollary 2.4 and Proposition 2.2.

In [21] Sarason found necessary and sufficient conditions for H^{∞} to be the multipliers of H(b) in terms of the functions b and a, where a is the outer function satisfying |a|^{2}+|b|^{2} = 1 (so h = a/(1 − b)). Combining his last condition with Corollary 2.4, we get:

Corollary 2.5. Suppose a is outer, b is in H^{∞}, and |a|^{2} + |b|^{2} = 1. Then

(i) a and b form a Corona pair, and |a|^{2} satisfies A_{2} is equivalent to (ii) |\frac{a}{1−b}|^{2} satisfies A_{2}.

3. Adding a singular measure

In general, there are no inclusion relations between T (μ_{1}+μ_{2}) and T (μ_{1}), T (μ_{2}), except for the obvious one T (μ_{1}) ∩ T (μ_{2}) ⊆ T (μ_{1} + μ_{2}). However, if μ_{1} is absolutely continuous and μ_{2} is singular, then T (μ_{1} + μ_{2}) ⊆ T (μ_{1}) (unless μ_{1} is zero, in which case T (μ_{2}) is just the constants - see 5.3).

Proposition 3.1. If μ is a measure on 𝕋 with μ_{a} non-zero, and m is any function in T (μ), then m is in T (μ_{a}).

Proof. Let χ_{a} be a measurable function that is 1 σ-almost everywhere, and 0 μ_{s}-almost everywhere. By Forelli’s lemma ([8,p43]), there exist polynomials q_{n} that tend to χ_{a} (σ + μ)-almost everywhere, and whose infinity-norms remain bounded. By hypothesis, there is some constant C such that

∫_{𝕋} |Tm̄p|^{2}dμ ≤ C ∫_{𝕋} |p|^{2}dμ

for all polynomials p. Therefore

∫_{𝕋} |Tm̄(q_{n}p)|^{2}dμ_{a} ≤ ∫_{𝕋} |Tm̄(q_{n}p)|^{2}dμ ≤ C ∫_{𝕋} |q_{n}p|^{2}d(μ_{a} + μ_{s}).  (3.2)

Since m is in T (μ), Tm̄ has a continuous extension, R say, to H^{2}(μ). Claim: χ_{a}R(χ_{a}p) = Tm̄(p) μ_{a}-almost everywhere.

Given the claim, (3.2) then yields, as n tends to infinity,

∫_{𝕋} |Tm̄p|^{2}dμ_{a} ≤ C ∫_{𝕋} |p|^{2}dμ_{a},

which is the desired conclusion.

To prove the claim, we shall show that χ_{a}R(χ_{a}p)\sqrt{\frac{dμ}{dσ}} and Tm̄(p)\sqrt{\frac{dμ}{dσ}}, both functions in L^{2}(σ), have the same Fourier coefficients, and therefore are equal σ-almost everywhere. To this end, note that

lim_{n→∞} ∫_{𝕋} |[R(q_{n}p) − R(χ_{a}p)]\sqrt{\frac{dμ}{dσ}}|^{2}dσ = 0.

Therefore

[χ_{a}R(χ_{a}p)\sqrt{\frac{dμ}{dσ}}] (l) = ∫_{𝕋} χ_{a}R(χ_{a}p)\sqrt{\frac{dμ}{dσ}}z̄^{l}dσ

= lim_{n→∞} ∫_{𝕋} χ_{a}R(q_{n}p)\sqrt{\frac{dμ}{dσ}}z̄^{l}dσ

= lim_{n→∞} ∫_{𝕋} Tm̄(q_{n}p)\sqrt{\frac{dμ}{dσ}}z̄^{l}dσ

= lim_{n→∞} ∫_{𝕋}(Πm̄q_{n}p)\overline{(\sqrt{\frac{dμ}{dσ}}z^{l})}dσ

= lim_{n→∞} ∫_{𝕋}(m̄q_{n}p)\overline{(Π\sqrt{\frac{dμ}{dσ}}z^{l})}dσ

= ∫_{𝕋}(m̄p)\overline{(Π\sqrt{\frac{dμ}{dσ}}z^{l})}dσ

= ∫_{𝕋} Tm̄(p)\sqrt{\frac{dμ}{dσ}}z̄^{l}dσ

= [Tm̄p\sqrt{\frac{dμ}{dσ}}] (l)

as required. □

Adding a point mass to a measure μ also cannot increase T (μ), even if μ is not absolutely continuous:

MULTIPLIERS OF DE BRANGES SPACES 9

Proposition 3.3. Let μ be any measure on 𝕋, δ_{λ} be the point mass at λ, for some λ on 𝕋. If m is in T (μ + δ_{λ}), then m is in T (μ).

