Riesz Composition Operators

Paul S. Bourdon and Joel H. Shapiro

Abstract. We give a sufficient condition for a univalently induced composition operator on the Hardy space H^{2} to be a Riesz operator. We then establish that every Riesz composition operator has a Koenigs model and explore connections our work has with the model theory and spectral theory of composition operators.

1 Introduction

If φ is a holomorphic function on the unit disc U with φ(U) ⊂ U then the composition operator C_{φ} induced by φ is the linear map:

C_{φ}f = f ∘ φ (f holomorphic on U).

Although C_{φ} is initially defined on the full space H(U) of functions holomorphic on U, a famous result of J.E. Littlewood shows that it takes the Hardy space H^{2} into itself, and its restriction to H^{2} is a bounded operator [9]. The connection between operator-theoretic properties of C_{φ} and function-theoretic properties of the inducing map φ has led to a rapidly expanding body of research, some major threads of which you can find developed in the recent books [6] and [16].

Much of the recent interest in composition operators arises from connections with function theory that emerge from the study of compactness of these operators (see for example [11], [15] and [17]). In this paper the motivating concept from operator theory is the closely related concept of “Rieszness”.

A Riesz operator on a Hilbert space is a bounded operator whose essential spectrum is the singleton {0}. We will discuss this condition in more detail in the next section, but right now it is enough to know that the class of Riesz operators includes all operators that have a compact positive power, and that the Riesz operators are “spectrally indistinguishable” from the compact ones in that their nonzero spectral points occur as isolated, finite-multiplicity eigenvalues.

We show here that if C_{φ} is a Riesz operator on H^{2} then the mapping φ has to arise from the geometric model associated with Koenigs’s solution to Schroeder’s functional equation (the eigenfunction equation for C_{φ}). The corresponding result for compact composition operators has long been known, and forms the basis for the investigation begun in [17] connecting the compactness problem with the geometry of Schroeder-Koenigs models. Our work suggests that in this study it is the question of Rieszness, rather than of compactness that should be the main operator-theoretic issue.

The research of both authors was supported in part by the National Science Foundation

Bourdon & Shapiro  Riesz Composition Operators

Our interest in Riesz operators began with a question posed by Michael Neumann, who asked if the Hardy space H^{2} supports a Riesz composition operator that is not power-compact. We answer Neumann’s question affirmatively by deriving an easily checked sufficient condition for a univalently induced composition operator on H^{2} to be Riesz. The connection with Schroeder-Koenigs models arises when we apply our condition to show that certain non-power-compact examples that motivated the work of [17] are in fact Riesz operators. These matters occupy sections 3 and 4, with the necessary background material collected for the reader’s convenience in section 2.

In section 5 we make the general connection between Riesz operators and Schroeder-Koenigs models by showing that if C_{φ} is Riesz, then φ must have an attractive fixed point a in U. Koenigs’ work ([8]) on Schroeder’s equation then shows that if, in addition, φ is univalent, then φ has the following model:

φ = σ^{−1} ∘ (φ′(a)σ),

where σ is a univalent mapping on U, which is uniquely determined (up to a constant multiple). Our work suggests that the geometry of σ(U) in this Koenigs model for φ determines Rieszness; more specifically it leads us to conjecture that the “no-twisted-sectors” property introduced by Shapiro, Smith, and Stegenga in [17] is equivalent to Rieszness. (After this work was completed Pietro Poggi-Corradini established the validity of this conjecture [12].)

In the final section of this paper we discuss the “no-twisted-sectors” conjecture in more detail, and point out a connection between our work and a recent result of Cowen and MacCluer [5, Corollary 19] concerning composition-operator spectra.

2 Background

For completeness of exposition we summarize here some necessary prerequisites.

2.1 The Hardy space H^{2}. This is the space of functions that are analytic in the unit disc U and whose Taylor coefficients in the expansion about the origin are square summable. Thus H^{2} is a Hilbert space isomorphic to ℓ^{2}, where the norm is defined by:

∥f∥^{2} = ∑_{n=0}^{∞} |f̂(n)|^{2},

with f̂(n) denoting the n-th Taylor coefficient of f.

2.2 Riesz operators. For our purposes the class of Riesz operators is best defined to be those bounded operators T on Hilbert space for which the distance between T^{n} and the compacts tends to zero super-exponentially. To be more precise it helps to recall that the distance (in the operator norm) from an operator T to the space of compact operators is called the essential norm of T, denoted by ∥T ∥_{e}. To say that T is a Riesz operator means that

lim_{n→∞} ∥T^{n}∥_{e}^{1/n} = 0.

This definition makes it clear that every power-compact operator is Riesz.

