COMPACT COMPOSITION OPERATORS ON THE SMIRNOV CLASS

JUN SOO CHOA, HONG OH KIM, AND JOEL H. SHAPIRO

Abstract. We show that a composition operator on the Smirnov class N^{+} is compact if and only if it is compact on some (equivalently: every) Hardy space H^{p} for 0 < p < ∞. Along the way we show that for composition operators on N^{+} both the formally weaker notion of boundedness, and a formally stronger notion we call metric compactness, are equivalent to compactness.

1. Introduction

Every holomorphic self-map φ of open unit disc D of the complex plane induces, by means of the equation

C_{φ}f(z) = f(φ(z)) (f ∈ H(D), z ∈ D),

a linear transformation C_{φ} called a composition operator on the vector space H(D) of all functions holomorphic on D. The true appeal of composition operators comes from Littlewood’s Subordination Principle, which implies that each such operator takes every Hardy space H^{p} continuously into itself (see [3, Theorem 2.22, page 30], [4, Theorem 1.7, page 10], [16, Chapter 1]).

The H^{p}-continuity of composition operators raises the issue of how the functiontheoretic properties of φ influence the operator-theoretic behavior of C_{φ} on various F -spaces (complete, metrizable topological vector spaces) of analytic functions. The past three decades have witnessed a flowering of research on this subject, much of which is summarized in two recent books: [3] and [16], with some of the most recent directions outlined in the conference proceedings [7].

The problem most immediately suggested by Littlewood’s theorem, that of determining which composition operators are compact on the Hardy spaces, has been extensively studied, beginning with the 1969 dissertation of H. J. Schwartz [13]. A few years later Shapiro and Taylor [19] showed that the compactness problem for H^{p} does not depend on the index 0 < p < ∞, and in 1987 Shapiro showed that it could be solved in terms of the asymptotic behavior of the Nevanlinna counting function of φ [15] (see also [16, Chapter 10]).

The goal of this paper is to show that the compactness problem’s robustness with regard to the underlying Hardy space (meaning “H^{p} space for 0 < p < ∞”) persists even at the natural upper limit of these spaces: the Smirnov class N^{+}. Furthermore, we show that the apparently weaker property of boundedness (which is, for general F -spaces not the same as continuity) is, for composition operators on N^{+}, actually equivalent to compactness. The chain of argument leads naturally to an apparently stronger property that we call “metric compactness,” which has come up in the context of the full Nevanlinna class N, and which we also show is equivalent, for composition operators on N^{+}, to compactness.

Date: September 3, 1998.

1991 Mathematics Subject Classification. Primary 47B38; Secondary 30D55.

Key words and phrases. Composition operator, Smirnov class, compact operator.

Research supported in part by BSRI, KOSEF, and NSF.

In the next section we carefully define the spaces N and N^{+}, as well as the accompanying notions of boundedness and compactness. Granting these definitions, here is the statement of our main result:

1.1. Main Theorem. For a holomorphic self-map φ of the unit disc, the following conditions are equivalent:

(a) C_{φ} is a bounded operator on N^{+}.

(b) C_{φ}(N) ⊂ N^{+}.

(c) C_{φ} is compact on H^{p} for some (equivalently: for all) 0 < p < ∞.

(d) C_{φ} is metrically compact on N.

(e) C_{φ} is metrically compact on N^{+}.

(f) C_{φ} is compact on N^{+}.

We prove this result in §4, after setting out the relevant definitions, prerequisites, and preliminary results in the next two sections.

1.2. Remarks. The equivalence (c) ↔ (d) was recently established by Choa and Kim [1]; the proof of our Main Theorem provides a more elementary argument.

About twenty years ago Nakamura and Yanagihara [10] showed that composition operators take N^{+} into itself, and characterized the compact ones in a manner that anticipated more sophisticated results of Sarason [12] on which our proof that (b) ↔ (c) is based. Nakamura and Yanagihara were the first to ask if the compactness problem for composition operators on N^{+} might be equivalent to the one for Hardy spaces. Earlier Yanagihara studied N^{+} as an F -space, showing that it is not locally convex, and identifying its dual space [22].

An abstract analogue of the Smirnov class, called the Hardy Algebra, plays an important role in the general theory of function algebras (see [5, Chapter V]).

Acknowledgment. We thank Alec Matheson whose detailed criticisms and insightful suggestions greatly improved this paper. In particular, Matheson suggested the line of argument given here for the implication (b) → (d) of the Main Theorem; our original proof was considerably more complicated.

