Abstract

Let φ be an analytic map that takes the open unit disk of ℂ into itself. The composition operator C_{φ}, acting on either the Hardy space H^{2} or one of the weighted Bergman spaces A_{α}^{2}, is defined by the rule

C_{φ}(f) = f ∘ φ.

Every composition operator is bounded on each of these spaces, but the exact value of its norm is rarely known. In view of this situation, we consider several problems relating to this basic, yet elusive question. Our strategy for studying ∥C_{φ}∥ centers around determining the spectrum of the operator C_{φ}^{∗}C_{φ}. We obtain our most substantial results in the context of H^{2}, in particular for a composition operator induced by a linear fractional map:

• We determine the conditions under which ∥C_{φ}∥ is given by the action of either C_{φ} or C_{φ}^{∗} on the reproducing kernel functions of H^{2}.

• Assuming a specific set of conditions on φ, we describe ∥C_{φ}∥ in terms of the zeros of a particular polynomial.

• We obtain several results, in the case where φ(0) = 0, pertaining to the norm of C_{φ} acting on various subspaces of H^{2}.

Acknowledgements

While I owe a debt of gratitude to many people, I wish to express my particular thanks to three individuals: to Barbara MacCluer, for introducing me to this area of analysis, sharing the benefit of her expertise, and patiently following my progress during the development of these results; to Paul Bourdon, for his enthusiastic interest in my research; and to my wife Kelli, whose continuing support is invaluable to all my endeavors, mathematical and otherwise.

Contents

Introductory Remarks 1

1 Composition Operators on Functional Hilbert Spaces 3 1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 The norm of an operator . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 The quantities S_{φ} and S_{φ}^{∗} . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Hardy Spaces and Weighted Bergman Spaces 11 2.1 The Hardy space H^{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 The weighted Bergman spaces A_{α}^{2} . . . . . . . . . . . . . . . . . . . . 16 2.3 Compactness and essential norms . . . . . . . . . . . . . . . . . . . . 19 2.4 The quantities S_{φ} and S_{φ}^{∗} . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Multiplication operators . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.6 Functional Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Composition Operators with Linear Fractional Symbol 35 3.1 Cowen’s adjoint formula . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 The operator C_{φ}^{∗}C_{φ} . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3 The quantities S_{φ} and S_{φ}^{∗} . . . . . . . . . . . . . . . . . . . . . . . . 40 4 The Spectrum of a Weighted Composition Operator 44

4.1 The spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2 An application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5 Norms of Composition Operators with Linear Fractional Symbol 52 5.1 The spectrum of C_{φ}^{∗}C_{φ} . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.2 The eigenfunctions of C_{φ}^{∗}C_{φ} . . . . . . . . . . . . . . . . . . . . . . . 61 5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.4 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6 Norms of Composition Operators on Subspaces of H^{2} 74 6.1 The essential norm of C_{φ} on H_{(n)}^{2} . . . . . . . . . . . . . . . . . . . . 75 6.2 The norm of C_{φ} on H_{(n)}^{2} . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.3 The linear fractional case . . . . . . . . . . . . . . . . . . . . . . . . . 78

Bibliography 85

List of Symbols 89

Index of Terminology 90

Introductory Remarks

This thesis considers questions relating to the norm of a composition operator C_{φ} acting on various Hilbert spaces of analytic functions.

• Chapter 1 defines our basic objects of study: functional Hilbert spaces, reproducing kernel functions, and composition operators. We outline our general strategy for determining the norm of an operator. The quantities S_{φ} and S_{φ}^{∗} are discussed for the first time.

• Chapter 2 introduces the particular Hilbert spaces in which we are most interested: the Hardy space H^{2} and the weighted Bergman spaces A_{α}^{2}. We discuss the essential norm of a composition operator acting on one of these spaces, as well as the quantities S_{φ} and S_{φ}^{∗}. We briefly consider multiplication operators and define the Banach spaces H^{p} and A_{α}^{p}.

• Chapter 3, which deals with composition operators induced by linear fractional maps, provides the foundation for much of our subsequent work. Having derived Cowen’s adjoint formula, we establish a useful representation for the operator C_{φ}^{∗}C_{φ} : H^{2} → H^{2}. We apply this result to obtain information about S_{φ} and S_{φ}^{∗}.

• Chapter 4 considers the spectrum of an operator which “locally resembles” a weighted composition operator. We use these results to determine the norm of a composition operator whose symbol has the form φ(z) = az + b.

• Chapter 5 contains the most significant results of this thesis. We introduce a class of linear fractional maps and determine the norm of a corresponding operator C_{φ} : H^{2} → H^{2}. We identify the elements of H^{2} on which such an operator attains its norm and give specific examples of maps belonging to this class. Several related results are also discussed.

• Chapter 6 considers the norm of a composition operator acting on various subspaces of H^{2}. We pay particular attention to the situation where φ is a linear fractional map.

Chapter 1

Composition Operators on

Functional Hilbert Spaces

This chapter introduces the general concepts considered in this thesis: functional Hilbert spaces, reproducing kernel functions, and composition operators. Section 1.2 outlines our basic strategy for obtaining results about the norm of an operator. The chapter concludes with a brief discussion of S_{φ} and S_{φ}^{∗}, the quantities we use to describe the action of a composition operator (and its adjoint) on the reproducing kernel functions of a functional Hilbert space.

1.1 Definitions

Let H be a nontrivial Hilbert space whose elements are functions on a set U. We call H a functional Hilbert space if every point-evaluation functional is bounded on H; that is, for every point w in U, the map λ_{w} : H → ℂ defined by the rule

λ_{w}(f) = f(w)

is continuous. In this case, for each w in U, the Riesz representation theorem guarantees the existence of a unique element K_{w} of H such that

⟨f, K_{w}⟩ = f(w)

for every f in H. These elements K_{w} are called the reproducing kernel functions for H. Provided that we have an explicit representation for K_{w}, it is always easy to compute its norm in H:

∥K_{w}∥ = \sqrt{⟨K_{w}, K_{w}⟩} = \sqrt{K_{w}(w)}.

We write k_{w} to denote the normalized kernel function K_{w}/∥K_{w}∥.