2.3 The essential spectrum. Suppose H is a Hilbert space. The quotient B/K of the algebra B of bounded operators on H by the closed ideal K of compact operators is called the Calkin Algebra of H. The essential norm of an operator T ∈ B is thus revealed as the Calkin Algebra norm of the coset T +K. The spectrum of this coset is called the essential spectrum of T. The spectral radius formula, applied to the Calkin Algebra, reveals a spectral interpretation of our definition of Riesz operator:

An operator is Riesz if and only if its essential spectrum is the singleton {0}.

This interpretation, along with a little operator theory, leads to the assertion made in the Introduction that Riesz operators are spectrally indistinguishable from compacts (see [7, Chapter 3] or [18]).

2.4 Essential norms and angular derivatives. We rely heavily on the formula obtained in [15] for the essential norm of a composition operator. We will discuss this in more detail in section 3, but for now it is enough to know that if φ is univalent then this formula reduces to:

∥C_{φ}∥_{e} = lim inf_{|z|→1−} [\frac{1 − |φ(z)|}{1 − |z|}]^{− \frac{1}{2}} .

This equation relates the study of composition operators with classical work on boundary properties of holomorphic self-maps of the disc. In these studies φ is said to have an angular derivative at a point ζ ∈ ∂U if two things happen:

(a) φ′(z) has a finite angular (nontangential) limit as z → ζ, and

(b) The angular limit of φ (which necessarily exists at ζ because of condition (a)) has modulus 1.

We denote the angular limit in (a) by φ′(ζ). More generally, whenever f is a complexvalued function defined on U that has an angular limit at ζ ∈ ∂U, then we denote this limit by f(ζ). The Julia-Carathéodory theorem asserts that φ has an angular derivative at ζ ∈ ∂U if and only if

lim inf_{z→ζ} \frac{1 − |φ(z)|}{1 − |z|} < ∞.  (1)

The theorem also asserts that when the difference quotient in (1) is finite, then:

• The left-hand side of (1) is precisely |φ′(ζ)|, and

• The angular limit of the difference quotient exists, and equals |φ′(ζ)| (see e.g. [2, §298–300], [6, page 51] or [16, page 57] for the full story).

The Julia-Carathéodory theorem allows us to think of |φ′(ζ)| as a function on ∂U with values in the extended interval [0, ∞]. Just think of |φ′(ζ)| as defined by the left-hand side of (1), and make the connection with angular limits of the derivative by interpreting “|φ′(ζ)| = ∞” to mean “the angular limit of φ′ either doesn’t exist at ζ, or it exists and the corresponding angular limit of φ has modulus < 1.” Note that that |φ′| is bounded away from zero on ∂U: for example, if φ(0) = 0 then by the Schwarz Lemma and (1), |φ′| ≥ 1 on ∂U.

2.5 Lemma. The function |φ′| : ∂U → (0, ∞] is lower semicontinuous.

Proof. We have to show that for each M > 0 the set {|φ′| > M} is open, or equivalently,

E_{M} = {ζ ∈ ∂U : |φ′(ζ)| ≤ M}

is closed.

To this end suppose {ζ_{n}} is a sequence of points in E_{M} that converges to ζ. We have to show ζ ∈ E_{M}. Given ϵ > 0, the Julia-Carathéodory Theorem guarantees that for each n there exists a point z_{n} ∈ U at distance less than 1/n from ζ_{n} such that

\frac{1 − |φ(z_{n})|}{1 − |z_{n}|} < M + ϵ .

Since z_{n} → ζ, another application of the Julia-Carathéodory theorem shows that the angular derivative of φ exists at ζ, and shows that |φ′(ζ)| ≤ M + ϵ. Since ϵ is an arbitrary positive number, |φ′(ζ)| ≤ M, hence ζ ∈ E_{M}. ///

Angular derivatives of iterates of φ may be computed via the usual chain rule.

2.6 Lemma. (The Chain Rule) Suppose φ is a holomorphic self-map of U and ζ ∈ U. Suppose further that for some positive integer n the iterate φ_{n} has an angular derivative at ζ. Then:

(a) For each 1 ≤ j ≤ n the iterate φ_{j} has an angular derivative at ζ.

(b) φ has an angular derivative at φ_{j}(ζ) for 1 ≤ j < n.

(c) Letting ζ_{j} = φ_{j}(ζ), we have φ′_{n}(ζ) = ∏_{j=0}^{n−1} φ′(ζ_{j}) .

We omit the proof, which is a routine exercise in applying the Julia-Carathéodory Theorem. Note that the “j = n” part of statement (a) is just the hypothesis on φ_{n}, and that existence of the angular limits ζ_{j} = φ_{j}(ζ) used in (c) is insured by (a), as is the fact that these limits lie on the unit circle.