2. Prerequisites

2.1. Boundedness and compactness. A linear transformation of a topological vector space X is called compact ([8, Chapter 5, Problem A, page 206]) if it takes some neighborhood of zero into a relatively compact set. This definition of compactness is equivalent to the usual one if X is a Banach space, and in 1954 J. H. Williamson showed that it renders the entire Riesz theory of compact operators valid for any topological vector space—even those that are not locally convex [21] (see also [8, Chapter 5, Problem B, page 207]).

Somewhat weaker than compactness is the notion of boundedness. A subset A of a topological vector space X is called bounded if, for every neighborhood V of the origin, there is a positive number t such that A ⊂ tV (see, for example, [8, page 44]). A bounded linear transformation on X is one that takes some neighborhood of zero to a bounded set. It is easy to check that relatively compact sets are bounded, hence compact linear transformations are bounded. Moreover, every bounded linear transformation is continuous, and conversely for any Banach space, or more generally for any linear topological space possessing a bounded neighborhood of zero. But in spaces having no bounded neighborhood of zero (e.g. the space H(D) in its compact-open topology) the converse fails: the identity map is not bounded.

2.2. Notation. We employ the following simplifying notations:

• m denotes Lebesgue measure on ∂D, normalized to have total mass one. Thus if f is a continuous complex-valued function on ∂D then

∫_{∂D} f dm = \frac{1}{2π} ∫_{∂D} f(e^{iθ}) dθ.

• L^{1} denotes real L^{1}(m), M is the space of real-valued finite Borel measures on ∂D, and H^{∞} is the space of bounded analytic functions on D.

• If f is a complex-valued function on D and 0 ≤ r < 1, then the dilate of f by r, f_{r} : \overline{D} → ℂ, is defined by f_{r}(z) = f(rz) for |z| ≤ 1.

• For x a positive number, log^{+} x = max(log x, 0), so that log x = log^{+} x − log^{+}(1/x) for every x > 0.

2.3. The Smirnov and Nevanlinna classes. The Nevanlinna class N is the collection of functions f ∈ H(D) for which

sup_{0≤r<1} ∫_{∂D} log^{+} |f_{r}| dm < ∞.

Because of the numerical inequalities

(1)  log^{+} x ≤ log(1 + x) ≤ log^{+} x + log 2  (x ≥ 0)

the same definition results if the integrand log^{+} |f| is replaced by log(1 + |f|). We define

||f|| =^{def} sup_{0≤r<1} ∫_{∂D} log(1 + |f_{r}|) dm = lim_{r→1−} ∫_{∂D} log(1 + |f_{r}|) dm

(the last equality here reflects the subharmonicity of log(1 + |f|), which causes its integral means to increase with r), and set d(f, g) = ||f − g|| for each pair f, g ∈ N. The functional d thus defined is a natural translation-invariant metric that makes the vector space N into an additive topological group. However N is not a topological vector space—its scalar multiplication is discontinuous (see [17], and §3.3 below).

N contains each Hardy space, and just as for Hardy spaces, for each f ∈ N the radial limit

f^{∗}(e^{iθ}) =^{def} lim_{r→1−} f(re^{iθ})

exists for a.e. real θ [4, Theorem 2.2, page 17]. The Hardy spaces actually lie in the Smirnov Class N^{+}, which is the subspace of N consisting of functions f for which

∫_{∂D} log^{+} |f^{∗}| dm = lim_{r→1−} ∫_{∂D} log^{+} |f_{r}| dm,

or, equivalently, for which the family of functions {log^{+} |f_{r}| : 0 ≤ r < 1} is uniformly integrable on the unit circle. Thanks once again to inequalities (1) it makes no difference in this definition if log^{+} |f| is replaced by log(1 + |f|). Because of this and the Vitali Convergence Theorem [11, Chapter 6, page 133, Problem 10] we have:

||f|| = ∫_{∂D} log(1 + |f^{∗}|) dm (f ∈ N^{+}),

from which the Dominated Convergence Theorem can be used to show that scalar multiplication is continuous on N^{+}. Thus N^{+} is a metrizable topological vector space, and standard arguments show that its metric is complete, hence it is an F -space (see [17, page 919] for details).

More is true: N^{+} is an algebra under pointwise multiplication. This follows immediately from the characterization of N^{+} in terms of uniform integrability, and the subadditivity of log^{+}:

(2)  log^{+} xy ≤ log^{+} x + log^{+} y  (x, y ≥ 0).