Proof. Without loss of generality, we can assume μ(λ) = 0. By hypothesis, there is a constant C such that

∫_{𝕋} |Tm̄p|^{2}d(μ + δ_{λ}) ≤ C ∫_{𝕋} |p|^{2}d(μ + δ_{λ}).  (3.4)

Let q_{n}(z) = [\frac{1}{2}(1 + λ̄z)]^{n}; q_{n} converges to 1 at λ, and zero everywhere else. Claim: lim_{n→∞}[Tm̄((1 − q_{n})p)](ζ) = [Tm̄(p)](ζ), for all ζ ≠ λ on the circle. Given the claim, (3.4), with (1−q_{n})p instead of p, yields, in the limit as n tends to infinity,

∫_{𝕋} |Tm̄p|^{2}dμ ≤ C ∫_{𝕋} |p|^{2}dμ,

as desired.

The claim is equivalent, by writing p as a sum of monomials and applying the backward shift to m, to showing that, for all ζ ≠ λ, lim_{n→∞}[Tm̄q_{n}](ζ) = 0. Now

[Tm̄q_{n}](ζ) = \frac{1}{2^{n}} ∑_{l=0}^{n} λ̄^{l}(_{l}^{n}) ∑_{k=0}^{l} \overline{m̂(k)}ζ^{l−k}
= \frac{λ̄^{n}}{2^{n}} ∑_{k=0}^{n} \overline{m̂(k)}ζ^{n−k} ∑_{j=0}^{n−k}(_{j}^{n})(λζ̄)^{j}.  (3.5)

First, observe that if m is a monomial, m(z) = z^{k}, then the absolute value of (3.5) is less than or equal to \frac{1}{2^{n}} \{|1 + λζ̄|^{n} + ∑_{i=0}^{k}(_{i}^{n})\}, which tends to zero as n tends to infinity. So we can add any polynomial to m and not change the behaviour of (3.5) in the limit; let us therefore fix ϵ > 0, and subtract a finite number of terms from m so that ∫_{𝕋} |m|^{2}dσ < (\frac{ϵ}{C(λζ̄)})^{2}, where C(λζ̄) is the constant from Lemma 3.6.

Applying Lemma 3.6, (3.5) becomes

|(Tm̄q_{n})(ζ)| ≤ \frac{C(λζ̄)}{\sqrt{n}} ∑_{k=0}^{n} |m̂(k)|
≤ C(λζ̄)∥m∥_{2}
< ϵ.

10 B. MARK DAVIS AND JOHN E. MCCARTHY

So for large enough n, (3.5) can be made arbitrarily small, so [Tm̄q_{n}](ζ) tends to zero, as required. □

In the proof, we used the following computational lemma: Lemma 3.6. Suppose ξ is on the unit circle, but not equal to 1. Then there is a constant C(ξ) such that

|\frac{1}{2^{n}} ∑_{j=0}^{l}(_{j}^{n})ξ^{j}| ≤ C(ξ)\frac{1}{\sqrt{n}},

for all l ≤ n. Proof. The quantity on the left-hand side is a partial sum of a Fourier series, and can be written using the Dirichlet kernel as

\frac{1}{2^{n}} ∑_{j=0}^{l}(_{j}^{n})ξ^{j} = \frac{1}{2π} ∫_{0}^{2π}(\frac{1 + ξe^{−it}}{2} )^{n} \frac{Sin(l + \frac{1}{2})t}{Sin\frac{1}{2}t}dt.  (3.7)