2.7 The Denjoy-Wolff Theorem. If a holomorphic self-map φ of U fixes a point a ∈ U then the Schwarz Lemma guarantees that |φ′(a)| ≤ 1, with equality if and only if φ is an elliptic automorphism (i.e. conformally equivalent to a rotation). In case |φ′(a)| < 1 the fixed point is attractive in the sense that the sequence (φ_{n} : n ≥ 1) converges to the constant function a uniformly on compact subsets of U.

The Denjoy-Wolff Theorem asserts that something similar happens even if φ fixes no point of U. In this case there is a unique point α ∈ ∂U, called the Denjoy-Wolff point of φ such that φ_{n} → α uniformly on compact subsets of U. In addition, φ has remarkable regularity at α. For example, φ(α) = α in the sense of angular limits, so α serves as a “boundary fixed point” for φ. Moreover, φ has some degree of smoothness at α in the sense that the angular derivative exists there. Although not required for the sequel, it is worth noting the angular derivative of φ at the Denjoy-Wolff point is a strictly positive number that is ≤ 1. For complete details see e.g. [16, Chapter 5].

3 Sufficient Condition for Rieszness

As usual, the symbol φ denotes a holomorphic self-map of U, and as in the proof of Lemma 2.5, E_{M} = {ζ ∈ ∂U : |φ′(ζ)| ≤ M}. We also continue with the convention of extending φ to ∂U by defining φ(ζ) to be the angular limit of φ at ζ ∈ ∂U, whenever this limit exists. Recall that by Fatou’s theorem, this limit exists at (Lebesgue) almost every point of ∂U, and (as pointed out in the preceding section) at every point where the map φ has an angular derivative.

3.1 Theorem. Suppose φ is univalent, and that for every M > 0 there exists a positive integer N = N(M) such that

n ≥ N ⇒ φ_{n}(E_{M}) ∩ E_{M} = ∅ .

Then C_{φ} is a Riesz operator (i.e., its essential spectrum is {0}).

Proof. We know from [15] that the essential norm of C_{φ} on H^{2} is given by the formula

∥C_{φ}∥_{e}^{2} = lim sup_{|w|→1−} \frac{N_{φ}(w)}{log \frac{1}{|w|}}  (2)

where N_{φ} is the Nevanlinna counting function of φ:

N_{φ}(w) = ∑_{z∈φ^{−1}\{w\}} log \frac{1}{|z|}

the sum being interpreted as 0 when w ∉ φ(U). Now if φ is univalent, then the sum has just one term; hence, using the notation w = φ(z) (and interpreting log(1/|z|) to be 0 whenever w ∉ φ(U)), we have, as advertised in section 2.4,

∥C_{φ}∥_{e}^{2} = lim sup_{|w|→1−} \begin{matrix}log & \frac{1}{|z|} \\ log & \frac{1}{|w|} \end{matrix} = lim sup_{|z|→1−} \frac{1 − |z|}{1 − |φ(z)|} = [lim inf_{|z|→1−} \frac{1 − |φ(z)|}{1 − |z|}]^{−1} .

Upon applying the Julia-Carathéodory Theorem to the term on the right, and noting that by lower semicontinuity |φ′| attains its infimum on ∂U, we obtain

∥C_{φ}∥_{e}^{2} = [min_{ζ∈∂U} |φ′(ζ)|]^{−1} .  (3)

Our goal is to show that the essential spectral radius of C_{φ} is zero, i.e.;

lim_{n→∞} ∥C_{φ}^{n}∥_{e}^{1/n} = {0},

which in view of (3) above, means

lim_{n→∞} min_{ζ∈∂U} |φ′_{n}(ζ)|^{1/n} = ∞ .  (4)

To prove (4) for our mapping φ, fix M > 1 and recall that by hypothesis there exists a positive integer N such that φ_{n}(E_{M}) is disjoint from E_{M} whenever n ≥ N.

Now fix n ≥ 2N. We are going to show that the quantity on the left-hand side of (4) is > \sqrt{M}, which will establish (4), and prove that C_{φ} is a Riesz operator.

To this end, we need only consider those ζ ∈ ∂U for which |φ′_{n}(ζ)| < ∞. The idea is to use the Chain Rule (part (c) of Lemma 2.6) to compute |φ′_{n}(ζ)|. At this point it is convenient to assume that φ(0) = 0, so that (as we observed just before stating Lemma 2.5) |φ′| ≥ 1 on ∂U. We will show in Theorem 5.3 that no loss of generality results from the assumption φ(0) = 0.

