ALGEBRAS GENERATED BY TWO BOUNDED HOLOMORPHIC FUNCTIONS

MICHAEL I. STESSIN PASCAL J. THOMAS

Abstract. We study the closure in the Hardy space or the disk algebra of algebras generated by two bounded functions, of which one is a finite Blaschke product. We give necessary and sufficient conditions for density or finite codimension (of the closure) of such algebras. The conditions are expressed in terms of the inner part of some function which is explicitly derived from each pair of generators. Our results are based on identifying z-invariant subspaces included in the closure of the algebra.

1. Introduction

Let g be an inner function in the unit disk. It is possible to prove that if h ∈ H^{2} satisfies the condition h(z_{1}) = h(z_{2}) for every pair z_{1}, z_{2} ∈ 𝔻 such that g(z_{1}) = g(z_{2}), then h is in the H^{2}-closure of the algebra of polynomials in g, ℂ[g], and vice versa (the proof rests on the fact that, in the case where g(0) = 0, the self-mapping g of the circle is Lebesgue measure-preserving). In fact, this result holds for a class of functions much broader than inner functions. It means that the H^{2}-closure of ℂ[g] is determined by the family of level sets of the function g. In particular, if g is a Riemann mapping of 𝔻 onto a polynomially convex domain Ω with a reasonable boundary, the algebra ℂ[g] is dense in H^{2}(𝔻) (on this topic, the interest reader may consult [5]).

It is natural to ask if a similar result holds for algebras generated by two bounded functions. In particular, if a pair of functions g_{1}, g_{2} separates points of the unit disk, under what condition is the algebra ℂ[g_{1}, g_{2}] dense in H^{2}?

The case of closures of algebras generated by two disk-algebra functions (closure is taken in the disk-algebra metric) was intensively studied (cf [11], [1], [10]). Some conditions for the density of such an algebra were found (we will discuss some of those results in comparison with those obtained in the present paper in section 7 below).

In a special case when both g_{1} and g_{2} are polynomials the result similar to the one for algebras with one generator holds. Namely, the following Theorem follows directly from Proposition 1 of [3].

Theorem A. Let p_{1} and p_{2} be two polynomials such that |p′_{1}(z)| + |p′_{2}(z)| > 0 for all z ∈ ℂ and their degrees are mutually prime. Then the algebra \overline{ℂ[p_{1}, p_{2}]} has finite codimension in H^{2}. If, in addition, p_{1} and p_{2} separate points of the unit disk, then ℂ[p_{1}, p_{2}] is dense in H^{2}.

01991 AMS subject classification: primary 30D55, secondary 46E25, 30D50

0The second-named author would like to thank the University at Albany, Harvard University, and Brown University for their hospitality during the completion of this work.

The conditions that the derivatives of p_{1} and p_{2} do not vanish simultaneously and the degrees of p_{1} and p_{2} are mutually prime are too restrictive. As we will show below, they could be replaced with one much less restrictive condition.

If the generators are not polynomials, the condition of point separation is not sufficient for the density of the algebra in H^{2}. The following simple example illustrates this. Let χ(z) = exp\{\frac{z+1}{z−1}\} be the singular inner function determined by the unit point mass at 1, and ψ(z) = zχ(z). Clearly, χ and ψ separate points of the unit disk. At the same time, every polynomial in χ and ψ belongs to the subspace L = {1} + χH^{2}, where {1} is the one-dimensional subspace of H^{2} consisting of constants, and χH^{2} is the z-invariant subspace generated by χ. Since the codimension of \overline{χH^{2}} is infinite, this implies that ℂ[χ, ψ] is not dense in H^{2}.

In this example both generators have non-trivial singular parts. What happens if one of them is analytic in the closed disk? The following argument pertains to the case when one of the generators is a Blaschke product of order two. It shows that even in this case the H^{2} codimension of the closure of an algebra could be infinite, while its generators separate points of the unit disk. The study of this simple case also gives the flavor of the work which we carry out in the body of the paper, where we deal with the case where one generator is a Blaschke product of any finite order n ≥ 2.

Let |a| < 1 and B(z) = czφ(z), where φ(z) = \frac{a−z}{1−āz} is a Möbius involution of the disk, and c a constant of modulus 1. Given any z ∈ 𝔻, there is at most another point w of the disk such that B(w) = B(z), given by w = φ(z). There is exactly one z_{0} ∈ 𝔻 such that z_{0} = φ(z_{0}), and it is the only zero of B′. In the simplest case where B(z) = z^{2}, φ(z) = −z and z_{0} = 0.

Any function h ∈ H^{p}(𝔻) which satisfies h∘φ(z) = h(z) can be rewritten h = h_{1}∘B, where h_{1} is well-defined from the set-theoretic point of view because of the hypothesis, is holomorphic outside of the critical point of B by taking local inverses, and is actually holomorphic everywhere by removing the isolated singularity. If p = ∞, h_{1} is clearly bounded. If p < ∞, since the mapping from the unit circle to itself induced by B is C^{∞} with nonvanishing derivative and 2-to-1, we have h_{1} ∈ H^{p}(𝔻). We then write h ∈ H^{p}(B).