Let θ be the argument of ξ, and let α = min(\frac{θ}{2}, \frac{2π−θ}{2} , \frac{π}{4} ). Choose n so large that n(n + 1)(\sqrt{\frac{1+Cosα}{2}})^{n} < 1. The integral in (3.7) can be written as the sum of the integral from θ − α to θ + α, and the integral over the rest of the circle. For the latter integral, |\frac{1+ξe^{−it}}{2} | ≤ \sqrt{\frac{1+Cosα}{2}}, so the modulus of the integral is less than 2(l + \frac{1}{2})(\sqrt{\frac{1+Cosα}{2}})^{n}, which is less than \frac{2}{n}. In the former integral, we have control over the second term:

| ∫_{θ−α}^{θ+α}(\frac{1 + ξe^{−it}}{2} )^{n} \frac{Sin(l + \frac{1}{2})t}{Sin\frac{1}{2}t}dt| ≤ \frac{1}{Sin\frac{α}{2}} \frac{1}{2^{n}} ∫_{0}^{2π} |1 + e^{it}|^{n}dt

= \frac{1}{Sin\frac{α}{2}} \frac{1}{2^{n}} ∫_{0}^{2π}(2 + 2Cos t)^{\frac{n}{2}} dt

= \frac{2}{Sin\frac{α}{2}} ∫_{0}^{π} |Cos^{n}u|du.

This last integral can be worked out explicitly; applying Stirling’s formula to the result, one gets that it is bounded by a constant times \frac{1}{\sqrt{n}}, as desired. □

Before stating our next theorem, we must say what we mean by Πf/h̄, when h and f are in H^{2}, and h is outer. In general, f/h̄ will not be integrable on the circle; however, 1/h̄(z) = ∑_{n=0}^{∞} β_{n}z̄^{n} is a harmonic function on the unit disk, with a power series that converges almost uniformly. Suppose f(z) = ∑_{n=0}^{∞} a_{n}z^{n}. Then f(z)/h̄(z) =

MULTIPLIERS OF DE BRANGES SPACES 11

∑_{n=0}^{∞} ∑_{m=0}^{∞} a_{n}β_{m}z^{n}z̄^{m}, where the series converges almost uniformly on the disk. Truncate this by annihilating those terms in z^{n}z̄^{m} where m is greater than n, to get the function g̃(z) = ∑_{n=0}^{∞} ∑_{m=0}^{n} a_{n}β_{m}z^{n}z̄^{m}; let g(z) = ∑_{n=0}^{∞} ∑_{m=0}^{n} a_{n}β_{m}z^{n−m} (one could think of this as the radial limit in the sense of Abel of g̃). We define Πf/h̄ to be this function g; this will agree with the usual definition when f/h̄ is summable, but will be defined (at least as an analytic function on the disk) for all h and f.

Observe that defining truncations via passage to the disk means that Πh̄Πf = Πh̄f, whenever h is analytic; so, in particular, Π\frac{1}{h̄}Πh̄f = Πf. Moreover, an analytic function f is in the range of Th̄ (i.e. can be written Πh̄k, for some H^{2} function k) if and only if Πf/h̄ is in H^{2}.

We can now say what additional restrictions the addition of a finite number of atoms to a Szegö measure imposes on the Toeplitz operators. The special case of the measure σ + 2δ_{1} (which corresponds to the function b(z) = \frac{1+z}{2} ) has been studied in [21] and [16].

Theorem 3.8. Let μ = |h|^{2}σ + ∑_{n=1}^{N} r_{n}δ_{λ_{n}}, where h is an outer function in H^{2}, and each r_{n} is positive. Then the following are equivalent:

(i) m is in T (μ). (ii) For each n, m is in T (|h|^{2}σ + δ_{λ_{n}}). (iii) (a) m is in T (|h|^{2}σ)

(b) For each n, the Fourier series of m at λ_{n} converges, to some number m(λ_{n}) say.

(c) For each n,

∫_{𝕋} |Π\frac{1}{h̄(z)} \frac{m(z) − m(λ_{n})}{z − λ_{n}}|^{2}dσ(z) < ∞.  (3.9)

(iv) (a) m is in T (|h|^{2}σ)

(b) For each n, the Fourier series of m at λ_{n} converges, to some number m(λ_{n}) say. (c′) For each n, m lies in Π(λ̄_{n} − z̄)h̄H^{2}.

Proof. We will show (i) ⟺ (ii), (ii) ⟺ (iii), and (c) ⟺ (c′).

(i) ⟹ (ii): This follows from Proposition 3.3.

(ii) ⟹ (i): Since there are only a finite number of atoms, there is some constant C such that, for all polynomials p and for each n,

∫_{𝕋} |Tm̄p|^{2}(|h|^{2}dσ + dδ_{λ_{n}}) ≤ C ∫_{𝕋} |p|^{2}(|h|^{2}dσ + dδ_{λ_{n}}).