Any element f of H that is orthogonal to each K_{w} has the property that

f(w) = ⟨f, K_{w}⟩ = 0

for every w in U, which means that f is identically 0; in other words, the kernel functions span a dense subset of H. In general, for a subset W of U, we write K_{W} to denote the closed linear span of the kernel functions {K_{w} : w ∈ W }. The orthogonal complement K_{W}^{⊥} is precisely the set of all functions in H that vanish on W.

The term reproducing kernel Hilbert space is a common synonym for functional Hilbert space. If the elements of a functional Hilbert space H happen to be analytic functions on an open subset U of ℂ^{n}, we refer to H as a Hilbert space of analytic functions.

Consider a functional Hilbert space H on a set U. Let φ be a self-map of U, that is, a map taking U into U. The composition operator C_{φ} on H is defined by the rule

C_{φ}(f) = f ∘ φ.

The map φ is referred to as the symbol of the composition operator C_{φ}. It is easy to see that every composition operator is linear; moreover, we make the following observation:

Proposition 1.1 Any composition operator that takes H into H is automatically bounded.

Proof. We shall appeal to the closed graph theorem. Suppose that {f_{n}} is a sequence of functions in H converging in norm to an element f; suppose also that {C_{φ}(f_{n})} converges to some element g. Since each point-evaluation functional is bounded, the functions f_{n} converge pointwise to f; therefore the functions C_{φ}(f_{n}) = f_{n} ∘φ converge pointwise to C_{φ}(f) = f ∘ φ. The sequence {C_{φ}(f_{n})} must also converge pointwise to g, which implies that g = C_{φ}(f); hence our assertion follows.

While it is easy to describe a composition operator C_{φ}, it is often difficult to represent the adjoint C_{φ}^{∗} explicitly. The adjoint does, however, have one particularly useful property. Suppose that C_{φ} is bounded on a functional Hilbert space H; observe that

⟨f, C_{φ}^{∗}(K_{w})⟩ = ⟨C_{φ}(f), K_{w}⟩ = ⟨f ∘ φ, K_{w}⟩ = f(φ(w)) = ⟨f, K_{φ(w)}⟩

for each f in H. Consequently C_{φ}^{∗}(K_{w}) = K_{φ(w)} for every kernel function K_{w}.

1.2 The norm of an operator

Suppose that T is a bounded operator on a Hilbert (or Banach) space H. As long as there is no ambiguity regarding the space on which T is acting, we simply write ∥T ∥ to denote the norm of T; we write ∥T : H → H∥ whenever we want to specify the underlying space explicitly. The main results of this thesis pertain to norms of composition operators acting on various Hilbert spaces of analytic functions. In this section, we develop a general method for dealing with questions relating to norms.

Let T be a bounded operator on a Hilbert space H. One reasonable strategy for determining ∥T ∥ is to investigate the spectrum of the operator T^{∗}T. Since T^{∗}T is self-adjoint, its spectral radius equals ∥T^{∗}T ∥ = ∥T ∥^{2}. The following observation underscores the connection between the spectrum of T^{∗}T and the norm of T:

Proposition 1.2 Let h be an element of H; then ∥T (h)∥ = ∥T ∥ ∥h∥ if and only if (T^{∗}T)(h) = ∥T ∥^{2} h.

This proposition can be proved with a straightforward Hilbert space argument, as shown below, or can be deduced from other well-known results (e.g. Theorem III.43 in [17]).

Proof of Proposition 1.2. Suppose that (T^{∗}T)(h) = ∥T ∥^{2} h; then

∥T (h)∥^{2} = ⟨T (h), T (h)⟩ = ⟨(T^{∗}T)(h), h⟩ = ⟨∥T ∥^{2} h, h⟩ = ∥T ∥^{2} ∥h∥^{2} ,

as we had hoped to show.

Conversely, suppose that ∥T (h)∥ = ∥T ∥ ∥h∥; then

∥T ∥^{2} ∥h∥^{2} = ∥T (h)∥^{2} = ⟨T (h), T (h)⟩ = ⟨(T^{∗}T)(h), h⟩
≤ ∥(T^{∗}T)(h)∥ ∥h∥ ≤ ∥T ∥^{2} ∥h∥^{2} .

In particular, the first inequality is an equality, which means that (T^{∗}T)(h) is a scalar multiple of h; since ∥(T^{∗}T)(h)∥ = ∥T ∥^{2} ∥h∥ and T^{∗}T is a positive operator, we conclude that (T^{∗}T)(h) = ∥T ∥^{2} h.

Whenever ∥T (h)∥ = ∥T ∥ ∥h∥ for h ≠ 0, we say that the operator T attains its norm on the element h.

Let B(H) denote the Banach algebra consisting of all bounded operators on H; let B_{0}(H) denote the closed ideal consisting of the compact operators. The quotient B(H)/B_{0}(H) is often called the Calkin algebra. We write ∥T ∥_{e} (or ∥T : H → H∥_{e}) to denote the essential norm of an operator T in B(H); that is,

∥T ∥_{e} = inf_{K∈B_{0}(H)} ∥T − K∥.

The essential norm represents the distance between T and the compact operators; in particular, ∥T ∥_{e} = 0 if and only if T itself is compact. One can also view ∥T ∥_{e} as being the norm of the coset T + B_{0}(H) in the Calkin algebra. The essential spectrum of T is simply defined to be the spectrum of the element T + B_{0}(H) in B(H)/B_{0}(H); the essential spectral radius of T, which we write r_{e}(T), is the largest modulus belonging to a point in the essential spectrum of T. One can easily show that the essential spectrum of T is contained in the spectrum of T. Whenever T is a normal operator, any point in the spectrum of T that does not belong to the essential spectrum must be an eigenvalue of finite multiplicity (see Proposition XI.4.6 in [8]).

It is not difficult to show that ∥T^{∗}T ∥_{e} = ∥T ∥_{e}^{2} for any bounded operator T; from this fact, we can deduce that r_{e}(T) = ∥T ∥_{e} whenever T is self-adjoint. Similarly, recall that r(T), the spectral radius of T, equals ∥T ∥ for any self-adjoint operator. In light of Proposition 1.2, our next observation follows easily: Proposition 1.3 If ∥T ∥_{e} < ∥T ∥, then T attains its norm on an element of H. Proof. Consider the positive operator T^{∗}T; observe that

r_{e}(T^{∗}T) = ∥T^{∗}T ∥_{e} = ∥T ∥_{e}^{2} < ∥T ∥^{2} = ∥T^{∗}T ∥ = r(T^{∗}T).