To estimate the size of the product on the right-hand side of the chain rule formula, let

J(ζ) = {0 ≤ j < n : φ_{j}(ζ) ∉ E_{M}} .

That is, J(ζ) is the set of indices in [0, n−1] for which |φ′_{j}(ζ)| > M. Observe that our choice of N guarantees that the number of indices in J(ζ) is ≥ n−N for every ζ ∈ ∂U. Thus by the Chain Rule (using the notation ζ_{j} = φ_{j}(ζ), and writing J = J(ζ)),

|φ′_{n}(ζ)| = ∏_{j=0}^{n−1} |φ′(ζ_{j})| ≥ ∏_{j∈J} |φ′(ζ_{j})| ≥ M^{n−N}

from which follows

|φ′_{n}(ζ)|^{1/n} ≥ M^{1− \frac{N}{n}}

so |φ′_{n}(ζ)|^{1/n} ≥ \sqrt{M} if n ≥ 2N, as promised. ///

Here is an easy-to-use special case of Theorem 3.1 that we will apply in the next section to construct a Riesz composition operator that is not power-compact.

3.2 Corollary. Suppose φ is a holomorphic self-map of U that is univalent on U and extends continuously to \overline{U} (the closed unit disc). Suppose further that

(a) At each fixed point on the unit circle, φ does not have an angular derivative, and

(b) If ζ ∈ ∂U is not a fixed point of φ then φ_{n}(ζ) ∈ U for some positive integer n.

Then C_{φ} is a Riesz operator on H^{2}.

Proof. Let F P denote the (possibly empty) set of fixed points of φ on ∂U. Fix M > 0 and note that by hypothesis (a) we have E_{M} ⊂ ∂U\F P. Note that for each nonnegative integer n, the set

G_{n} = {ζ ∈ ∂U : φ_{n}(ζ) ∈ U} = φ_{n}^{−1}(U) ∩ ∂U

is open (continuity of φ), and that (G_{n}) is an increasing sequence whose union contains ∂U\F P (by hypothesis (b)). Thus this union contains the compact set E_{M} and there is a positive integer N such that G_{n} ⊃ E_{M} whenever n ≥ N. In particular:

n ≥ N ⇒ |φ′| ≡ ∞ on φ_{n}(E_{M}) ⇒ φ_{n}(E_{M}) ∩ E_{M} = ∅,

so C_{φ} is Riesz by Theorem 3.1. ///

4 A Non-Power-Compact Riesz Composition Operator

4.1 A strip with a bulge. Our example is a holomorphic self-map of U that played a key role in the work of [17]. Let G be the union of the open unit disc and the open horizontal strip of unit width whose lower boundary is the real axis. So G is a strip with a bulge that contains the origin, and \frac{1}{2}G ⊂ G. Let σ be the univalent holomorphic mapping of U onto G that fixes the origin and has derivative > 0 there (Riemann Mapping Theorem). Define φ on U by

φ(z) = σ^{−1}(\frac{1}{2}σ(z)) (z ∈ U).  (5)

We think of φ in the following way: its action on U is conformally similar, via σ, to the action of the mapping M_{1/2} of “multiplication by 1/2” acting on G. We think of M_{1/2} : G → G as a “model” for φ : U → U, and deduce the properties of φ from properties of the model. In particular, by going back and forth between U and G it is easy to check that φ extends continuously to \overline{U} and has two fixed points on ∂U, which correspond to “the points at plus and minus infinity on the boundary of G”.

4.2 Theorem. For the mapping φ just described, C_{φ} is a Riesz operator on H^{2} that is not power-compact.

Proof. The fact that C_{φ} is not power-compact was noted in [17]. For completeness we review the argument. Let L denote the half-line (2, ∞) on the real axis, and note that M_{1/2}(L) ⊂ ∂G. Let I = σ^{−1}(L), an open arc of ∂U. The behavior of the model translates into the fact that φ(I) ⊂ ∂U, in particular φ has radial limits of modulus one on the arc I. It is a standard result about composition operators that whenever the inducing map has radial limits of modulus one on a subset of ∂U having positive measure, the induced composition operator is not compact (see e.g. [17, §2.5, page 32]). Thus our operator C_{φ} is not compact.

Now if n is a positive integer, then the n-th iterate φ_{n} of φ is obtained by changing the multiplier 1/2 in the definition (5) of φ to 1/2^{n}. The same analysis shows that C_{φ}^{n} = C_{φ_{n}} is not compact. Thus C_{φ} is not power-compact.