Since B(0) = 0, the composition operator C_{B} (C_{B}(f) = f ∘ B) is an isometry of H^{p} into H^{p}. Therefore, \overline{ℂ[B]} = H^{p}(B), where ℂ[B] stands for the polynomials in B, and the closure is taken either with respect to H^{p} convergence when p < ∞, or to bounded pointwise convergence when p = ∞.

In particular, let g ∈ H^{∞}(𝔻), then g + g ∘ φ and

Γ(g)(z) := \frac{g(z) − g ∘ φ(z)}{z − φ(z)}

are in H^{∞}(B) (the possible singularity is easily seen to be removable). Furthermore,

g − g ∘ φ = 2g − (g + g ∘ φ)

shows that g − g ∘ φ ∈ ℂ[H^{∞}(B), g] ⊂ \overline{ℂ[B, g]}.

We claim that the following statement is true.

Proposition. Let 0 < p < ∞ and g ∈ H^{∞}(𝔻). When the closure is taken with respect to H^{p}, \overline{ℂ[B, g]} = \overline{ℂ[B]} + I(g)H^{p}(𝔻) = H^{p}(B) + I(g)H^{p}(𝔻), where I(g) is the inner factor in the canonical inner-outer factorization of Γ(g).

The proof of this result is as follows. Let h ∈ H^{p}(𝔻). Then

h(z)Γ(g)(z) = \frac{1}{2}(h(z) + h ∘ φ(z))Γ(g)(z) + \frac{1}{2} \frac{h(z) − h ∘ φ(z)}{z − φ(z)}(g(z) − g ∘ φ(z)).

By applying the above decomposition to the terms of a sequences of polynomials h_{ν} tending to h in H^{p}, it is then easy to see that the first term is in \overline{ℂ[B]} ⊂ \overline{ℂ[B, g]}, while \frac{h(z)−h∘φ(z)}{z−φ(z)} is multiplied by g −g ∘φ, which has been show above to be in \overline{ℂ[B, g]}. This proves that Γ(g)H^{p}(𝔻) ⊂ \overline{ℂ[B, g]}, and \overline{Γ(g)H^{p}(𝔻)} = I(g)H^{p}(𝔻), by Beurling’s theorem.

Conversely, consider a monomial of the form M = g^{α}B^{β}. If α = 0, then M ∈ ℂ[B]. If α ≥ 1,

g^{α} = (\frac{1}{2}(g − g ∘ φ) + \frac{1}{2}(g + g ∘ φ))^{α}

= \frac{1}{2^{α}} ∑_{l=0}^{α} ( \begin{matrix}α \\ l \end{matrix} ) (g − g ∘ φ)^{l}(g + g ∘ φ)^{α−l} ∈ H^{∞}(B) + (g − g ∘ φ)H^{∞}(B),

as can be seen by splitting the last sums into terms with l even or odd respectively. Now g − g ∘ φ = (z − φ)Γ(g) ∈ Γ(g)H^{∞}(𝔻).

This proves that ℂ[B, g] ⊂ ℂ[B]+I(g)H^{p}(𝔻). To prove the corresponding inclusion for the H^{p} closures, it will be enough to show that \overline{ℂ[B]} + I(g)H^{p}(𝔻) is closed in H^{p}.

Each term in the sum is a closed subspace. For any f ∈ H^{p}(𝔻), write

f = π_{1}(f) + π_{2}(f) := \frac{1}{2}(f + f ∘ φ) + \frac{1}{2}(f − f ∘ φ).

Then the maps π_{1} and π_{2} are continuous from H^{p}(𝔻) to itself, π_{1} is a projection onto \overline{ℂ[B]} = H^{p}(B), and H^{p}(B) ⊂ Ker π_{2}. Suppose that f = lim_{n}(x_{n} + y_{n}), with x_{n} ∈ H^{p}(B) and y_{n} ∈ I(g)H^{p}(𝔻). Then

π_{2}(f) = lim_{n→∞} (π_{2}(x_{n}) + π_{2}(y_{n})) = lim_{n→∞} π_{2}(y_{n}),

which proves the existence of the last limit. Since Γ(g) ∘ φ = Γ(g), we have I(g) ∘ φ = I(g), and if y_{n} = I(g)w_{n}, with w_{n} ∈ H^{p}(𝔻), we have

π_{2}(y_{n}) = I(g)(w_{n} − w_{n} ∘ φ) ∈ I(g)H^{p}(𝔻),

therefore π_{2}(f) ∈ I(g)H^{p}(𝔻) and f ∈ \overline{ℂ[B]} + I(g)H^{p}(𝔻), q.e.d.