A closer analysis shows that N^{+} is a topological algebra, i.e., pointwise multiplication is a continuous mapping N^{+} × N^{+} → N^{+}. This is an easy consequence of Vitali’s Convergence Theorem and the following useful convergence criterion, which itself follows immediately from inequality (1).

2.4. Lemma. Suppose {f_{n}} is a sequence in N^{+} and f ∈ N^{+}. Then f_{n} → f in N^{+} if and only if on ∂D: f_{n}^{∗} → f^{∗} in measure and {log^{+} |f_{n}^{∗}|} is uniformly integrable.

For more on uniform integrability see [5, Chapter V].

2.5. Factorization in the Nevanlinna class. We require some standard factorization theorems, dating back to 1929 work of Smirnov [20], for functions in the Nevanlinna class. Suppose f ∈ N. Then Jensen’s formula guarantees that the zeros {z_{n}} of f (counted according to multiplicity) satisfy the Blaschke condition ∑_{n}(1 − |z_{n}|) < ∞ (see [11, Theorems 15.18 and 15.23]). Thus f = BG where B is the Blaschke product formed from these zeros, and G is holomorphic and nevervanishing on D. Since lim_{r→1−} ∫_{∂D} log |B_{r}| dm = 0 [11, Theorem 15.24], it follows easily that G ∈ N and ∥G∥ = ∥f∥. This decomposition f = BG is our first fundamental factorization of functions in N.

In this factorization G vanishes nowhere on D, so log |G| is harmonic on D. Now sup_{0≤r<1} ∫_{∂D} log^{+} |G_{r}| dm < ∞ because G ∈ N, so in particular if r_{k} → 1− then the measures log^{+} |G_{r_{k}}| dm have a cluster point μ^{+} in the weak star topology of M. It is not difficult to show that P [μ^{+}] is the least harmonic majorant of log^{+} |G| (see, e.g., [6, Theorem 5.3, page 70]), hence μ^{+} is independent of the sequence {r_{k}}. Thus lim_{r→1−} log^{+} |G_{r}| dm = μ^{+} in the weak star topology.

Now the mean value property of harmonic functions (and the fact that log |G(0)| > −∞) insures that the conclusions of the last paragraph still hold when log^{+} |G| is replaced by log^{+}(1/|G|). Thus there is another finite positive Borel measure μ^{−} on ∂D for which lim_{r→1−} log^{+}(1/|G_{r}|) dm = μ^{−} (weak star limit). These observations lead to the following additional weak star limits:

lim_{r→1−} log |G_{r}| dm = dμ^{+} − dμ^{−} and lim_{r→1−} | log |G_{r}|| dm = dμ^{+} + dμ^{−}.

Thus, associated with each f ∈ N there is the non-vanishing function G ∈ N defined by G = f/B, and the real Borel measure μ = μ^{+} − μ^{−} for which log |G| = P [μ] (Poisson integral) on D. Upon holomorphically completing the Poisson integrals of μ^{+} and μ^{−}, and then exponentiating the result, we obtain our second fundamental factorization:f = g/h, where g and h belong to H^{∞}, both have moduli bounded by 1 on D, and h vanishes at no point of D.

The third factorization of f comes from decomposing μ into singular and absolutely continuous parts. It turns out that the absolutely continuous part of μ is just log |G^{∗}|, where G^{∗} is the radial limit function of G (see once again [6, Theorem 5.3]), so dμ = log |G^{∗}| dm + dσ where σ is singular with respect to m. Now σ = δ −ν, where δ and ν are positive finite Borel measures on ∂D that are mutually singular (as well as being singular with respect to m). The measure δ is particularly important in our development. Following the terminology in [17] we call it the denominator measure of f.

To produce the desired factorization of f, let γ[δ] denote the exponential of the holomorphic completion of the Poisson integral of δ. Employing similar notation for the other two measures discussed in this paragraph we have

f = \frac{ω · B · γ[log |G^{∗}| dm] · γ[−ν]}{γ[−δ]}, (3)

where B is the Blaschke product formed with the zeros of f and ω is a complex number of modulus one.

Observe that the numerator on the right-hand side of (3), being a constant multiple of the product of three N^{+}-functions, also belongs to N^{+}, while the denominator has the form

γ[−δ] = exp \{∫_{∂D} \frac{z + ζ}{z − ζ} dδ(ζ)\} (z ∈ D).