To show that C_{φ} is a Riesz operator we check that φ satisfies the hypotheses of Corollary 3.2. We have already noted that φ is univalent on U and extends continuously to \overline{U}. Since every (finite) point of ∂G is eventually taken into G by successive multiplications by 1/2, we see that each non-fixed point of ∂U is taken into U by some iterate of φ, so hypothesis (b) of Corollary 3.2 is satisfied.

It remains to show that hypothesis (a) is satisfied, i.e. that φ does not have an angular derivative at either of its boundary fixed points. For this we require a crucial result of [17], which states that whenever a self-map φ of U defined by a formula like (5) has an angular derivative at a boundary fixed point, then the “model domain” G contains a “twisted sector”

S_{ϵ}(Γ) = ⋃_{w∈Γ}{z : |z − w| < ϵ|w|},

for some ϵ > 0 and some simple arc Γ that connects the origin to ∞ ([17, Proposition 3.3], see also [16, §9.6]). We will say more about twisted sectors in the final section, but for now it is enough to note that the strip-like region G in our present example doesn’t contain one. Thus φ does not have an angular derivative at either of its boundary fixed points.

We have shown that C_{φ} satisfies all the hypotheses of Corollary 3.2, so it is therefore a Riesz operator. ///

4.3 Remarks. The construction of φ can be refined to create non-power-compact Riesz composition operators whose inducing functions have infinitely many fixed points on the unit circle. For this, take a sequence of rays emanating from the origin, and making angles with the real axis that decrease to zero. For definiteness, suppose also that these angles are < π/2 (you can also symmetrize the situation with respect to the real axis to increase its aesthetic appeal). Make each ray the top edge of a little open strip, taking care to make the strips thin enough so that they don’t intersect outside the unit disc. Let G be the union of this countable collection of strips and the unit disc, and proceed just as before.

A bit more work, using Theorem 3.1 directly, shows that the examples of nonpower-compact composition operators constructed in [17] from “jellyfish models” also induce Riesz operators.

5 Riesz Operators and Koenigs Models

5.1 Koenigs Models. The method we used in §4 to construct our non-power-compact Riesz composition operator is actually part of a much more general scheme. If λ is any complex number of modulus < 1 and G a simply connected plane domain with 0 ∈ G and λG ⊂ G, then the Riemann Mapping Theorem provides a univalent map σ of U onto G with σ(0) = 0, and just as in the last section we can use σ to define a holomorphic self-map φ of U. This time the formula is

φ(z) = σ^{−1}(λσ(z)) (z ∈ U) .  (6)

Once again σ establishes a conformal similarity between the action of φ on U and that of the mapping M_{λ} of “multiplication-by-λ” on G. We call M_{λ} : G → G the Koenigs Model for φ, and observe that the subtleties of how φ acts on U are now coded into the geometry of G.

Classical work of Gabriel Koenigs [8, 1884] shows that:

Every univalent self-map of U with an attractive fixed point in U has a Koenigs model, which is uniquely determined (up to a constant multiple of the model domain G).

More generally Koenigs showed that if φ is any holomorphic self-map of U with a fixed point in U, say φ(0) = 0 without loss of generality, and if 0 < |φ′(0)| < 1, then there is a holomorphic function σ on U such that

σ(φ(z)) = φ′(0)σ(z) (z ∈ U).  (7)

Koenigs showed further that σ is unique up to multiplication by a constant, and is univalent whenever φ is univalent (see e.g., [16, §6.1] for an exposition of these matters). Thus if φ is univalent then (7), which is usually called Schroeder’s equation, can be rewritten in the form (6), with λ = φ′(0). Note that the Schwarz Lemma guarantees that |φ′(0)| ≤ 1 for any holomorphic self-map of U that fixes the origin— by insisting on strict inequality we simply prevent φ from being a rotation.

5.2 Koenigs eigenfunctions. We continue to assume that φ is a holomorphic self-map of U that fixes the origin, and that 0 < |φ′(0)| < 1. The map σ discussed above is often called the Koenigs eigenfunction of φ. This terminology comes from observing that equation (7) can be rewritten

C_{φ}σ = φ′(0)σ ,

hence σ is an eigenfunction of the operator C_{φ} : H(U) → H(U). The essential uniqueness of σ says that the eigenvalue φ′(0) has multiplicity one. Upon raising both sides of (7) to the n-th power we see that σ^{n} is an eigenfunction and φ′(0)^{n} the corresponding eigenvalue (n = 0, 1, 2, …). Koenigs showed that all these eigenvalues have multiplicity one.