Corollary. ℂ[B, g] is dense in H^{p} if and only if Γ(g) is an outer function.

In the case where B(z) = z^{2} and Γ(g) is not an outer function, it is easy to find an annihilator for \overline{ℂ[B]} + I(g)H^{p}(𝔻) directly. Here φ(z) = −z. By Beurling’s theorem, there exists h ∈ H^{q}(𝔻) such that ∫_{∂𝔻} h\overline{f} = 0, for any f ∈ I(g)H^{p}(𝔻). We claim h_{1} := h − h ∘ φ is the annihilator we are looking for. Indeed, for any f ∈ I(g)H^{p}(𝔻),

∫_{∂𝔻}(h ∘ φ)\overline{f} = ∫_{∂𝔻} h\overline{f ∘ φ} = 0,

since f ∘ φ ∈ I(g)H^{p}(𝔻). So ∫_{∂𝔻} h_{1}\overline{f} = 0.

Furthermore, if f ∈ H^{p}(B), then f ∘ φ = f and

∫_{∂𝔻}(h − h ∘ φ)\overline{f} = ∫_{∂𝔻} h\overline{f} − ∫_{∂𝔻}(h ∘ φ)\overline{f} = ∫_{∂𝔻} h\overline{f} − ∫_{∂𝔻} h\overline{f ∘ φ} = 0.

The case of a general Blaschke product B of degree 2 can be reduced (as far questions of density and the computation of I(g) are concerned) to the case of z^{2} by using a Möbius map from the disk to itself taking the unique critical point of B to 0. We omit the details.

If B(z) = z^{2} and g(z) = χ(z) = exp \frac{z+1}{z−1} as above, then a computation shows that Γ(χ) has no zeroes, and χ is continuous up to the boundary and non zero for any ζ ∈ ∂𝔻 \ {1}. It is then easy to see that Γ(χ) is continuous on ∂𝔻 \ {+1, −1}, so that any singular inner factor of Γ(χ) would have to have its singular measure concentrated on those two points; but Γ(χ) admits a finite non-zero radial limit at each of those points, which precludes any singular inner factor. Thus, ℂ[z^{2}, χ(z)] is dense in H^{2}.

On the other hand, still for B(z) = z^{2}, if we pick g(z) = zχ(z^{2}), then Γ(g) = χ(z^{2}), which is singular inner. Thus, ℂ[B, g] is not dense in H^{2}. At the same time z^{2} and g(z) obviously separate points of 𝔻 (which follows also from the fact that Γ(g) does not vanish inside 𝔻). Moreover, by the above Proposition, for every function f ∈ H^{2} which is orthogonal to the z-invariant subspace χH^{2}(𝔻) the function h(z) = zf(z^{2}) is orthogonal to ℂ[B, g], so \overline{ℂ[B, g]} has infinite codimension in H^{2}.

Notations and statements of results

In this paper we study algebras with two generators, one of which is a finite Blaschke product of degree n:

B(z) := ∏_{j=1}^{n} \frac{a_{j} − z}{1 − ā_{j}z} , (1)

where a_{1}, ..., a_{n} ∈ 𝔻. Given a point z ∈ 𝔻, there are exactly n points in the unit disk (counting multiplicities), z := φ_{0}(z), φ_{1}(z), ..., φ_{n−1}(z) such that B(φ_{j}(z)) = B(z), j = 1, ..., n − 1. For f ∈ H^{∞} we define Γ_{B}(f) by

Γ_{B}(f)(z) = \frac{∏_{j=1}^{n−1}(f(z) − f ∘ φ_{j}(z))}{∏_{j=1}^{n−1}(z − φ_{j}(z))}.

In what follows we usualy omit the subscript for Γ_{B} and write Γ(f) instead of Γ_{B}(f). It will be shown in section 2 that for all f ∈ H^{∞}, Γ(f) is analytic and, since it is obviously bounded, is an H^{∞} function. Moreover, for all 0 < p ≤ ∞ the mapping f ⟼ Γ(f) can be extended as a continuous mapping from H^{p} to H^{p/(n−1)}.

The main result of this paper is the following Theorem.

Theorem B. Let A stand for the algebra of functions ℂ[B, g] where g ∈ H^{∞}. The closure of A, denoted \overline{A}, is taken with respect to the H^{p} norm when p < ∞, and with respect to bounded pointwise convergence when p = ∞.

1. For every p > 0 the closure of A in H^{p} contains the z-invariant subspace Γ(g)H^{p}.

2. \overline{A} = H^{p} if and only if Γ(g) is outer.

3. \overline{A} has finite codimension in H^{p} if and only if the inner part of Γ(g) is a finite Blaschke product.

As a corollary to the proof of Theorem B we obtain the following generalization of Theorem A (though p_{1} is not a Blaschke product, Γ_{p_{1}}(p_{2}) below can be defined in a similar way. The details may be found in section 4).