The usual notation for this function is S_{δ}; it is a bounded analytic function on D, with modulus bounded by 1 there, and has radial limits of modulus 1 at almost every point of ∂D. In other words, like B, it is an inner function [4, §2.4, page 24] (of course S_{δ} differs from B in that it vanishes at no point of D). Note that in the trivial case δ ≡ 0 we have S_{δ} ≡ 1. Similar remarks apply, of course, to γ[−ν] = S_{ν}. The remaining factor H = γ[log |G^{∗}| dm] on the right-hand side of (3) is called the outer factor of f. This function belongs to N^{+}, vanishes nowhere, and |G^{∗}| = |f^{∗}| a.e. on ∂D. Thus f = ωBS_{ν}H/S_{δ}, where the numerator is in N^{+} and S_{ν} and S_{δ} have no common inner factor. This is our third fundamental factorization of f. It turns out that this factorization into the product of unimodular constant, a Blaschke product, an outer function, and a quotient of relatively prime singular inner functions is unique (see [11, §17.14–17.19], [4, Theorem 2.9, page 25], or [6, Theorem 5.5, page 74] for more details).

The singular measure δ in this last factorization is particularly important: if it is not the zero-measure then it acts as an obstruction preventing f from belonging to N^{+}. Indeed, it is easy to check that the family of functions {log^{+} |f_{r}| : 0 ≤ r < 1} is uniformly integrable on ∂D if and only if δ = 0 (this story continues in §3.2 and §3.3). In particular, a function in N^{+} is invertible if and only if it is outer.

It follows immediately from the definition given above that for every outer function H,

log |H(0)| = ∫_{∂D} log |H^{∗}| dm. (4)

It is not difficult to see from the Blaschke-singular-outer factorization of functions in N^{+} that this extension of the mean value property to the boundary characterizes the outer ones.

2.6. Composition operators on N and N^{+}. The fact that every composition operator maps N continuously into itself is most easily seen by recalling that f ∈ N if and only if the subharmonic function log(1 + |f|) has a harmonic majorant (see [11, Exercise 4, page 352] for the corresponding result with log^{+} |f| in place of log(1 + |f|)—the proof in our case is the same). Moreover if f ∈ N then the least harmonic majorant Λ of log(1 + |f|) is the Poisson integral of the measure that is the weak star limit of the measures log(1 + |f_{r}|) dm as r → 1−. In particular, ∥f∥ = Λ(0).

Now if φ is a holomorphic self-map of D, then Λ ∘ φ is a harmonic majorant for log(1 + |f ∘ φ|), hence f ∘ φ ∈ N, and

∥f ∘ φ∥ ≤ Λ(φ(0)) ≤ \frac{1 + |φ(0)|}{1 − |φ(0)|} Λ(0) = \frac{1 + |φ(0)|}{1 − |φ(0)|} ∥f∥,

where the second inequality is an application of Harnack’s inequality. This shows that C_{φ} acts continuously on N.

To show that every composition operator takes N^{+} into itself, we need only observe that N^{+} is the closure in N of H^{∞}. The key here is that the Vitali Convergence Theorem guarantees that for each f ∈ N^{+} the dilates f_{r} defined by f_{r}(z) = f(rz) converge in N to f as r → 1−. Since each of these dilates is bounded on D, this shows that H^{∞} is dense in N^{+}. But composition operators preserve H^{∞}, so their N-continuity insures that they also preserve N^{+}.

2.7. The harmonic Hardy space h^{1}. This is the space of real-valued harmonic functions u on D for which

∥u∥_{1} =^{def} sup_{0≤r<1} ∫_{∂D} |u_{r}| dm < ∞.

Each u ∈ h^{1} is the Poisson integral of a finite real Borel measure μ on ∂D, and the correspondence u → μ is a linear isometry taking h^{1} onto M, the space of all such measures on ∂D, regarded as a Banach space in its variation norm. In particular, h^{1} is a Banach space. We denote by h_{0}^{1} the subspace of h^{1} consisting of Poisson integrals of measures that are absolutely continuous with respect to Lebesgue measure. Thus h_{0}^{1} is isometrically isomorphic with L^{1}(∂D), and is therefore a closed subspace of h^{1}. h_{0}^{1} can be characterized as those u ∈ h^{1} for which the family of functions {|u_{r}| : 0 ≤ r < 1} is uniformly integrable on ∂D.