Therefore ∥T ∥^{2}, the largest element of the spectrum of T^{∗}T, does not belong to the essential spectrum; since any self-adjoint operator is normal, ∥T ∥^{2} must be an eigenvalue of finite multiplicity. Consequently T^{∗}T has an eigenvector corresponding to ∥T ∥^{2}, on which the operator T attains its norm.

It is helpful to remember Proposition 1.3 when studying composition operators, particularly since it is often easier to compute ∥C_{φ}∥_{e} than ∥C_{φ}∥. As it happens, in most of the cases where we already know ∥C_{φ}∥ exactly (see Sections 2.1 and 2.2), the operator has the property that ∥C_{φ}∥_{e} = ∥C_{φ}∥, a condition sometimes called extremal noncompactness.

1.3 The quantities S_{φ} and S_{φ}^{∗}

Let H be a functional Hilbert space on a set U; suppose that φ is a self-map of U that induces a bounded composition operator on H. Recall that C_{φ}^{∗}(K_{w}) = K_{φ(w)} for any kernel function K_{w}; it seems reasonable, when investigating ∥C_{φ}∥, to consider the action of C_{φ}^{∗} on the reproducing kernel functions of H.

Define N to be the set of all w in U such that f(w) = 0 for each f in H. (For many spaces, including those introduced in Chapter 2, the set N is empty. We shall only encounter a few situations where N is nonempty: the spaces H_{(n)}^{2}, as defined in Chapter 6, have the property that N = {0} whenever n ≥ 1.) Since C_{φ} takes H into H, the set φ(N) must be contained in N. The kernel function K_{w} is identically zero if and only if the point w belongs to N. We define the quantity

S_{φ}^{∗} = sup_{w∈U\N} \{∥C_{φ}^{∗}(k_{w})∥\} = sup_{w∈U\N} \{\frac{∥C_{φ}^{∗}(K_{w})∥}{∥K_{w}∥}\}
= sup_{w∈U\N} \{\frac{∥K_{φ(w)}∥}{∥K_{w}∥}\} = sup_{w∈U\N} \{\sqrt{\frac{K_{φ(w)}(φ(w))}{K_{w}(w)}}\}.

Since C_{φ}^{∗}(K_{w}) = K_{φ(w)}, the set {K_{w} : w ∈ φ^{−1}(N)} consists precisely of those reproducing kernel functions which belong to the kernel of C_{φ}^{∗}. Therefore

S_{φ}^{∗} = sup_{w∈U\N} \{\frac{∥C_{φ}^{∗}(K_{w})∥}{∥K_{w}∥}\} = sup_{w∈U\φ^{−1}(N)} \{\frac{∥C_{φ}^{∗}(K_{w})∥}{∥K_{w}∥}\},

where the latter supremum is taken to be 0 if U = φ^{−1}(N).

In similar fashion, we define

S_{φ} = sup_{w∈U\N} \{∥C_{φ}(k_{w})∥\} = sup_{w∈U\N} \{\frac{∥C_{φ}(K_{w})∥}{∥K_{w}∥}\}.

The values of S_{φ} and S_{φ}^{∗} serve to quantify the actions of C_{φ} and C_{φ}^{∗} on the set of reproducing kernel functions. Whenever S_{φ} = ∥C_{φ}∥, the norm of C_{φ} is given by the action of the operator on the kernel functions; likewise, when S_{φ}^{∗} = ∥C_{φ}∥, the norm is given by the action of C_{φ}^{∗} on the kernel functions. While S_{φ}^{∗} is generally easier to compute, the quantity S_{φ} is more likely to equal ∥C_{φ}∥. In fact, we have the following result:

Proposition 1.4 Let C_{φ} be a bounded composition operator on a functional Hilbert space H; then S_{φ}^{∗} ≤ S_{φ}.

Proof. We begin with a general observation. Let T be a bounded operator on H; then, for any element f not in the kernel of T,

\frac{∥T (f)∥}{∥f∥} = \begin{matrix}∥T (f)∥ & 2 \\ ∥T (f)∥ & ∥f∥ \end{matrix} = \frac{⟨(T^{∗}T)(f), f⟩}{∥T (f)∥ ∥f∥} ≤ \begin{matrix}∥(T^{∗}T)(f)∥ & ∥f∥ \\ ∥T (f)∥ & ∥f∥ \end{matrix} = \frac{∥T^{∗}(T (f))∥}{∥T (f)∥}. (1.1)

If U = φ^{−1}(N), then S_{φ}^{∗} = 0; thus the claim of the proposition is trivially true. Otherwise, applying (1.1) with T = C_{φ}^{∗} and f from the set {K_{w} : w ∈ U \ φ^{−1}(N)}, we see that

S_{φ}^{∗} = sup_{w∈U\φ^{−1}(N)} \{\frac{∥C_{φ}^{∗}(K_{w})∥}{∥K_{w}∥}\} ≤ sup_{w∈U\φ^{−1}(N)} \{\frac{∥C_{φ} (C_{φ}^{∗}(K_{w}))∥}{∥C_{φ}^{∗}(K_{w})∥}\}
= sup_{w∈U\φ^{−1}(N)} \{\frac{∥C_{φ} (K_{φ(w)})∥}{∥K_{φ(w)}∥}\} ≤ S_{φ},

as we had hoped to show.

The quantities S_{φ} and S_{φ}^{∗}, under various names, have received a good deal of attention in recent years. Several authors ([1], [2], [6], and [23]) have studied these quantities in the context of the Hardy space H^{2}. Proposition 1.4 in this section is a straightforward generalization of Proposition 3.1 in [6]; the calculation of line (1.1) appears as Proposition 2.15 in [23]. We shall revisit the quantities S_{φ} and S_{φ}^{∗} in Sections 2.4 and 3.3.