Theorem C. Let p_{1} and p_{2} be two polynomials. If Γ_{p_{1}}(p_{2}) is not identically equal to zero, then the algebra \overline{ℂ[p_{1}, p_{2}]} has finite codimension in H^{2}. Furthermore, ℂ[p_{1}, p_{2}] is dense in H^{2} if and only if Γ_{p_{1}}(p_{2}) does not vanish in the open unit disk.

In the case when Γ(g) ≡ 0, Theorem B (as well as Theorem C) does not give much information about the size of the closure of the algebra. It is shown in section 5 below that the identical vanishing of Γ(g) implies a strong functional relation between B and g. The following Theorem gives a necessary and sufficient condition for the identical vanishing of Γ(g).

Theorem D. The function Γ(g) vanishes identically in the unit disk if and only if there exist a function φ from the disk to itself and a discrete subset A of the disk such that for any z ∈ 𝔻 \ A, #φ^{−1}(z) > 1, such that B = B_{1} ∘ φ and g = g_{1} ∘ φ, where B_{1} and g_{1} are functions defined on the disk.

A direct corollary to this result is

Theorem D′. The function Γ(g) vanishes identically if and only if the closure of ℂ[B, g] does not contain a non-trivial z-invariant subspace.

As another corollary to Theorem D we obtain the following result.

Corollary. If the order of the Blaschke product B is a prime number, then, whenever the bounded analytic function g is not in H^{∞}[B], Γ(g) does not vanish identicaly.

Finally, though the disk algebra A(𝔻̄) is not our prime focus, our methods yield similar disk algebra results. It is easily seen that if g is a disk algebra function, Γ(g) is also in the disk algebra. The following Theorem holds.

Theorem E.

1. Let g ∈ A(𝔻̄). Then the closure of ℂ[B, g] in A(𝔻̄) contains Γ(g)A(𝔻̄).

2. ℂ[B, g] is dense in A(𝔻̄) if and only if Γ(g) has no zeros in the closed unit disk.

3. \overline{ℂ[B, g]} has finite codimension in A(𝔻̄) if and only if Γ(g) has no singular inner factor and only a finite number of zeros in the closed unit disk.

The structure of this paper is as follows. Section 2 is devoted to some auxiliary results related to level sets of the Blaschke product B. Here we introduce the function Γ(g) and prove its analyticity. In section 3 we prove the “necessary” part of Theorem 1. In section 4 we show how Theorem C follows from Theorem B. In this section we also give an estimate of the number of generators of a closed submodule of H^{2} over a polynomial algebra expressed in terms of the function Γ. Section 5 deals with the case when Γ(g) ≡ 0. Theorems D and D′ are proved in this section. In section 6 we discuss z-invariant subspaces contained in the algebra’s closure. Section 7 is devoted to applications to the disk algebra case. Here we also compare our results with previous ones obtained by J. Wermer and R. G. Blumenthal without the assumption that one of the generators be a finite Blaschke product. Finally, in section 8 we conclude the proofs of Theorems B and E.

Acknowledgements. The authors would like to thank John Wermer for stimulating discussions and the references [4], [12], and Emma Previato for pointing out and explaining [2]. We thank the referee for his careful comments and corrections.

2. Auxiliary results

2.1. Level sets of B. For any z ∈ \overline{𝔻}, the set B^{−1}(B(z)) can be written as \{z =: φ_{0}(z), φ_{1}(z), …, φ_{n−1}(z)}, which are the n solutions (possibly equal) of a polynomial equation of degree n; the maps φ_{j} may be chosen holomorphic in z near any point z^{0} such that φ_{j}(z^{0}) is a simple zero of B − B(z^{0}) (equivalently, such that B′(z^{0}) ≠ 0), but cannot be chosen to be globally continuous.

Notice in particular that B′ cannot vanish on the boundary ∂𝔻 of the disc, and that, since they all map the circle into itself and never coincide, the maps (φ_{j}, 1 ≤ j ≤ n − 1) can be ordered by circular order, and thus are all individually well-defined in a neighborhood of the unit circle. Explicitly, we can write φ_{j}(θ) = exp(iα_{j}(θ)), where θ < α_{j}(θ) < θ + 2π. A priori, those choices only hold locally. Then label the n − 1 distinct points of the unit circle {φ_{1}(1), …, φ_{n−1}(1)} by demanding α_{j}(0) < α_{j+1}(0), 1 ≤ j ≤ n − 2. Because B′(e^{iθ}) ≠ 0 for any θ, we never have lim_{θ→θ_{0}} α_{j}(θ) = lim_{θ→θ_{0}} α_{j+1}(θ), so the set

U_{j} := {θ ∈ [0, 2π) : α_{j}, α_{j+1} ∈ C^{0}[0, θ] and ∀θ′ ∈ [0, θ], α_{j}(θ) < α_{j+1}(θ)}

is both open and closed, so is equal to the whole interval [0, 2π) and passing to the limit and using B′(e^{iθ}) ≠ 0 again, we will have

2π < α_{j}(2π) < α_{j+1}(2π) < 4π, 1 ≤ j ≤ n − 2,

and therefore α_{j}(2π) = α_{j}(0) + 2π, 0 ≤ j ≤ n − 1, and we have liftings of continuous maps.