Therefore

∑_{n=1}^{N} r_{n} ∫_{𝕋} |Tm̄p|^{2}(|h|^{2}dσ + dδ_{λ_{n}}) ≤ C ∑_{n=1}^{N} ∫_{𝕋} r_{n}|p|^{2}(|h|^{2}dσ + dδ_{λ_{n}}),

and so

∫_{𝕋} |Tm̄p|^{2}(|h|^{2}dσ + d ∑_{n=1}^{N} r_{n}δ_{λ_{n}}) ≤ \frac{max(1, ∑_{n=1}^{N} r_{n})}{min(1, ∑_{n=1}^{N} r_{n})}C ∫_{𝕋} |p|^{2}(|h|^{2}dσ + d ∑_{n=1}^{N} r_{n}δ_{λ_{n}}).

(ii) ⟹ (iii): (a) follows from Proposition 3.1.

(b): Let χ_{n} be the function that is one at λ_{n}, zero elsewhere. In H^{2}(|h|^{2}σ + δ_{λ_{n}}), the functions λ̄_{n}^{l}z^{l} − χ_{n} tend weakly to zero as l tends to infinity, so, since Tm̄ is continuous and therefore weakly continuous,

(Tm̄(λ̄_{n}^{l}z^{l}), χ_{n})_{H^{2}(|h|^{2}σ+δ_{λn})} → (Tm̄χ_{n}, χ_{n})_{H^{2}(|h|^{2}σ+δ_{λn})}.

But the left-hand side is just λ̄_{n}^{l}[Tm̄z^{l}](λ_{n}), which is \overline{m̂(0)}+\overline{m̂(1)}λ̄_{n}+...+\overline{m̂(l)}λ̄_{n}^{l}. Therefore the Fourier series of m at λ_{n} converges to (χ_{n}, Tm̄χ_{n})_{H^{2}(|h|^{2}σ+δ_{λn})}.

(c): There is some constant C such that, for all polynomials p,

|[Tm̄p](λ_{n})| ≤ C√ ∫_{𝕋} |p|^{2}(|h|^{2}dσ + dδ_{λ_{n}}).

Therefore there is some function g in H^{2}(|h|^{2}σ + δ_{λ_{n}}) such that

[Tm̄p](λ_{n}) = (p, g)_{H^{2}(|h|^{2}σ+δ_{λn})}.

g can be written as g′ + αχ_{n}, for some g′ in H^{2}(|h|^{2}σ), and g′ in turn can be written as k/h, for some k in H^{2}. Therefore we have

< p, \frac{m}{1 − λ̄_{n}z} > =< p, h̄k > + ᾱp(λ_{n}),

and so

< p, \frac{m − α}{1 − λ̄_{n}z} > =< p, h̄k > .

Setting p(z) = z^{l}, this says that the positive Fourier coefficients are the same, i.e.

\frac{m − α}{1 − λ̄_{n}z} = Πh̄k.

Letting l tend to infinity, one also gets that α = m(λ_{n}). Therefore

Π\frac{1}{h̄(z)} \frac{m(z) − m(λ_{n})}{z − λ_{n}} = −Π\frac{λ_{n}}{h̄} Πh̄k
= −λ_{n}k,

and this is square-integrable, as required.

(iii) ⟹ (ii): It is sufficient to prove that there exists some constant C such that

|[T_{m̄−m̄(λ_{n})}p](λ_{n})| ≤ C√ ∫_{𝕋} |p|^{2}|h|^{2}dσ  (3.10)

for all polynomials p, for then the norm of Tm̄ on H^{2}(|h|^{2}σ + δ_{λ_{n}}) is less than or equal to (C + |m(λ_{n})| + ∥Tm̄∥_{H^{2}(|h|^{2}σ)}).

The right-hand side of (3.10) is C| < ph, ph > |, and the left-hand side is

| < p, \frac{m − m(λ_{n})}{1 − λ̄_{n}z} > | = | < ph, \frac{1}{h̄} \frac{m − m(λ_{n})}{1 − λ̄_{n}z} > |

= | < ph, Π\frac{1}{h̄} \frac{m − m(λ_{n})}{1 − λ̄_{n}z} > |.