The work of §2.6 reveals a close connection between h^{1} and N. Suppose f ∈ N and write f = BG, where B is the Blaschke product formed with the zeros of f. In §2.6 we noted that log |G| ∈ h^{1}, and the remarks of the last paragraph show that f ∈ N^{+} if and only if log |G| ∈ h_{0}^{1}. Furthermore, h^{1} can be characterized as the collection of real-valued harmonic functions on D whose absolute values have harmonic majorants. Once this is done, arguments similar to those of §2.6 show that every composition operator maps h^{1} continuously into itself, and preserves h_{0}^{1}.

In [12] Sarason proved that a composition operator is compact on h^{1} if and only if it is compact on h_{0}^{1}, and that this happens if and only if it maps h^{1} into h_{0}^{1}. Now for composition operators, compactness on h^{1} implies compactness on the (holomorphic) Hardy space H^{1}, and therefore on every H^{p} for 0 < p < ∞. Conversely, in [18], Shapiro and Sundberg proved that if a composition operator is compact on a Hardy space then it is compact on h^{1}. Thus for each holomorphic self map φ of D: C_{φ} is compact on a Hardy space ⟺ C_{φ}(h^{1}) ⊂ h_{0}^{1}.

Matheson has pointed out that from Sarason’s results and the third fundamental factorization of §2.5 one can derive an alternate proof that composition operators preserve N^{+}. Indeed, because composition operators preserve h_{0}^{1} they also preserve outer functions, and the desired result now follows from the characterization of N^{+} as quotients of bounded functions, where the denominator is outer.

We remark in passing that in [12] Sarason asked how one might derive the result of Shapiro and Sundberg using purely function theoretic means, and this motivated subsequent work of Cima and Matheson, and of Jonathan Shapiro (see [2] and [14]). The earlier paper [10] of Nakamura and Yanagihara can be regarded as a forerunner of this entire body of work.

2.8. Metric compactness. Our work leads naturally to a somewhat stronger concept of operator compactness that was introduced for the Nevanlinna class by Masri [9]. We call an operator on N (or, equally well, on N^{+}) metrically compact if it takes every metric ball into a relatively compact set (recall that for N^{+} the usual definition of compactness requires only that the image of some ball be relatively compact). Relative to a bounded metric only the zero-operator is metrically compact, however for composition operators on N our Main Theorem shows the metrically compact ones to be just those that restrict compactly to N^{+} (hence to H^{2}).

3. Preliminary results

We use the symbol κ for the compact-open topology on H(D), and write f_{α} →^{κ} f for convergence in this topology (uniform convergence on compact subsets of D) of a net (f_{α}) to f. We write B_{R} for the closed ball in N of radius R > 0, centered at the origin, and B_{R}^{+} for the corresponding ball in N^{+}; that is: B_{R} = {f ∈ N : ||f|| ≤ R} and B_{R}^{+} = B_{R} ∩ N^{+}.

3.1. Lemma. B_{R} is κ-compact in N for every R > 0.

The proof is a consequence of Montel’s theorem and the fact that each function in the Nevanlinna class obeys a growth condition that can be expressed in terms of its distance in N to the origin [17, Prop. 1.1, page 917]. It is a routine extension of arguments familiar to all Hardy space enthusiasts, so we omit it.

The next result shows that the denominator measure δ in the Third Fundamental Factorization of §2.5 plays an important role in the computation of distances in N.

3.2. Lemma. Suppose f ∈ N and δ is its denominator measure, so that f = F/S_{δ} where F ∈ N^{+} and S_{δ} is relatively prime to the inner factor of F. Then ||f|| = ||F || + δ(∂D).

Proof. From the work of §2.5 we know that the least harmonic majorant of log^{+} |f| is u^{+} = P [μ^{+}], where dμ = log^{+} |F^{∗}| dm+dδ. Let v be the least harmonic majorant of log(1 + |f|), so that v = P [η] where η is the weak star limit of the measures log(1 + |f_{r}|) dm as r → 1−. Then the absolutely continuous part of η is log(1 + |f^{∗}|) dm = log(1+|F^{∗}|) dm, so that dη = log(1+|F^{∗}|) dm+dσ, where σ is a positive measure on ∂D that is singular with respect to m. Thus

∥f∥ = lim_{r→1−} ∫_{∂D} log(1 + |f_{r}|) dm = ∫_{∂D} dη = ||F || + σ(∂D),

so we will be finished if we can prove that σ = δ.