Furthermore, the quantities |φ_{j}(e^{iθ}) − φ_{k}(e^{iθ})|, 0 ≤ θ ≤ 2π, 0 ≤ j ≠ k ≤ n − 1, are uniformly bounded away from 0.

Near a multiple point z^{0}, so that B − B(z_{0}) has a zero of order N, the φ_{j} can be expressed as series in (z − z^{0})^{1/N} (so they are actually continuous at the point z^{0}, if not in any neighborhood of it) (cf. for example [7]).

Symmetric polynomial (resp. rational) functions of φ_{j}(z), 0 ≤ j ≤ n − 1 are welldefined and holomorphic (resp. meromorphic) on the whole disc. For a multiple point z^{0}, if A ⊂ {0, …, n − 1} is such that φ_{j}(z^{0}) = φ_{k}(z^{0}) for any j, k ∈ A, and φ_{j}(z^{0}) ≠ φ_{k}(z^{0}) for any j ∈ A, k ∉ A, then the symmetric polynomial functions of (φ_{j}(z), j ∈ A) are well-defined and holomorphic in some neighborhood of z^{0} (as can be seen by factoring the equation).

In the special case that is of interest to us, since φ_{0}(z) = z is always globally welldefined, the same is true of the symmetric functions of (φ_{j}(z), 1 ≤ j ≤ n − 1). More precisely, we have the following easy lemma. From now on, we shall always assume that B vanishes at the origin, so in (1) we have a_{1} = 0. This is no loss of generality as it can be obtained by composing our functions by a Möbius map of the disk.

Lemma 1. Any symmetric polynomial (resp. rational function, resp. holomorphic function in the disk) in φ_{j}, 0 ≤ j ≤ n − 1, can be written as a polynomial (resp. rational function) in B(z).

Any symmetric polynomial (resp. rational function) in φ_{j}, 1 ≤ j ≤ n − 1, can be written as a polynomial (resp. rational function, resp. holomorphic function in the disk) in z and B(z).

Proof. Denote the elementary symmetric functions in n (resp n − 1) indeterminates by

σ_{k}(X_{0}, … X_{n−1}) := ∑_{A⊂{0,... ,n−1},#A=k} ∏_{j∈A} X_{j},
resp. σ′_{k}(X_{1}, … X_{n−1}) := ∑_{A⊂{1,... ,n−1},#A=k} ∏_{j∈A} X_{j}.

It is clear that the σ_{k}(φ_{0}(z), …, φ_{n−1}(z)) depend only on B(z). The polynomial equation which admits {φ_{0}(z), …, φ_{n−1}(z)} as solutions is

∏_{j=1}^{n}(a_{j} − X) = B(z) ∏_{j=1}^{n}(1 − ā_{j}X).

Since a_{1} = 0, the coefficient of X^{n} is exactly (−1)^{n}, and all the coefficients of X^{m}, 0 ≤ m ≤ n, are affine functions of B(z), which proves the first assertion.

To prove the second assertion of the lemma, it will be enough to show that the σ′_{k} are polynomials in X_{0} and σ_{k}. But

σ_{k}(X_{0}, … X_{n−1}) = X_{0}σ′_{k−1}(X_{1}, … X_{n−1}) + σ′_{k}(X_{1}, … X_{n−1}),

with σ_{0} = σ′_{0} = 1, so the relation σ′_{k} = σ_{k} − X_{0}σ′_{k−1} allows us to conclude by induction.

2.2. Auxiliary functions. For any f ∈ O(𝔻), define

D(f)(z) := ∏_{j=1}^{n−1} (f(z) − f ∘ φ_{j}(z)) .

This is symmetric in (φ_{j}(z), 1 ≤ j ≤ n − 1), and, therefore, by Lemma 1 is globally holomorphic in z. In particular, D(z) := ∏_{j=1}^{n−1} (z − φ_{j}(z)) is a function which vanishes at z^{0} if and only if B′(z^{0}) = 0 (it can be seen to be a discriminant).