Caughran and Schwartz [3] (proof of their Theorem 3) showed that the point φ′(0)^{n} always belongs to the spectrum of C_{φ} : H^{2} → H^{2} (see also [4, Theorem 4.1] and [1]). The subtlety here is that in general σ^{n} may not belong to H^{2}, so φ′(0)^{n} need not always be an eigenvalue of C_{φ} : H^{2} → H^{2}. However if C_{φ} is a Riesz operator on H^{2}, then φ′(0)^{n}, being a non-zero spectral point, must be an eigenvalue—of multiplicity one by the work of Koenigs. This forces the corresponding eigenfunction σ^{n} to lie in H^{2}. So if C_{φ} is Riesz, then σ^{n} ∈ H^{2} for every positive integer n. Equivalently:

C_{φ} Riesz on H^{2} ⇒ σ ∈ ⋂_{p<∞} H^{p}.  (8)

For power-compact operators this result is implicit in the work of Caughran and Schwartz [3, Theorem 3].

In the context of Riesz operators no restriction is imposed by assuming, as we have done above, that φ has an attractive fixed point in U. We show now that this condition is, in fact, necessary for C_{φ} to be Riesz. Consequently:

Every univalently induced Riesz composition operator has a Koenigs model.

5.3 Theorem. Suppose φ is a holomorphic self-map of U and C_{φ} is a Riesz operator on H^{2}; then φ fixes a point a ∈ U, and |φ′(a)| < 1.

Proof. Suppose first that φ does not fix any point of U. Will show that the essential norm of C_{φ}^{n} does not go to zero super-exponentially, and so C_{φ} is not a Riesz operator on H^{2} (section 2.2).

For this we employ the boundary fixed point α guaranteed for φ by the Denjoy-Wolff Theorem (see section 2.7). Recall that φ has an angular derivative at α, and so by the Julia-Carathéodory Theorem (section 2.4)

lim_{r→1−} \begin{matrix}log & \frac{1}{r} \\ log & \frac{1}{|φ(rα)|} \end{matrix} = lim_{r→1−} \frac{1 − r}{1 − |φ(rα)|} = \frac{1}{|φ′(α)|}  (9)

For 0 ≤ r < 1 write w(r) = φ(rα), so that

N_{φ}(w(r)) ≥ log \frac{1}{r} .  (10)

The essential-norm formula (2) combined with (9) and (10) yields:

∥C_{φ}∥_{e}^{2} = lim sup_{|w|→1−} \frac{N_{φ}(w)}{log \frac{1}{|w|}} ≥ lim_{r→1−} \begin{matrix}log & \frac{1}{r} \\ log & \frac{1}{|w(r)|} \end{matrix} = \frac{1}{|φ′(α)|}  (11)

Now for each positive integer n we have C_{φ}^{n} = C_{φ_{n}}. Since the Denjoy-Wolff point of φ is the limit of the sequence of successive iterates of φ, we see that φ_{n} also does not fix any points of U, and that α must be its Denjoy-Wolff point. Thus φ_{n} has an angular derivative at α, and our Chain Rule (Lemma 2.6) shows that φ′_{n}(α) = φ′(α)^{n}. This allows us to apply the essential-norm estimate (11) to φ_{n}, obtaining

∥C_{φ}^{n}∥_{e}^{2} = ∥C_{φ_{n}}∥_{e}^{2} ≥ \frac{1}{|φ′(α)|^{n}} ,

from which follows

lim_{n} ∥C_{φ}^{n}∥_{e}^{1/n} ≥ \frac{1}{|φ′(α)|^{1/2}} > 0.

Thus, by our definition (section 2.2), C_{φ} is not a Riesz operator.

Summary: If φ has no fixed point in U then C_{φ} is not a Riesz operator on H^{2}. Suppose now that C_{φ} is a Riesz operator. Then φ fixes a point of U, which by a

standard similarity argument we may assume to be the origin: φ(0) = 0. It remains to show that |φ′(0)| < 1.

The Schwarz Lemma insures that |φ′(0)| ≤ 1, and in case of equality that φ has to be a rotation of U. Thus if we do not have the desired strict inequality on φ′(0), then C_{φ} must be a unitary operator on H^{2}. We claim this contradicts the assumption that C_{φ} is Riesz.

More generally:

Every linear isometry of Hilbert space into itself has essential spectral radius = 1.