Chapter 2

Hardy Spaces and Weighted

Bergman Spaces

Let 𝔻 denote the open unit disk in the complex plane. For the remainder of this thesis, we work with various Hilbert spaces of analytic functions on 𝔻. In particular, we consider composition operators acting on the Hardy space H^{2} (the space for which we obtain the most interesting results) and the weighted Bergman spaces A_{α}^{2}. Sections 2.1 and 2.2 serve as a brief introduction to this material, providing relevant definitions and general results. Section 2.3 summarizes much of what is known about the essential norms of composition operators acting on these spaces. In Section 2.4, we address the quantities S_{φ} and S_{φ}^{∗} in the particular context of H^{2} and A_{α}^{2}; much of this section is a generalization of the work of Bourdon and Retsek [6]. The last two sections of this chapter deal with topics which, although somewhat detached from the principal concerns of this thesis, pertain to the overall issues in which we are interested.

2.1 The Hardy space H^{2}

The Hardy space H^{2} is the Hilbert space consisting of all analytic functions f(z) = ∑_{n=0}^{∞} a_{n}z^{n} on 𝔻 with

∥f∥_{2}^{2} = ∑_{n=0}^{∞} |a_{n}|^{2} < ∞,

where the inner product of two functions f(z) = ∑_{n=0}^{∞} a_{n}z^{n} and g(z) = ∑_{n=0}^{∞} b_{n}z^{n} is defined

⟨f, g⟩ = ∑_{n=0}^{∞} a_{n}\overline{b_{n}}.

There are several other ways to express the Hardy space inner product: for example,

⟨f, g⟩ = lim_{r↑1} ∫_{0}^{2π} f(re^{iθ})\overline{g(re^{iθ})} \frac{dθ}{2π}

or

⟨f, g⟩ = ∫_{0}^{2π} f(e^{iθ})\overline{g(e^{iθ})} \frac{dθ}{2π},

where f and g are defined almost everywhere on ∂𝔻 via their radial limits (see, for example, Sections 2.1 and 2.3 of [13]).

Every point-evaluation functional is bounded on H^{2}, which means that H^{2} is a Hilbert space of analytic functions on 𝔻. The reproducing kernel functions for the Hardy space are easy to describe:

K_{w}(z) = ∑_{n=0}^{∞} \overline{w}^{n}z^{n} = \frac{1}{1 − \overline{w}z}.

Therefore

∥K_{w}∥_{2} = \sqrt{K_{w}(w)} = \sqrt{\frac{1}{1 − |w|^{2}}}

and

k_{w}(z) = \frac{K_{w}(z)}{∥K_{w}∥_{2}} = \frac{\sqrt{1 − |w|^{2}}}{1 − \overline{w}z}.

The Hardy space has proved to be an especially popular environment for the study of composition operators. The first publications on the subject ([20] and [25]) dealt exclusively with composition operators on H^{2} (or the related Banach spaces H^{p}). While mathematicians have subsequently considered composition operators acting on a variety of spaces, the Hardy space still remains a particular focus of attention. As we shall see, the techniques we use to prove our most interesting results depend on properties which are peculiar to H^{2}.

Any analytic map φ : 𝔻 → 𝔻 induces a bounded composition operator on H^{2}; in fact,

\sqrt{\frac{1}{1 − |φ(0)|^{2}}} ≤ ∥C_{φ} : H^{2} → H^{2}∥ ≤ \sqrt{\frac{1 + |φ(0)|}{1 − |φ(0)|}}.  (2.1)

(See, for example, Corollary 3.7 in [11].) The upper bound follows from the Littlewood subordination theorem [18]; the lower bound can be deduced from an argument similar to that of Lemma 2.12. In general, though, there is no known procedure for computing the norm of C_{φ} on H^{2}. We see from (2.1) that ∥C_{φ}∥ = 1 whenever φ(0) = 0. There are only a few other examples where we can determine the norm exactly; prior to the author’s research, ∥C_{φ}∥ was known for the following cases:

(1) φ is inner; that is, lim_{r↑1} |φ(re^{iθ})| = 1 for almost all θ in [0, 2π),
(2) φ(z) = az + b where |a| + |b| ≤ 1,
(3) φ(z) = \frac{(r+s)z+(1−s)}{r(1−s)z+(1+sr)} where 0 < s < 1 and 0 ≤ r ≤ 1.

Case (1) was established by Nordgren [20]. Cowen [9] was the first to calculate the norm in case (2); we provide a new proof of his result in Section 4.2. We can compute the norms in case (3) because the adjoints of the corresponding composition operators are subnormal, a result due to Cowen and Kriete [10]. Cases (1) and (3) have the property that ∥C_{φ}∥ = ∥C_{φ}∥_{e}, as does case (2) when |a| + |b| = 1.

The inequalities in (2.1) are sharp; for any value of φ(0), there are particular examples of φ for which ∥C_{φ}∥ equals the upper bound and examples for which ∥C_{φ}∥ equals the lower bound. As it turns out, there are only a few maps φ for which ∥C_{φ}∥ achieves either extreme. When φ(0) = 0, the norm of C_{φ} equals both the upper and lower bound; suppose then that φ(0) ≠ 0. In this case, Joel Shapiro [28] showed that ∥C_{φ}∥ equals the upper bound if and only if φ is an inner function. On the other hand, ∥C_{φ}∥ equals the lower bound if and only if φ is a constant map, as we shall show in Lemma 2.12.

In the aforementioned cases where we know ∥C_{φ}∥, the norm is given by the action of the operator on the set of reproducing kernel functions; that is, ∥C_{φ}∥ = S_{φ}. Moreover, ∥C_{φ}∥ = S_{φ}^{∗} in those cases where φ is univalent. This situation, however, does not hold in general, a fact first proved by Appel, Bourdon, and Thrall [1]. These authors considered the linear fractional map φ(z) = 2/(3 − z); while they did not calculate ∥C_{φ}∥ exactly, they did demonstrate that S_{φ}^{∗} < S_{φ} < ∥C_{φ}∥ for this particular φ. Bourdon and Retsek [6], in fact, showed that it is quite rare for S_{φ}^{∗} to equal ∥C_{φ}∥, a result we shall restate as Proposition 2.10.

Our main results for composition operators on H^{2} pertain to the situation where

φ(z) = \frac{az + b}{cz + d}

is a nonconstant linear fractional self-map of 𝔻. For such φ, we determine the conditions under which either S_{φ} = ∥C_{φ}∥ or S_{φ}^{∗} = ∥C_{φ}∥ (Theorem 3.7). We introduce a new set of conditions on φ under which, at least in principle, we can calculate ∥C_{φ}∥ (Theorem 5.11). Assuming these conditions, we identify the elements of H^{2} on which C_{φ} attains its norm, each of which is a finite linear combination of kernel functions. We also obtain some interesting results, in the case where φ(0) = 0, pertaining to the norm of C_{φ} acting on various subspaces of H^{2} (Theorems 6.10 and 6.12).