The supremum of this over all polynomials with | < ph, ph > | = 1 is just

√ ∫_{T} |Π\frac{1}{\overline{h(z)}} \frac{m(z) − m(λ_{n})}{1 − λ̄_{n}z}|^{2}dσ,

which is finite as required.

(c) ⟹ (c′): As remarked earlier, condition (3.9) is equivalent to saying that

\frac{m(z) − m(λ_{n})}{z − λ_{n}} = Πh̄(z)k(z),

for some function k in H^{2}. Therefore

m(z) − m(λ_{n}) = (z − λ_{n})Πh̄(z)k(z)

= Π(z̄ − λ̄_{n})(−λ_{n}z)Πh̄(z)k(z)

= Π(z̄ − λ̄_{n})h̄(z)\{(−λ_{n})[zk(z) − \frac{1}{\overline{ĥ(0)}} < zk(z), h(z) >]\}.

Therefore m is in Π(λ̄_{n} − z̄)h̄H^{2}.

(c′) ⟹ (c): Suppose m = Π(z̄ − λ̄_{n})h̄f, for some f in H^{2}. Then

m(z) + λ̄_{n} < f, h >= (z − λ_{n})Πh̄[−λ_{n}S^{∗}f].  (3.11)

Noting that the k^{th} partial sum of the Fourier series of m at λ_{n} is −λ_{n} < h̄f, 1−λ̄_{n}z^{k+1} >, (3.11) implies that

\frac{m(z) − m(λ_{n})}{z − λ_{n}} = Πh̄[−λ_{n}S^{∗}f],

from which (3.9) follows. □

Remark: The problem in trying to generalise the theorem to an infinite number of atoms is that one loses control of the norms when trying to localise: a sufficient condition for Tm̄ to be bounded on H^{2}(|h|^{2}σ + ∑ r_{n}δ_{λ_{n}}) is that ∥Tm̄∥_{H^{2}(|h|^{2}σ+δ_{λn})} ≤ C for all n; a necessary condition is that ∥Tm̄∥_{H^{2}(|h|^{2}σ+r_{n}δ_{λn})} ≤ C for all n. One can construct examples that show that the former condition is not necessary; we do not know any examples that prove that the latter condition is insufficient.

Nevertheless, this (strong) sufficient condition extends to all singular measures, and the following result enables us, in the next section, to characterise universal multipliers:

Theorem 3.12. Suppose that μ = μ_{a} + μ_{s} is a measure on 𝕋, and that there is a set D, dense in the support of μ_{s}, and a constant C, such that ∥Tm̄∥_{H^{2}(μ_{a}+δ_{λ})} ≤ C for all λ in D. Then m is in T (μ).

Proof. Let ν_{k} be a sequence of atomic measures, with atoms lying in D, that converge weak-star to μ_{s} (such a sequence exists because any continuous function that vanishes on

MULTIPLIERS OF DE BRANGES SPACES 15

D must be annihilated by μ_{s}). By the argument in the proof of (ii) ⟹ (i) in Theorem 3.8,

∥Tm̄∥_{H^{2}(μ_{a}+ν_{k})} ≤ \frac{max(1, ∥ν_{k}∥)}{min(1, ∥ν_{k}∥)}C ≤ C′,

for some C′.

For any polynomial p,

∫_{𝕋} |Tm̄p|^{2}dμ = lim_{k→∞} ∫_{𝕋} |Tm̄p|^{2}d(μ_{a} + ν_{k})

≤ C^{′2} lim_{k→∞} ∫_{𝕋} |p|^{2}d(μ_{a} + ν_{k})

= C^{′2} ∫_{𝕋} |p|^{2}dμ,

so m is in T (μ), as required. □

4. Universal Multipliers

In Section 2, we defined the dual H^{2}(μ)^{∗} to be the set of analytic functions whose formal (H^{2}) inner product with a polynomial is bounded by the H^{2}(μ)-norm of the polynomial, i.e. the set of f for which

| < p, f > | ≤ C√ ∫_{𝕋} |p|^{2}dμ.

If μ = |h|^{2}σ is Szegö, this means there is some g in H^{2}(μ) = H^{2}/h such that

< p, f > = (p, g)_{H^{2}(μ)}
= ∫_{𝕋} p\overline{h̄(hg)}dσ,

so f is in Πh̄H^{2} (which is the range of Th̄, where, if h is not in H^{∞}, Th̄ is thought of as mapping H^{2} into the set of functions analytic on 𝔻); conversely, any f in Πh̄H^{2} is in H^{2}(μ)^{∗}.