This is well-known, but to keep our exposition complete, here is a proof. Suppose T is a Hilbert-space isometry. We begin by showing that the essential norm of T is 1. Since the essential norm is just the distance to the subspace of compact operators, it is clear that ∥T ∥_{e} ≤ ∥T ∥ ≤ 1. To prove the opposite inequality, fix a compact operator K, and an orthonormal sequence (e_{n}). Since T is an isometry, ∥T e_{n}∥ = 1 for each n, so it follows that for each positive integer n:

∥T − K∥ ≥ ∥(T − K)e_{n}∥ ≥ ∥T e_{n}∥ − ∥Ke_{n}∥ = 1 − ∥Ke_{n}∥ .  (12)

Now every orthonormal sequence converges weakly to zero, so because K is compact, ∥Ke_{n}∥ → 0 as n → ∞. Thus upon letting n → ∞ in the right-hand side of (12), we see that ∥T − K∥ ≥ 1, hence ∥T ∥_{e} ≥ 1, as desired.

Since every Hilbert-space isometry T has essential norm one, and since every power of T is again an isometry, it follows from the last paragraph that ∥T^{n}∥_{e} = 1 for every positive integer n; hence, T cannot be a Riesz operator. In particular, no rotation of the disc can induce a Riesz composition operator. Thus, if φ fixes the origin and induces a Riesz composition operator, then |φ′(0)| < 1. ///

5.4 Remarks. (a) The argument given in the first part of Theorem 5.3 shows that if φ fixes no point of U, the essential spectral radius of C_{φ} : H^{2} → H^{2} is bounded below by |φ′(α)|^{−1/2}, where α ∈ ∂U is the Denjoy-Wolff point of φ. For this we could also have referenced [1, Lemma 5.3], where we showed that for this situation r_{e}(C_{φ}) is precisely |φ′(α)|^{−1/2}.

However the argument given above, in addition to offering a more self-contained exposition, can be easily modified to show that for any holomorphic self-map φ of U:

r_{e}(C_{φ}) ≥ max_{ζ∈F P} |φ′(ζ)|^{−1/2} ,

where on the right, F P denotes the collection of boundary fixed points of φ. For this we only need to use the general version of the Chain Rule given in [16, Chapter 4, Exercise 10, page 74] to show that if φ has an angular derivative at a boundary fixed point ζ then so does every iterate φ_{n}, and φ′_{n}(ζ) = φ′(ζ)^{n}.

Thus if C_{φ} is a Riesz operator on H^{2} then φ cannot have an angular derivative at any boundary fixed point. Since every positive power of a Riesz operator is also Riesz, the same holds for each iterate of φ. Now boundary fixed points of iterates of φ are, in an obvious way, “boundary periodic points” of φ, so we have:

If C_{φ} : H^{2} → H^{2} is a Riesz operator, then φ does not have an angular derivative at any boundary periodic point.

Poggi-Corradini [13] has recently proved the converse of this result for univalent maps φ. In general the converse is false, as shown by the fact that there exist inner functions with no angular derivative at any point of ∂U (see [16, section 10.2] for example)—the last part of the proof of Theorem 5.3 shows that no inner function can induce a Riesz composition operator. (Every inner function with a fixed point in U is conjugate, via a conformal disc automorphism, to an inner function that vanishes at the origin, and by [10, section 1] such inner functions induce isometries on H^{2}. Thus every composition operator induced by an inner function is similar to an isometry, and therefore is not Riesz.)

(b) There is a parallel between the results discussed here and the corresponding ones for compactness. Theorem 5.3 was first obtained in the context of power-compact operators by Caughran and Schwartz [3, 1975]. It has long been known that nonexistence of the angular derivative at every point of ∂U is necessary for C_{φ} to be compact on H^{2}, and that this condition is sufficient if φ is univalent (see [16, Chapter 2] for details and historical notes). Our results, along with those of Poggi-Corradini show that the same is true for Rieszness if non-existence of the angular derivative is required not everywhere on the boundary, but only at the boundary periodic points of the inducing map.

(c) Shapiro and Taylor [19, 1973] proved that for 0 < p < ∞, a composition operator is compact on H^{p} if and only if it is compact on H^{2}. The same is true for Riesz operators. The notions of essential norm and essential spectral radius, make sense for any Banach space, and more generally for any p-Banach space where 0 < p ≤ 1. Thus Riesz operators can be defined in these more general settings, and it turns out that all the spectral theory that makes Riesz-operator spectra identical with compact-operator spectra still holds (see [18]).

Now in [1] we have shown that for any holomorphic self-map φ of U, and any 0 < p < ∞,

r_{e}(C_{φ} : H^{p} → H^{p}) = [r_{e}(C_{φ} : H^{2} → H^{2})]^{2/p} .

It follows from this identity that for 0 < p < ∞:

C_{φ} is a Riesz operator on H^{p} if and only if it is a Riesz operator on H^{2}.