We take the argument from [17, Proof of Theorem 3.1]. The numerical inequality (1) implies that u^{+} ≤ v ≤ u^{+} + log 2 at each point of D. These are inequalities on Poisson integrals, and by the uniqueness of Poisson integrals they imply the corresponding inequalities on measures: dμ^{+} ≤ dη ≤ dμ^{+} + (log 2) dm. Now inequalities between measures carry over to their singular parts, and since the first and last measures above both have singular part δ, the same holds for η.

3.3. Scalar multiplication on N. Suppose f = F/S_{δ} as in the proof above. If a is a complex number then the corresponding factorization of af is (aF)/S_{δ}, so by Lemma 3.2 ||af|| = ||aF || + δ(∂D). But on N^{+} scalar multiplication is continuous, hence lim_{a→0} ||aF || = 0. Thus lim_{a→0} ||af|| = δ(∂D), which shows that scalar multiplication on N is discontinuous at the origin. For more on the strange topology of N, see [17].

The following two corollaries of Lemma 3.2 are essential to the proof of our Main Theorem. For the first recall that the definition of “bounded set” given in §2.1, is stronger than that of “metrically bounded set.”

3.4. Proposition. Every bounded subset of N^{+} is relatively κ-compact.

Proof. Suppose A ⊂ N^{+} is bounded. Fix a sequence (f_{n}) in A. Our goal is to find a subsequence (f_{n_{k}}) and an f ∈ N^{+} such that f_{n_{k}} →^{κ} f.

By the definition of boundedness, for each ε > 0 there exists t_{ε} > 0 such that A ⊂ t_{ε}B_{ε}^{+}. In particular, the entire sequence (f_{n}) lies in the κ-compact (by Lemma 3.1) subset t_{1}B_{1} of N. Thus f_{n_{k}} →^{κ} f for some subsequence (f_{n_{k}}) and some f ∈ N. We will be done if we can show that f ∈ N^{+}.

Since f ∈ N, §2.5 provides the factorization f = F/S_{δ}, where F ∈ N^{+} and S_{δ} is the singular inner function associated with the singular denominator measure δ of f. We will be done if we can show that δ = 0 (so that S_{δ} ≡ 1).

For this, fix ε > 0 and observe that t_{ε}^{−1}f_{n_{k}} →^{κ} t_{ε}^{−1}f, and that each t_{ε}^{−1}f_{n_{k}} belongs to B_{ε}^{+}, hence also to B_{ε}. Thus the κ-compactness of B_{ε} insures that t_{ε}^{−1}f ∈ B_{ε}. As in §3.3, the denominator measure associated with t_{ε}^{−1}f is the same as the one associated with f, namely δ, so Lemma 3.2 shows that δ(∂D) ≤ ||t_{ε}^{−1}f|| ≤ ε. Because ε is an arbitrary positive number, this implies δ(∂D) = 0, as desired.

3.5. Lemma. If f ∈ N and ε > 0 then f = f_{0}f_{1} ⋯ f_{n}, where f_{0} ∈ N^{+} and f_{1}, …, f_{n} ∈ N with ||f_{j}|| < ε for 1 ≤ j ≤ n.

Proof. Consider once again the Third Fundamental Factorization f = F/S_{δ} provided by §2.5. Let f_{0} = F. Choose a positive integer n so that δ(∂D)/n ≤ ε, and set f_{1}, …, f_{n} all equal to (S_{δ})^{−1/n} = 1/(S_{δ/n}). Then f has the desired product representation, with Lemma 3.2 providing the estimate ||f_{j}|| = δ(∂D)/n ≤ ε.

We close this section with an important result about inversion in the Smirnov class that will be needed in the last step of our proof of the Main Theorem.

3.6. Proposition. The mapping f → 1/f is continuous on the set of invertible elements of the Smirnov class.

Proof. Suppose {f_{n}} is a sequence of invertible elements of N^{+}, f ∈ N^{+} is invertible, and f_{n} → f in N^{+}. Our task is to show that 1/f_{n} → 1/f in N^{+}. Now f_{n}^{∗} → f^{∗} in measure on ∂D, so (1/f_{n})^{∗} = 1/f_{n}^{∗} → 1/f^{∗} in measure, hence by Lemma 2.4 we need only show that {log^{+}(1/|f_{n}^{∗}|)} is uniformly integrable on ∂D.