Define Γ(f)(z) := D(f)(z)/D(z). This is clearly meromorphic in the disc, but actually each singularity is removable. Indeed, near any point z^{0} where for some j, φ_{j}(z^{0}) = z^{0}, we have

f(z) = ∑_{m≥0} α_{m}(z − z^{0})^{m}, f(φ_{j}(z)) = ∑_{m≥0} α_{m}(φ_{j}(z) − z^{0})^{m},

where the second series converges in a neighborhood of z^{0} since |φ_{j}(z) − z^{0}| = |φ_{j}(z) − φ_{j}(z^{0})| ≤ C|z − z^{0}|^{1/N}. Therefore,

f(z) − f(φ_{j}(z)) =

(z − φ_{j}(z)) ∑_{m≥1} α_{m} ((z − z^{0})^{m−1} + (z − z^{0})^{m−2}(φ_{j}(z) − z^{0}) + ⋯ + (φ_{j}(z) − z^{0})^{m−1}) , and the infinite series converges in a neighborhood of z^{0} because its general term is bounded by Cm|α_{m}||z − z^{0}|^{(m−1)/N}. Let A := {j ≥ 1 : φ_{j}(z^{0}) = z^{0}}, then ∏_{j∉A} (f(z) − f ∘ φ_{j}(z)) / (z − φ_{j}(z)) is clearly holomorphic in a neighborhood of z^{0}, and

∏_{j∈A} \frac{(f(z) − f ∘ φ_{j}(z))}{(z − φ_{j}(z))}

= ∏_{j∈A} ∑_{m≥1} α_{m} ((z − z^{0})^{m−1} + (z − z^{0})^{m−2}(φ_{j}(z) − z^{0}) + ⋯ + (φ_{j}(z) − z^{0})^{m−1}) .

The coefficients of the product series are easily seen to be symmetric in (φ_{j}(z), j ∈ A), therefore well-defined and holomorphic in a neighborhood of z^{0}, so the product on the left hand side of the equation, and therefore the function Γ(f), has a removable singularity at z^{0}, as required.

The zeros of the function Γ(f) have a simple interpretation in terms of self-inter-section or cusps of the analytic graph {(B(ζ), f(ζ)), ζ ∈ 𝔻} ⊂ ℂ^{2}.

Lemma 2. Let z ∈ 𝔻, then Γ(f)(z) = 0 if and only if one of the following occurs: (i) There exists z′ ≠ z such that B(z′) = B(z) and f(z′) = f(z); (ii) B′(z) = f′(z) = 0.

Proof. The product defining Γ(f) vanishes if and only if one its factors does, i.e. there exists j, 1 ≤ j ≤ n − 1, such that

\frac{f(z) − f ∘ φ_{j}(z)}{(z − φ_{j}(z))} = 0.

If z ≠ φ_{j}(z), this simply means that f(z) = f(φ_{j}(z)), and we have case (i). If z = φ_{j}(z), B is not locally invertible at z, therefore B′(z) = 0, and the vanishing of the factor means that

0 = lim_{z′→z} \frac{f(z′) − f ∘ φ_{j}(z′)}{(z′ − φ_{j}(z′))} = lim_{z′→z} ∫_{0}^{1} f′(tz′ + (1 − t)φ_{j}(z′)) dt,

which happens exactly when f′(z) = 0.

3. Necessary conditions for density or finite codimension

We shall now see how the function Γ(g) defined above relates to the properties of the algebra ℂ[B, g]. The main result of this section is Theorem 5.

Lemma 3. For 0 < p ≤ ∞, the mapping defined by f ↦ Γ(f) is continuous from H^{p} to H^{p/(n−1)}.

Proof. We have seen that Γ(f) is holomorphic on the whole disc. By the remarks at the beginning of §2.1, in a neighborhood of the unit circle the maps φ_{j} are all globally defined, holomorphic, with a non-vanishing derivative, it is clear that each function f ∘ φ_{j} is an H^{p} function : holomorphic on a neighborhood of the unit circle in 𝔻, bounded for p = ∞, and for p < ∞, f ∘ φ_{j}(rζ) tends to f ∘ φ_{j}(ζ) in the L^{p} norm on the unit circle. The H^{p}-norm of f ∘ φ_{j} is bounded by C∥f∥_{H^{p}} = C∥f∥_{L^{p}(∂𝔻)}.

Since |z − φ_{j}(z)| is bounded from below in a neighborhood of the unit circle, a similar H^{p} bound holds for (f − f ∘ φ_{j})/(z − φ_{j}(z)). Now the map Γ, seen as a map from H^{p} to L^{p/(n−1)}(∂𝔻), is a composition of linear and (n − 1)-linear maps, so to check that it is continuous it is enough to see that

∥Γ(f)∥_{L^{p/(n−1)}(∂𝔻)} ≤ C∥f∥_{p}^{n−1},

which follows from the above estimates and repeated use of Hölder’s inequality.

For p = ∞, the required estimate follows immediately from |z − φ_{j}(z)| being bounded from below.

Proposition 4. Let A := ℂ[B, g], and \overline{A} be the closure of A with respect to the H^{p} norm, for some p ∈ (0, ∞). Let Γ(g) = B_{g}S_{g}F_{g} be the canonical factorization of Γ(g) into, respectively, Blaschke product, singular inner factor, and outer factor. Then for any f ∈ \overline{A}, Γ(f) ∈ B_{g}S_{g}H^{p/(n−1)}.