Before we turn our attention to other Hilbert spaces of analytic functions, we mention a few results which are specific to composition operators on H^{2}. The first, which we shall require on several occasions, is originally due to Nordgren [20]:

Proposition 2.1 Let φ : 𝔻 → 𝔻 be an inner function with φ(0) = 0. The operator C_{φ} : H^{2} → H^{2} is an isometry of H^{2}.

Our proof is identical to that presented in [11].

Proof of Proposition 2.1. Since φ is an inner function, \overline{φ(e^{iθ})} = (φ(e^{iθ}))^{−1} for almost all θ in [0, 2π). Therefore ∥φ^{n}∥_{2} = 1 for all n and

⟨φ^{n}, φ^{m}⟩ = ∫_{0}^{2π}(φ(e^{iθ}))^{n}\overline{(φ(e^{iθ}))^{m}} \frac{dθ}{2π} = ∫_{0}^{2π}(φ(e^{iθ}))^{n−m} \frac{dθ}{2π} = (φ(0))^{n−m} = 0

whenever n > m. Let f(z) = ∑_{n=0}^{∞} a_{n}z^{n} be an arbitrary element of H^{2}; observe that

∥f ∘ φ∥_{2}^{2} = ⟨∑_{n=0}^{∞} a_{n}φ^{n}, ∑_{m=0}^{∞} a_{m}φ^{m}⟩ = ∑_{n=0}^{∞} |a_{n}|^{2} = ∥f∥_{2}^{2} .

In other words, the operator C_{φ} : H^{2} → H^{2} is an isometry.

The next result pertains to the elements of H^{2} on which a composition operator attains its norm:

Proposition 2.2 Suppose that the operator C_{φ} : H^{2} → H^{2} attains its norm on an element g of H^{2}. If φ is not an inner function, then g cannot vanish at any point w of 𝔻.

Proof. Suppose that g(w) = 0 for some w in 𝔻. Consider the Blaschke factor

b_{w}(z) = \frac{w − z}{1 − \overline{w}z}.

This function is an isometric zero-divisor for H^{2}; that is, the function

g̃(z) = \frac{g(z)}{b_{w}(z)}

belongs to H^{2}, with \left∥{ \atop g}\right∥_{2} = ∥g∥_{2}. Because φ is not an inner function, neither is the composition b_{w} ∘ φ. Since lim_{r↑1} g(re^{iθ}) is nonzero for almost all θ, we see that

lim_{r↑1} |\frac{g(φ(re^{iθ}))}{b_{w}(φ(re^{iθ}))}| > lim_{r↑1} |g(φ(re^{iθ}))|

for θ in a set of positive measure. Hence ∥C_{φ}(g̃)∥_{2} > ∥C_{φ}(g)∥_{2}, contradicting our choice of g. Corollary 2.3 Suppose that φ is not inner; if g_{1} and g_{2} are functions on which C_{φ} attains its norm, then one is a scalar multiple of the other. Proof. Both g_{1} and g_{2} are eigenfunctions for C_{φ}^{∗}C_{φ} : H^{2} → H^{2} corresponding to the eigenvalue ∥C_{φ}∥^{2}; moreover, g_{1}(0) and g_{2}(0) are both nonzero. If g_{1} − (g_{1}(0)/g_{2}(0)) g_{2} were not identically 0, then it would be an eigenfunction corresponding to ∥C_{φ}∥^{2}, in other words a function on which C_{φ} attains its norm, that vanishes at 0. Therefore g_{1} = (g_{1}(0)/g_{2}(0)) g_{2}, as we had hoped to show.

2.2 The weighted Bergman spaces A_{α}^{2}

For α > −1, the weighted Bergman space A_{α}^{2} is the Hilbert space consisting of all analytic functions f on 𝔻 with

∥f∥_{2,α}^{2} = ∫_{𝔻} |f(z)|^{2} (α + 1) (1 − |z|^{2})^{α} \frac{dA(z)}{π} < ∞,

where dA denotes area measure on 𝔻. The inner product of two functions f and g in A_{α}^{2} is defined

⟨f, g⟩_{α} = ∫_{𝔻} f(z)\overline{g(z)}(α + 1) (1 − |z|^{2})^{α} \frac{dA(z)}{π}.

The space A_{0}^{2} = A^{2} is often referred to simply as the (unweighted) Bergman space. The inner product in A_{α}^{2} can also be expressed as a series; given f(z) = ∑_{n=0}^{∞} a_{n}z^{n} and g(z) = ∑_{n=0}^{∞} b_{n}z^{n} in A_{α}^{2}, we may write

⟨f, g⟩_{α} = ∑_{n=0}^{∞} a_{n}\overline{b_{n}}β_{α}(n),

where

β_{α}(n) = ∥z^{n}∥_{2,α}^{2} = \frac{(α + 1)Γ(n + 1)Γ(α + 1)}{Γ(n + α + 2)}.

(See, for example, Section 1 of [22].) In particular,

β_{0}(n) = \frac{1}{n + 1} and β_{1}(n) = \frac{2}{(n + 1)(n + 2)}.

Every weighted Bergman space A_{α}^{2} is a functional Hilbert space; the reproducing kernel functions for A_{α}^{2} may be written

K_{w}(z) = (\frac{1}{1 − \overline{w}z} )^{α+2} ,

where the power function is defined by taking the principal branch of the appropriate logarithm; note that

k_{w}(z) = \frac{K_{w}(z)}{∥K_{w}∥_{2,α}} = (\frac{1 − |w|^{2}}{(1 − \overline{w}z)^{2}} )^{(α+2)/2} .

As we can easily see, the kernel functions for the Hardy space H^{2} have the form corresponding to those for the nonexistent “α = −1” weighted Bergman space. This similarity will allow us to prove several results simultaneously for the Hardy space and the weighted Bergman spaces. For this reason, as a matter of convenience, we often write A_{−1}^{2} to denote the Hardy space H^{2} (with the usual norm and inner product).