Since all the Smirnov functions involved in this argument are invertible, they are—as we observed in §2.5—outer functions, hence by (4) we have for each n,

log |f_{n}(0)| = ∫_{∂D} log |f_{n}^{∗}| dm = ∫_{∂D} log^{+} |f_{n}^{∗}| dm − ∫_{∂D} log^{+}(1/|f_{n}^{∗}|) dm,

whereupon

∫_{∂D} log^{+}(1/|f_{n}^{∗}|) dm = ∫_{∂D} log^{+} |f_{n}^{∗}| dm − log |f_{n}(0)|.

Now the right-hand side of the last equation converges to

∫_{∂D} log^{+} |f^{∗}| dm − log |f(0)|

by the convergence criterion of Lemma 2.4 and the fact that f_{n}(0) → f(0) ≠ 0. Thus the sequence of non-negative functions {log^{+}(1/|f_{n}^{∗}|)} is uniformly integrable on ∂D, as we wished to show.

4. Proof of the Main Theorem.

We have now assembled all the ingredients needed for the proof of Theorem 1.1. We prove only the nontrivial implications, which are (a) → (b) → (c) → (d).

Proof that (a)→(b). Hypothesis (a) is that C_{φ}(B_{R}^{+}) is bounded for some R > 0, and we are supposed to prove that C_{φ}(N) ⊂ N^{+}.

The first step is to prove that C_{φ}(B_{R}) ⊂ N^{+} (remember that B_{R} is the closed ball of radius R in the full Nevanlinna class N). Given f ∈ B_{R} consider once more the family of dilates {f_{r} : 0 ≤ r < 1} defined in §2.6. Then f_{r} ∈ B_{R}^{+} for each r, and therefore, by our hypothesis, the image-family {f_{r} ∘ φ : 0 ≤ r < 1} is a bounded subset of N^{+}. Since f_{r} →^{κ} f, we also have f_{r} ∘ φ →^{κ} f ∘ φ, so by Proposition 3.4, f ∘ φ ∈ B_{R}^{+} ⊂ N^{+}.

So far we know that C_{φ}(B_{R}) ⊂ N^{+} for a particular R > 0. Our goal is to show that C_{φ}(N) ⊂ N^{+}. If we could scalar-multiply every function in N into B_{R}, there would be no problem. Unfortunately scalar multiplication is not continuous on N (recall §3.3), however Lemma 3.5 provides a substitute.

Suppose f ∈ N. By Lemma 3.5 we can factor f as f_{0}f_{1} ⋯ f_{n}, where f_{0} ∈ N^{+} and each f_{j} is in B_{R}. Thus f ∘φ = (f_{0} ∘φ)(f_{1} ∘φ) ⋯ (f_{n} ∘φ). By the first paragraph of this proof f_{j} ∘ φ ∈ N^{+} for 1 ≤ j ≤ n. Since f_{0} ∈ N^{+} we also have f_{0} ∘ φ ∈ N^{+}. Thus f ∘φ is a product of functions in N^{+}, and since N^{+} is an algebra, f ∘φ ∈ N^{+}, as desired.

Proof that (b)↔(c). Suppose first that C_{φ}(N) ⊂ N^{+}. We want to show that C_{φ} is compact on (say) H^{2}. For this it is enough, by the previously mentioned result of Sarason (see §2.7), to show that C_{φ}(h^{1}) ⊂ h_{0}^{1}. Suppose u ∈ h^{1}, and let ũ denote the harmonic conjugate of u that vanishes at the origin. Then F = e^{u+iũ} ∈ N, so by hypothesis (b), F ∘ φ ∈ N^{+}. Thus u ∘ φ = log |F ∘ φ| ∈ h_{0}^{1}, as desired.

Conversely, suppose C_{φ} is compact on H^{2}, so by the result of Shapiro and Sundberg, it is compact on h^{1}, and therefore by Sarason’s theorem, C_{φ}(h^{1}) ⊂ h_{0}^{1}. We want to show that C_{φ}(N) ⊂ N^{+}. For this, fix f ∈ N and consider its first fundamental factorization f = BG (§2.5), where B is the Blaschke product formed with the zeros of f and G ∈ N never vanishes. Then f ∘φ = (B ∘φ)(G∘φ). Since B is bounded, so is B ∘ φ, so we need only show that G ∘ φ ∈ N^{+}. Now the harmonic function u = log |G| belongs to h^{1}, so u ∘ φ ∈ h_{0}^{1}, hence G ∘ φ ∈ N^{+}, as desired.