Proof. Suppose first that f ∈ A. Then, using the fact that B ∘ φ_{j} = B for any j,

f = ∑_{α,β≥0} a_{αβ}g^{α}B^{β} implies

Γ(f) = Γ(g) ∏_{j=1}^{n−1} ( ∑_{α≥1,β≥0} a_{αβ}B^{β} [∑_{l=0}^{α−1} g^{α−1−l}(g ∘ φ_{j})^{l}]) ,

so Γ(f) ∈ Γ(g)ℂ[H^{∞}(B), g], because the last product involves a finite set of powers of B and polynomials which are symmetric in the {g ∘ φ_{j}, 1 ≤ n − 1}, and therefore can be expressed as polynomials of g and bounded holomorphic functions symmetric in {g ∘ φ_{j}, 0 ≤ n − 1}, which are then necessarily bounded holomorphic functions of B.

Suppose A ∋ f_{ν} → f in H^{p}. By the above we have Γ(f_{ν}) = Γ(g)q_{ν}, q_{ν} ∈ ℂ[H^{∞}(B), g]. Now in the canonical factorization, the harmonic function log |F_{g}| is the Poisson integral of its boundary values, while log |S_{g}| is the Poisson integral of a singular measure −σ_{g}, where σ_{g} is a positive measure (carried on a Borel set of Lebesgue measure 0). Factoring both Γ(f_{ν}) and q_{ν}, and using the essential uniqueness of the factorization, we see that

σ_{f_{ν}} ≥ σ_{g}

for all indices ν (their difference is a positive measure). Since Γ(f_{ν}) must converge to Γ(f) in H^{p/(n−1)}, σ_{f_{ν}} converges weakly to σ_{f}, so σ_{f} ≥ σ_{g}, which implies that S_{g} divides Γ(f).

On the other hand, H^{p/(n−1)} convergence implies convergence of the function and all its derivatives uniformly on any compactum, and therefore since each Γ(f_{ν}) has zeroes at the points of B_{g}^{−1}(0) with at least the same multiplicity, the same holds true for Γ(f). Since dividing by B_{g}S_{g} only affects the boundary values of log |Γ(f)| on a set of measure 0, the remaining function is still in H^{p/(n−1)}, q.e.d.

Theorem 5. 1. If A is dense in H^{p}, then Γ(g) is an outer function.

2. If \overline{A} has finite codimension in H^{p}, then the inner part of Γ(g) is a finite Blaschke product (and in particular there is no singular inner factor).

Proof. If z ∈ \overline{A}, then by Proposition 4, 1 = Γ(z) ∈ B_{g}S_{g}H^{p/(n−1)}, which implies that Γ(g) has no inner factor by Beurling’s theorem.

If \overline{A} has finite codimension m, then it must contain a non-zero polynomial P of degree m. Then Γ(P) is analytic across the unit circle, so cannot admit a singular inner factor. On the other hand, any zero a of Γ(g) in 𝔻 provides a continuous functional on H^{p} which vanishes on \overline{A} : in the case (i) of Lemma 2, the functional is f ↦ f(a) − f(a′), in the case (ii), f ↦ f′(a). It is easy to see that if the set of zeros of Γ(g) is infinite, this family of functionals is not of finite dimension, which contradicts the hypothesis.

4. Main results: sufficient conditions for density, generating sets for submodules

Theorem 6. For any p > 0, \overline{A} contains Γ(g)H^{p}.

Let us defer the proof of Theorem 6 to Section 8 in favor of a few observations. First, by Beurling’s theorem again, if Γ(g) is outer, then the closure of Γ(g)H^{p} is

H^{p} itself; and if the inner factor of Γ(g) reduces to a finite Blaschke product B_{g}, then the closure of Γ(g)H^{p} is B_{g}H^{p}. Since a function f ∈ H^{p} is divisible by B_{g} if and only if it vanishes on the zeroes of B_{g} to an order greater or equal than that of B_{g}, B_{g}H^{p} is then a subspace of finite codimension.

So Theorem 6 has as an immediate consequence the converse statements to those in Corollary 5, and together they make up Theorem B from the introduction.

Corollary 7. 1. A is dense in H^{p} if and only if Γ(g) is an outer function.

2. \overline{A} has finite codimension in H^{p} if and only if the inner part of Γ(g) is a finite Blaschke product (and in particular there is no singular inner factor).

An application of Theorem 6 to the polynomial case gives the following extension of the results proved in [3].

Let p(z) and q(z) be two polynomials. An argument similar to the one above alows to define D_{p}(q) and Γ_{p}(q) by

\begin{matrix}D_{p}(q)(z) := ∏_{j=1}^{n−1}(q(z) − q(φ_{j}(z)) \\ Γ_{p}(q)(z) := D_{p}(q)(z)/D_{p}(z) \end{matrix}

where again {z = φ_{0}(z), φ_{1}(z), ..., φ_{n−1}(z)} are n roots of the polynomial equation (in t) p(t) = p(z), and n = deg p. The same argument shows that D_{p}(q) and Γ_{p}(q) are polynomials.