Composition operators have also received a good deal of attention in the context of the weighted Bergman spaces. Any analytic φ : 𝔻 → 𝔻 induces a bounded composition operator on every space A_{α}^{2}; in fact, for α > −1, we have that

(\frac{1}{1 − |φ(0)|^{2}} )^{(α+2)/2} ≤ ∥C_{φ} : A_{α}^{2} → A_{α}^{2}∥ ≤ (\frac{1 + |φ(0)|}{1 − |φ(0)|})^{(α+2)/2} .  (2.2)

(See, for example, Lemma 1.2.2 in [24].) Comparing with line (2.1), we see that these bounds are analogous to those for composition operators acting on A_{−1}^{2} = H^{2}. As in the case of the Hardy space, there are only a few examples of φ (with φ(0) ≠ 0) for which we can calculate ∥C_{φ} : A_{α}^{2} → A_{α}^{2}∥. For instance, we know the norm of C_{φ} acting on the unweighted Bergman space A^{2} when

(1) φ is an automorphism of 𝔻; that is, φ(z) = s (\frac{w−z}{1−\overline{w}z}) where |w| < |s| = 1,

(2) φ(z) = az + b where |a| + |b| ≤ 1,

(3) φ(z) = \frac{(r+s)z+(1−s)}{r(1−s)z+(1+sr)} where 0 < s < 1 and −1/7 ≤ r ≤ 1. Case (2) was established by Hurst [15], who adapted Cowen’s [9] proof for C_{φ} : H^{2} → H^{2}; in fact, Hurst determined the norm of such a composition operator acting on any of the weighted Bergman spaces. Case (3) follows from the cosubnormality results in Richman’s thesis [24]. As one might expect, the maps in cases (1) and (3) induce extremally noncompact composition operators, as do the maps in case (2) when |a| + |b| = 1.

We conclude this section with a straightforward, but remarkably useful observation: Observation 2.4 Take α ≥ −1. Let λ be an eigenvalue for C_{φ}^{∗}C_{φ} : A_{α}^{2} → A_{α}^{2} with a corresponding eigenfunction g; since C_{φ} fixes the constant function K_{0}(z) = 1, we see that

g(φ(0)) = ⟨C_{φ}(g), K_{0}⟩_{α} = ⟨C_{φ}(g), C_{φ}(K_{0})⟩_{α}
= ⟨(C_{φ}^{∗}C_{φ}) (g), K_{0}⟩_{α} = λ ⟨g, K_{0}⟩_{α} = λg(0).

In particular, Proposition 1.2 dictates that

g(φ(0)) = ∥C_{φ}∥^{2} g(0)

whenever C_{φ} attains its norm on g.

2.3 Compactness and essential norms

While not much is known about the norm of C_{φ}, we do have a good deal of information about its essential norm. Joel Shapiro [26] obtained an explicit formula for ∥C_{φ} : H^{2} → H^{2}∥_{e}; a similar formula was later discovered by Poggi-Corradini [22] for composition operators acting on A_{0}^{2} = A^{2} and A_{1}^{2}. This section includes the statements of these results, as well as several facts pertaining to the compactness of C_{φ}. We begin with the following lemma: Lemma 2.5 Take α ≥ −1. Let {f_{n}} be a sequence of functions in A_{α}^{2}. The following three conditions are equivalent:

(a) {f_{n}} is bounded in norm and converges to 0 pointwise on 𝔻.

(b) {f_{n}} is bounded in norm and converges to 0 uniformly on compact subsets of 𝔻.

(c) {f_{n}} converges to 0 weakly in A_{α}^{2}. Proof. Corollary 1.3 in [11] shows that conditions (a) and (c) are equivalent; condition (b) certainly implies condition (a). Suppose then that {f_{n}} is a sequence which converges to 0 pointwise on 𝔻, with

∥f_{n}∥_{2,α} ≤ M < ∞

for all n. Let Δ be a compact subset of 𝔻; then the quantity

H_{Δ} = sup_{w∈Δ}{|w|}

is strictly less than 1. Hence, for every function f_{n},

|f_{n}(w)| = ⟨f_{n}, K_{w}⟩_{α} ≤ ∥f_{n}∥_{2,α} ∥K_{w}∥_{2,α}
= \begin{matrix}∥f_{n}∥ & 2,α \\ (1 − |w|^{2}) & (α+2)/2 \end{matrix} ≤ \frac{M}{(1 − (H_{Δ})^{2})^{(α+2)/2}}

for all w in Δ. It follows from Montel’s theorem that the sequence converges to 0 uniformly on every compact subset of 𝔻.

Using this characterization of weak convergence, we can easily prove a well-known result relating to the compactness of C_{φ}. The following proposition, stated for composition operators acting on H^{2}, first appeared in Schwartz’s thesis [25]:

Proposition 2.6 Take α ≥ −1. Let φ be an analytic self-map of 𝔻. If the closure of the set φ(𝔻) is contained in 𝔻, then the operator C_{φ} : A_{α}^{2} → A_{α}^{2} is compact.

Proof. Let {f_{n}} be a sequence in A_{α}^{2} that converges weakly to 0. Lemma 2.5 shows that the sequence converges to 0 uniformly on the closure of φ(𝔻). In other words, the functions C_{φ}(f_{n}) = f_{n} ∘ φ converge to 0 uniformly on 𝔻, which implies that they converge to 0 in norm. Hence C_{φ} is completely continuous; since A_{α}^{2} is a Hilbert space, the operator must be compact.

We now state Shapiro and Poggi-Corradini’s essential norm formulae. For a real number ρ ≥ 0, we define the Nevanlinna-type counting function N_{φ,ρ} by the rule

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

for any w in 𝔻 \ {φ(0)}. For α equal to −1, 0, or 1, Shapiro and Poggi-Corradini showed that

∥C_{φ} : A_{α}^{2} → A_{α}^{2}∥_{e}^{2} = lim sup_{|w|↑1} \frac{N_{φ,α+2}(w)}{(− log |w|)^{α+2}} .