6 Connections and Conjectures

In [17] Shapiro, Smith, and Stegenga study the relationship between the geometry of Koenigs models and compactness of composition operators on H^{2}. They establish, for example, that if the image σ(U) of a (univalent) Koenigs function σ contains a twisted sector, then σ cannot belong to H^{p} for all p < ∞. Thus in view of the work of the previous section:

If C_{φ} is Riesz, then the image σ(U) of its Koenigs eigenfunction cannot contain a twisted sector.

On the other hand, in [11] Pietro Poggi-Corradini shows that if σ(U) contains no twisted sectors, then σ ∈ H^{p} for all p < ∞, and raises the problem of describing an operator-theoretic property of C_{φ} that is equivalent to σ(U)’s containing no twisted sectors. As we have seen, Rieszness of C_{φ} is sufficient to ensure no twisted sectors lie inside σ(U). Analysis of models presented in [17] provides some evidence that the Riesz property is also necessary: each model having the no-twisted-sector property (including the “bulging-strip” model we discussed in detail in Section 4) can be shown by application of Theorem 3.1 to yield a Riesz composition operator on H^{2}. This observation and further results in papers [1], [17], and [11] lead us to conjecture the following:

Suppose φ is univalent and σ is the Koenigs eigenfunction of φ; then the composition operator C_{φ} on H^{2} is Riesz if and only if σ(U) does not contain a twisted sector.

More generally, we conjecture that for an arbitrary holomorphic self-map φ(U) (satisfying φ(0) = 0 and 0 < |φ′(0)| < 1), the implication (8) goes both ways:

C_{φ} : H^{2} → H^{2} is Riesz if and only if σ ∈ H^{p} for all p < ∞.

As we mentioned in the Introduction, Poggi-Corradini [12] has recently established the first of these conjectures. We have just learned that he has proved the second as well [14].

We conclude by noting a connection between our work and the study of composition-operator spectra. In [5], Cowen and MacCluer characterize the spectrum of C_{φ} given that φ is univalent but not an automorphism, and φ(0) = 0; in particular they show that the spectrum of C_{φ} must contain the disc {z : |z| ≤ r_{e}(C_{φ})}, where r_{e}(C_{φ}) denotes the essential spectral radius of C_{φ} on H^{2}. Our Theorem 3.1 provides a sufficient condition for this disc to be degenerate (i.e. = {0}), while Theorem 4.2 shows that this can happen non-trivially.

References

[1] P. S. Bourdon and J. H. Shapiro, Mean growth of Koenigs eigenfunctions, Journal Amer. Math. Soc., to appear.

[2] C. Carathéodory, Theory of Functions of a Complex Variable, Vol. 2, Chelsea, New York, 1954.

[3] J. G. Caughran and H. J. Schwartz, Spectra of compact composition operators, Proc. Amer. Math. Soc. 51 (1975) 127–130.

[4] C. C. Cowen, Composition operators on H^{2}, J. Operator Th. 9 (1983), 77–106.

[5] C. C. Cowen and B. D. MacCluer, Spectra of some composition operators, J. Functional Analysis 125 (1994), 223–251.

[6] C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995.

[7] H. Dowson, Spectral Theory of Linear Operators, Academic Press 1978.

[8] G. Koenigs, Recherches sur les intégrales de certaines équationes functionelles, Annales Ecole Normale Superior (3) 1 (1884), Supplément, 3–41.

[9] J. E. Littlewood, On inequalities in the theory of functions, Proc. London Math. Soc. 23 (1925), 481–519.

[10] E. Nordgren, Composition operators, Canadian J. Math. 20 (1968), 442–449.

[11] P. Poggi-Corradini, Hardy spaces and twisted sectors for geometric models, Trans. Amer. Math. Soc., 348 (1996), 2503–2518..

[12] P. Poggi-Corradini, The Hardy class of geometric models and the essential spectral radius of composition operators, preprint.

[13] P. Poggi-Corradini, Riesz composition operators and their Königs eigenfunctions, preprint.

[14] P. Poggi-Corradini, The Hardy class of Kœnigs maps, preprint.

[15] J. H. Shapiro, The essential norm of a composition operator, Annals of Math. 125 (1987), 375–404.

[16] J. H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, 1993.

[17] J. H. Shapiro, W. Smith, and D. A. Stegenga, Geometric models and compactness of composition operators, J. Functional Analysis, 127 (1995) 21–62.

[18] J. H. Shapiro, Essential Fredholm Theory, in preparation.

[19] J. H. Shapiro and P. D. Taylor, Compact, nuclear, and Hilbert-Schmidt composition operators on H^{p}, Indiana Univ. Math. J. 23 (1973), 471–496.

Department of Mathematics Washington and Lee University Lexington, VA 24450

Department of Mathematics Michigan State University East Lansing, MI 48824