Proof that (b) → (d). We are given that C_{φ} is compact on some Hardy space, and we want to show that it is metrically compact on N, i.e. that it takes any metric ball B_{R} into a relatively compact subset of N. For this, fix R > 0 and suppose {f_{n}} is a sequence of functions in B_{R}. We wish to produce a subsequence {f_{n_{k}}} such that {f_{n_{k}} ∘ φ} converges in N.

By the second fundamental factorization of §2.5 we may write f_{n} = g_{n}/h_{n} where g_{n} and h_{n} are holomorphic in D, both have modulus bounded by 1 in D, and h_{n} never vanishes on D. By a normal families argument we may, upon passing to an appropriate subsequence, assume that there exist functions g and h holomorphic on D such that g_{n} →^{κ} g and h_{n} →^{κ} h. Since each h_{n} vanishes nowhere on D, the argument principle demands that h either vanish everywhere or nowhere. We need to know that it vanishes nowhere.

To prove this, recall that while establishing the second fundamental factorization for Nevanlinna class functions in §2.5 we showed that − log |h_{n}| = P [μ_{n}^{+}] where, for each n, μ_{n}^{+} is a weak star limit of measures log^{+} |(G_{n})_{r}| dm as r → 1−, and f_{n} = B_{n}G_{n} is the first fundamental factorization of f_{n}. Thus

μ_{n}^{+}(∂D) = lim_{r→1−} ∫_{∂D} log^{+} |(G_{n})_{r}| dm ≤ lim_{r→1−} ∫_{∂D} log(1 + |(G_{n})_{r}|) dm,

and the limit on the right is just ∥G_{n}∥ = ∥f_{n}∥ ≤ R, for all n. Thus all the measures {μ_{n}^{+}} have total variation bounded by R, which guarantees a subsequence {μ_{n_{k}}^{+}} that converges weak star to a finite positive Borel measure η. Consequently − log |h_{n_{k}}| = P [μ_{n_{k}}] → P [η] pointwise on D. We conclude that |h| = exp{−P [η]} is never zero, as desired.

To this point we know that

f_{n} = g_{n}/h_{n} →^{κ} g/h =^{def} f,

where f ∈ N. We finish the proof by showing that ∥f_{n} ∘φ−f ∘φ∥ → 0. Since C_{φ} is compact on a Hardy space we know that the radial limit function φ^{∗} has modulus < 1 at almost every point of ∂D (see [16, §2.2, page 23]), so that (g_{n}∘φ)^{∗} = g_{n}∘φ^{∗} → g ∘ φ^{∗} a.e. on ∂D. Consequently the Bounded Convergence Theorem insures that g_{n} ∘ φ → g ∘ φ in the metric of N^{+}. In exactly the same way, h_{n} ∘ φ → h ∘ φ in N^{+}.

For the endgame, note that because neither h nor any of the h_{n} ever vanishes on D, their reciprocals all belong to N. Condition (b) insures that the composition of each of these reciprocals with φ belongs to N^{+}, so that each h_{n} ∘ φ, as well as h ∘ φ, is invertible in N^{+}. The continuity of inversion in N^{+} (Proposition 3.6), along with the fact that h_{n} ∘ φ → h ∘ φ, now guarantees that 1/(h_{n} ∘ φ) → 1/(h ∘ φ) in N^{+}. Because N^{+} is a topological algebra we therefore have

f_{n} ∘ φ = (g_{n} ∘ φ) · \frac{1}{h_{n} ∘ φ} ⟶ (g ∘ φ) · \frac{1}{h ∘ φ} = f ∘ φ,

where the convergence is in the metric of N^{+}. This finishes the proof that (b) → (d), and with it, that of the Main Theorem.

Final remark. The “linear topological” notion of compactness (wherein the image of some ball is relatively compact) can be imported into N, whereupon it joins the list of equivalences that make up our Main Theorem. Indeed, metric compactness on N trivially implies compactness, and conversely compactness on N clearly implies compactness on N^{+}, which by our Main Theorem, implies (for composition operators) metric compactness on N.

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Department of Mathematics Education, Sung Kyun Kwan University, Jongro-Gu, Seoul 110–745, Korea

E-mail address: jschoa@yurim.skku.ac.kr

Department of Mathematics, KAIST, Taejon 305–701, Korea

E-mail address: hkim@ftn.kaist.ac.kr

Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA

E-mail address: shapiro@math.msu.edu