Whenever φ is univalent, this formula simply reduces to

∥C_{φ} : A_{α}^{2} → A_{α}^{2}∥_{e}^{2} = lim sup_{|w|↑1} (\frac{1 − |w|^{2}}{1 − |φ(w)|^{2}} )^{α+2} .  (2.3)

In the event that φ is univalent on a neighborhood of the closed unit disk, we have an even more attractive description of the essential norm. In this case, the Julia– Carathéodory theorem (see Theorem 2.44 in [11]) dictates that

∥C_{φ} : A_{α}^{2} → A_{α}^{2}∥_{e}^{2} = max {|φ′(w)|^{−(α+2)} : |w| = |φ(w)| = 1},  (2.4)

where the maximum over an empty set is taken to be 0. In particular, such a φ induces a compact composition operator on one of these three spaces if and only if

∥φ∥_{∞} = sup_{w∈𝔻}{|φ(w)|} < 1.

At this point, we mention a particularly helpful consequence of Shapiro and Poggi-Corradini’s proofs. Both authors observe that

∥C_{φ}∥_{e}^{2} ≥ lim sup_{|w|↑1} ∥C_{φ}(k_{w})∥_{2,α}^{2} ,

which in turn they show to be greater than or equal to

lim sup_{|w|↑1} \frac{N_{φ,α+2}(w)}{(− log |w|)^{α+2}} ,

the quantity they eventually prove to equal the square of the essential norm. Consequently

∥C_{φ}∥_{e} = lim sup_{|w|↑1} ∥C_{φ}(k_{w})∥_{2,α}  (2.5)

whenever α equals −1, 0, or 1. This observation appears to have been made first in the context of H^{2} by Cima and Matheson [7].

2.4 The quantities S_{φ} and S_{φ}^{∗}

Having become acquainted with the Hardy space and the weighted Bergman spaces, we redirect our attention to the the quantities S_{φ} and S_{φ}^{∗} (as defined in Section 1.3). Take α ≥ −1; throughout this section, unless otherwise stated, we assume that C_{φ} is a composition operator acting on A_{α}^{2}. Since every such space contains the constant functions, the set N = {w ∈ 𝔻 : f(w) = 0 for all f ∈ A_{α}^{2}} is empty; therefore

S_{φ} = sup_{w∈𝔻}\{∥C_{φ}(k_{w})∥_{2,α}\} = sup_{w∈𝔻}\begin{cases}∥C_{φ}(K_{w})∥ & 2,α \\ ∥K_{w}∥ & 2,α \end{cases} \}

and

S_{φ}^{∗} = sup_{w∈𝔻}\{∥C_{φ}^{∗}(k_{w})∥_{2,α}\} = sup_{w∈𝔻}\begin{cases}∥C_{φ}^{∗}(K_{w})∥ & 2,α \\ ∥K_{w}∥ & 2,α \end{cases} \}
= sup_{w∈𝔻}\begin{cases}∥K_{φ(w)}∥ & 2,α \\ ∥K_{w}∥ & 2,α \end{cases} \} = sup_{w∈𝔻} \{(\frac{1 − |w|^{2}}{1 − |φ(w)|^{2}} )^{(α+2)/2}\}.

It follows from Proposition 1.4 that S_{φ}^{∗} ≤ S_{φ}.

The next few remarks simplify our discussion of S_{φ} and S_{φ}^{∗}.

Lemma 2.7 Suppose that {w_{j}} is a sequence of points converging to w in 𝔻; then the normalized kernel functions {k_{w_{j}}} converge to k_{w} in the norm of A_{α}^{2}.

Proof. Simply note that

∥k_{w_{j}} − k_{w}∥_{2,α}^{2} = ⟨k_{w_{j}} − k_{w}, k_{w_{j}} − k_{w}⟩_{α}
= ∥k_{w_{j}} ∥_{2,α}^{2} − 2 Re ⟨k_{w}, k_{w_{j}}⟩_{α} + ∥k_{w}∥_{2,α}^{2}
= 2 − 2 Re (\begin{matrix}⟨k_{w}, K_{w_{j}}⟩ & α \\ ∥K_{w_{j}} ∥ & 2,α \end{matrix} )
= 2 − 2 Re (k_{w}(w_{j}) (1 − |w_{j}|^{2})^{(α+2)/2}).

Since k_{w} is a continuous function,

lim_{j→∞} k_{w}(w_{j}) (1 − |w_{j}|^{2})^{(α+2)/2} = k_{w}(w) (1 − |w|^{2})^{(α+2)/2} = 1,

and our claim follows.

Since C_{φ} and C_{φ}^{∗} are both bounded operators, we obtain the following characterization of S_{φ} and S_{φ}^{∗}: Observation 2.8 Either S_{φ} = ∥C_{φ}(k_{w})∥_{2,α} for a particular w in 𝔻 or

S_{φ} = lim sup_{|w|↑1} ∥C_{φ}(k_{w})∥_{2,α}.

Likewise, either S_{φ}^{∗} = ∥C_{φ}^{∗}(k_{w})∥_{2,α} for some w in 𝔻 or

S_{φ}^{∗} = lim sup_{|w|↑1} ∥C_{φ}^{∗}(k_{w})∥_{2,α} = lim sup_{|w|↑1} (\frac{1 − |w|^{2}}{1 − |φ(w)|^{2}} )^{(α+2)/2} .

One of our main objectives is to determine the conditions under which either S_{φ} or S_{φ}^{∗} equals ∥C_{φ}∥. (If S_{φ}^{∗} = ∥C_{φ}∥, then of course S_{φ} = ∥C_{φ}∥ as well.) Bourdon and Retsek [6] considered this question for composition operators acting on H^{2}; much of this section is a generalization of the material from Section 3 of their paper.

If φ(0) = 0, then this question does not demand a great deal of attention. In this case, expressions (2.1) and (2.2) dictate that ∥C_{φ} : A_{α}^{2} → A_{α}^{2}∥ = 1. Therefore

∥C_{φ}∥ = ∥k_{0}∥_{2,α} = ∥k_{φ(0)}∥_{2,α} = ∥C_{φ}^{∗}(k_{0})∥_{2,α},

so that S_{φ}^{∗} = S_{φ} = ∥C_{φ}∥. Hence we need only consider the case where φ(0) ≠ 0.

Our next lemma pertains to the quantity S_{φ}^{∗}. Lemma 2.9 Suppose that φ : 𝔻 → 𝔻 is an analytic map with φ(0) ≠ 0. If the operator C_{φ}^{∗} : A_{α}^{2} → A_{α}^{2} attains its norm on a kernel function, then φ must have the form φ(z) = az + b.