A multiplier of H^{2}(μ)^{∗} is a function m such that mf lies in H^{2}(μ)^{∗} whenever f does; from the preceding paragraph, it coincides in the |h|^{2}σ case with multipliers of ranTh̄. The multipliers of H^{2}(μ)^{∗} are the same as the co-analytic Toeplitz operators on H^{2}(μ) (though in general H^{2}(μ_{b})^{∗} is not identical to H(b)).

16 B. MARK DAVIS AND JOHN E. MCCARTHY

Proposition 4.1. Let μ be a measure on the circle. Then m is a multiplier of H^{2}(μ)^{∗} if and only if m is in T (μ). Proof. If m is a multiplier of H^{2}(μ)^{∗}, then, by the closed graph theorem, multiplication by m, M_{m}, is a bounded linear operator on H^{2}(μ)^{∗}. The proof then consists of first observing that H^{2}(μ)^{∗} really is the dual of the Hilbert space H^{2}(μ) (since any functional is determined by its action on polynomials), and secondly that the adjoint of M_{m} is Tm̄, because if either Tm̄ or M_{m} is bounded, then

< M_{m}^{∗}p, f > = lim_{r↑1} ∫ p(re^{iθ})\overline{m(re^{iθ})f(re^{iθ})}dσ(θ)
=< Tm̄p, f > .

So if m is a multiplier of H^{2}(μ)^{∗}, Tm̄ is bounded; and if Tm̄ is bounded, mf is in H^{2}(μ)^{∗} whenever f is. □

In [18], it is proved that a function m is a multiplier of H^{2}(μ)^{∗} for every Szegö measure μ if and only if m is in every H^{2}(μ)^{∗}, which in turn holds if and only if there is some constant c such that m̂(n) = O(exp(−c\sqrt{n})). These functions are multipliers of every H(b) when b is non-extreme: if μ_{b} is absolutely continuous, this follows from Proposition 4.1; if it isn’t, we can use the fact that m̂(n) decays so rapidly to apply Theorem 3.12. Theorem 4.2. The H^{∞} function m is a multiplier of every H(b) space, when b is not an extreme point of the ball of H^{∞}, if and only if there is some constant c > 0 such that m̂(n) = O(exp(−c\sqrt{n})). Proof. Necessity follows from the preceding paragraph.

Let us fix some m such that m̂(n) = O(exp(−c\sqrt{n})); we must show that for every measure μ of the form |h|^{2}σ + μ_{s}, m lies in T (μ). By Theorem 3.12, it is sufficient to show that for every outer function h there is a constant C such that

∥Tm̄∥_{H^{2}(|h|^{2}σ+δ_{λ})} ≤ C;

this in turn is equivalent to showing

sup_{λ∈𝕋} ∥\frac{m(z) − m(λ)}{z − λ}∥_{H^{2}(|h|^{2}σ)^{∗}} < ∞.

MULTIPLIERS OF DE BRANGES SPACES 17

A straightforward computation shows that there is some constant M independent of λ such that the n^{th} Fourier coefficient of \frac{m(z)−m(λ)}{z−λ} is smaller in modulus than Me^{−c\sqrt{n}/2}. Therefore the family of functionals \{\frac{m(z)−m(λ)}{z−λ} : λ ∈ 𝕋\} is pointwise bounded on H^{2}(|h|^{2}σ), and, by the Banach-Steinhaus theorem, equicontinuous, as desired. □

Remark: In [23], Sarason shows that for every non-extreme b, μ_{λb} is absolutely continuous for almost every λ on the unit circle; since H(λb) = H(b), this result, together with Proposition 4.1, also proves 4.2.