The following result has Theorem C as a special case.

Theorem 8. Let p_{1}, ..., p_{m} ∈ ℂ[z]. Suppose that for some 1 ≤ j ≤ n, at least one of Γ_{p_{j}}(p_{k}), k = 1, 2, ..., m, k ≠ j, is not identically zero. Then \overline{A} = \overline{ℂ[p_{1}, ..., p_{m}]} has finite codimension in H^{2}, and every closed A-submodule of H^{2} has at most min_{k≠j}{N (Γ_{p_{j}}(p_{k}))} generators, where for a polynomial R, N (R) is the number of zeros of R in the unit disk.

Proof. Let M be a positive number large enough so that the interior Ω of some component of the lemniscate {z : |p_{j}(z)| = M} contains the unit disk, and p′(z) ≠ 0 whenever z ∈ ∂Ω. Denote by ψ the Riemann mapping from Ω onto the unit disk , which maps p_{j}(0) to the origin. Then ψ is analytic in Ω̄ and the Hardy space in Ω, H^{2}(Ω) (we equip ∂Ω with harmonic measure), is isometric to H^{2} in the unit disk: an isometry is being given by

F : f ⟼ f ∘ ψ.

Note that F (p_{j}) is a finite Blaschke product and F (p_{l}), l = 1, ..., m, are analytic in a neighborhood of the unit disk. This implies that Γ_{F (p_{j})}(F (p_{k})) = F (Γ_{p_{j}}(p_{k})) have trivial singular parts in their inner-outer factorization for all k = 1, …, m and their inner parts are finite Blaschke products. By Theorem 6, F (\overline{A}) contains z-invariant subspaces generated by these Blaschke products. In particular this implies that for every fixed k, A contains all polynomials vanishing with greater or equal multiplicity at all zeros of Γ_{p_{j}}(p_{k}) in Ω. Since the H^{2}(Ω) convergence implies H^{2}-convergence in the unit disk, Lemma 3 of [3] implies that the closure of A in H^{2} contains the z-invariant subspace determined by the zeros of Γ_{p_{j}}(p_{k}) which lie in the interior of the unit disk, and, therefore, has finite codimension. Let B_{k} be the inner part of Γ_{p_{j}}(p_{k}). Thus, since every A-submodule S of H^{2} is a ℂ[B_{k}]-submodule, the number of generators of S as A-module does not exceed the number of generators of S as ℂ[B_{k}]-module. By the Wold decomposition theorem this last number does not exceed the order of B_{k} (see [3] for the details).

Corollary 9. If Γ_{p_{j}}(p_{k}), k = 1, ..., m, k ≠ j have no common zeros in the unit disk, then A = ℂ[p_{1}, ..., p_{m}] is dense in H^{2}.

Proof. Let a_{1}, ..., a_{N} be common zeros in ℂ of the polynomials Γ_{p_{j}}(p_{k}). There are polynomials q_{k}, k = 1, ..., m, k ≠ j such that

∑_{k} Γ_{p_{j}}(p_{k})(z)q_{k}(z) = R(z) = ∏_{l=1}^{N}(z − a_{l}).

By Theorem 6 \overline{R(z)H^{2}} is in \overline{A}. Since |a_{l}| ≥ 1, R(z) is outer, and \overline{R(z)H^{2}} = H^{2}.

Note that in general the estimate given by Theorem 8 is sharp. As an example consider the algebra A = ℂ[p_{1}, p_{2}] where p_{1}(z) = z(z − a), p_{2}(z) = z^{2}(z − a) for some |a| < 1. It is easily seen that Γ_{p_{1}}(p_{2})(z) = p_{1}(z). Thus, by Theorem 8 every closed A-submodule of H^{2} has at most two generators. Let M be the subspace of H^{2} consisting of all functions vanishing at some point b in the unit disk, different from a and 0. Clearly M is a closed A-submodule which is not singly generated. Indeed, for every function g ∈ M such that g(0) ≠ 0 and g(a) ≠ 0, there exists a function h ∈ M such that g(0)h(a) ≠ h(0)g(a), and, therefore, g does not generate M as A-module. If g(0) = 0 or g(a) = 0, then since for any f ∈ A, f(0) = f(a) = 0, the A-module generated by g cannot contain z − b, for instance.

5. Case Γ(g) ≡ 0.

At the other end of the spectrum from the density cases, there are cases when \overline{A} cannot contain any non-trivial invariant subspace — the simplest case would be when B and g are both even functions, for instance: then any function in \overline{A} is also even, and multiplying by z takes us out of that subset. In that case, naturally, Γ(g) vanishes identically, so Theorem 6 becomes vacuous. One would then like to know when Γ(g) ≡ 0.