5. The extreme point case

The next theorem states that the multipliers of H(b), when b is extreme, cannot be non-cyclic for S^{∗} (the adjoint of the unilateral shift). A function f is non-cyclic for S^{∗} if and only if there is an inner function I such that f lies in H^{2} ⊖ IH^{2} (this follows immediately from Beurling’s theorem). In particular, no function that is a polynomial of an inner function is cyclic. For an analytic condition on whether a function is cyclic, in terms of pseudo-continuations, see [19]. We note that this theorem has also been proved independently by Lotto and Sarason [17]. Theorem 5.1. If b is an extreme point of the ball of H^{∞}, and m is a non-constant multiplier of H(b), then m is a cyclic vector of S^{∗}. Proof. Our hypotheses are that H^{2}(μ_{b}) = L^{2}(μ_{b}), and that Tm̄ extends to a continuous operator on H^{2}(μ_{b}). Let Z be the operator of multiplication by the independent variable on L^{2}(μ_{b}). The commutator of Tm̄ with Z is bounded, and on any polynomial p,

(Tm̄Z − ZTm̄)p =< p, S^{∗}m > ⊮.

By hypothesis, the norm of this latter (constant) function is bounded by some constant C times the norm of p, so

| < p, S^{∗}m > | ≤ C√ ∫_{𝕋} |p|^{2}dμ_{b}.

Letting n = S^{∗}m, this says that there is some function g in L^{2}(μ_{b}), such that

∫_{𝕋} pn̄dσ = ∫_{𝕋} pḡdμ_{b}.

Since (n̄σ −ḡμ_{b}) annihilates the polynomials, the F. and M. Riesz theorem says there must be some function k in H_{0}^{1} so that (n̄σ − ḡμ_{b}) = kσ, i.e.

ḡμ_{b} = (n̄ − k)σ.  (5.2)

Because g is in L^{2}(μ_{b}), and \frac{dμ_{b}}{dσ} is not log-integrable (with respect to σ), neither is \frac{dḡμ_{b}}{dσ}. But if n were not cyclic for S^{∗}, there would be some inner function I such that ΠĪn = 0; In̄ would then be in H^{∞}, so so would I(n̄−k), and the modulus of this, which is the same as the modulus of \frac{dḡμ_{b}}{dσ}, would have to be log-integrable. Therefore n, and so m, is cyclic, as desired. □

As an immediate consequence, we get the result of Sarason mentioned earlier, that a polynomial is a multiplier of H(b) if and only if b is not extreme. From the proof we also get a new proof of a theorem of Lotto [16], that if b is inner, H(b) has no non-constant multipliers:

Porism 5.3. If b is inner, H(b) has no non-constant multipliers.

Proof. If b is inner, μ_{b} is purely singular, so (5.2) can only hold if both sides are zero, so n must be zero, so m must be a constant. □

6. Questions

Many questions remain open. For example:

If μ = |h|^{2}σ is absolutely continuous, when is T (μ) = ranTh̄ ∩ H^{∞}? In [3], it is proved that if log|h_{0}/h_{1}| is in BMO, the space of functions of bounded mean oscillation on the circle, then T (|h_{0}h_{1}|σ) contains T (|h_{0}|^{2}σ) ∩ T (|h_{1}|^{2}σ). It follows from this, and Theorem 2.3, that a sufficient condition that m be in T (|z − 1|σ) is that m = T_{(\overline{z−1})^{(1/2)+ϵ}}f, for any ϵ > 0 and some f in H^{2}. Can the epsilon be removed, to get agreement with the necessary condition from Proposition 2.2?

Suppose one adds an atomic measure with an infinite number of atoms, ∑ r_{n}δ_{λ_{n}}, to |h|^{2}σ. Is the condition ∥Tm̄∥_{H^{2}(|h|^{2}σ+r_{n}δ_{λn})} ≤ C for all n a sufficient condition for m to be in T (|h|^{2}σ + Σr_{n}δ_{λ_{n}})? What if one adds a non-atomic singular measure? What is even

MULTIPLIERS OF DE BRANGES SPACES 19

a reasonable conjecture for a necessary and sufficient condition for Tm̄ to be bounded on H^{2}(μ_{a} + μ_{s})?

The space M(ā), defined to be the range of Tā, is intimately connected to H(b) - see [21]. Does it equal the intersection over all inner functions J of H^{2}(μ_{Jb})^{∗}?

When is μ_{λb} absolutely continuous for all λ on the circle? (This question is related to characterising exposed points in H^{1} - see [23]).

We note that Lotto and Sarason have studied the multipliers of de Branges spaces in [17], concentrating mainly on the extreme point case, and have obtained some very interesting results.

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Department of Mathematics University of California at Berkeley Berkeley CA 94720 USA Department of Mathematics Indiana University Bloomington IN 47405 USA