Proof. Suppose that C_{φ}^{∗} attains its norm on the kernel function K_{w}. Proposition 1.2 dictates that

∥C_{φ}∥^{2} K_{w} = (C_{φ}C_{φ}^{∗}) (K_{w}) = C_{φ} (K_{φ(w)}).

Hence

\begin{matrix}∥C_{φ}∥ & 2 \\ (1 − \overline{w}z) & α+2 \end{matrix} = (\frac{1}{1 − \overline{φ(w)}φ(z)})^{α+2}

for all z in 𝔻. Thus it must be the case that

\frac{1 − \overline{φ(w)}φ(z)}{1 − \overline{w}z} = c  (2.6)

for some constant c. If φ(w) happened to equal 0, then w would also be 0, contradicting the assumption that φ(0) ≠ 0. Hence φ(w) ≠ 0, so we can solve (2.6) explicitly for φ(z) to show that it has the desired form.

In light of this fact, it is not difficult to prove the natural generalization of Proposition 3.4 in [6]:

Proposition 2.10 Let φ : 𝔻 → 𝔻 be an analytic map with φ(0) ≠ 0 and which does not have the form φ(z) = az + b; consider the operator C_{φ} : A_{α}^{2} → A_{α}^{2}. If the quantity S_{φ}^{∗} equals ∥C_{φ}∥, then ∥C_{φ}∥ = ∥C_{φ}∥_{e}.

Proof. The normalized kernel functions k_{w} are bounded in norm and converge pointwise to 0 as |w| approaches 1; thus Lemma 2.5 dictates that the k_{w} converge weakly to 0 as |w| tends to 1. Let K be a compact operator on A_{α}^{2}; observe that

∥C_{φ}^{∗} − K∥ ≥ lim sup_{|w|↑1} ∥(C_{φ}^{∗} − K) (k_{w})∥_{2,α}
≥ lim sup_{|w|↑1} (∥C_{φ}^{∗}(k_{w})∥_{2,α} − ∥K(k_{w})∥_{2,α})
= lim sup_{|w|↑1} ∥C_{φ}^{∗}(k_{w})∥_{2,α}.

Taking the infimum over the set of compact operators, we see that

∥C_{φ}∥_{e} = ∥C_{φ}^{∗}∥_{e} ≥ lim sup_{|w|↑1} ∥C_{φ}^{∗}(k_{w})∥_{2,α}.

Lemma 2.9 tells us that ∥C_{φ}∥ > ∥C_{φ}^{∗}(k_{w})∥_{2,α} for every w in 𝔻. Therefore, if ∥C_{φ}∥ = S_{φ}^{∗}, Observation 2.8 shows that

∥C_{φ}∥ = S_{φ}^{∗} = lim sup_{|w|↑1} ∥C_{φ}^{∗}(k_{w})∥_{2,α},

which implies that ∥C_{φ}∥ = ∥C_{φ}∥_{e}.

Bourdon and Retsek [6], in fact, proved that S_{φ}^{∗} = S_{φ} = ∥C_{φ}∥ whenever φ has the form φ(z) = az + b. Their proof, which is stated for composition operators acting on H^{2}, can be easily adapted to the weighted Bergman spaces A_{α}^{2} for α > −1.

Proposition 2.10 demonstrates how uncommon it is for S_{φ}^{∗} to equal ∥C_{φ}∥. The following result, which is specific to H^{2}, further illustrates this point:

Proposition 2.11 Suppose that φ : 𝔻 → 𝔻 can be written as the composition μ ∘ ν, where μ is an analytic self-map of 𝔻 and ν is a non-univalent inner function. Consider the operator C_{φ} : H^{2} → H^{2}; if φ(0) ≠ 0, then S_{φ}^{∗} < ∥C_{φ}∥.

This proposition is a generalization of Theorem 3.5 in [6], which gives the result in the case where μ(z) = z. The main idea of our proof is derived from Theorem 7.9 in [5]. The following argument requires a certain amount of geometric function theory; we refer the reader to Section 2.3 of [11] for the prerequisite material.

Proof of Proposition 2.11. Without loss of generality, we may assume that the inner function ν has the property that ν(0) = 0. If ν(0) were nonzero, we could introduce the involutive automorphism

b(z) = \frac{ν(0) − z}{1 − \overline{ν(0)}z},

observing that φ = μ ∘ ν = (μ ∘ b) ∘ (b ∘ ν). The map μ ∘ b is an analytic self-map of the disk; b ∘ ν is a non-univalent inner function that fixes the origin.

Since φ is not univalent, it cannot have the form φ(z) = az + b. Therefore, if S_{φ}^{∗} were to equal ∥C_{φ}^{∗}(k_{w})∥_{2} for some particular w, Lemma 2.9 would dictate that S_{φ}^{∗} < ∥C_{φ}∥. Suppose then that S_{φ}^{∗} = lim sup_{|w|↑1} ∥C_{φ}^{∗}(k_{w})∥_{2} (which means that the limit superior is positive). In this case, Proposition 2.46 in [11] shows that there is some ζ on ∂𝔻 such that

S_{φ}^{∗} = lim sup_{|w|↑1} \sqrt{\frac{1 − |w|^{2}}{1 − |φ(w)|^{2}}} = lim sup_{|w|↑1} \sqrt{\frac{1 − |w|}{1 − |φ(w)|}} = |φ′(ζ)|^{−1/2} .

Here φ′(ζ) denotes the (finite) angular derivative of φ at ζ.

Our first goal is to show that ν also has finite angular derivative at the point ζ. Since φ has finite angular derivative at ζ, it follows that

lim_{r↑1} |μ(ν(rζ))| = lim_{r↑1} |φ(rζ)| = 1,

and hence

lim_{r↑1} |ν(rζ)| = 1.

In other words, there is at least one sequence {w_{n}} (which converges to ζ nontangentially) such that lim_{n→∞} |ν(w_{n})| = 1. The Julia–Carathéodory theorem (see Theorem 2.44 in [11]) dictates that there is a sequence of points {z_{n}}, converging to ζ in a nontangential approach region, such that

lim_{n→∞} \frac{1 − |ν(z_{n})|}{1 − |z_{n}|} = lim inf_{z→ζ} \frac{1 − |ν(z)|}{1 − |z|}.

If the quantity

lim sup_{n→∞} |ν(z_{n})|

were strictly less than 1, then

lim inf_{z→ζ} \frac{1 − |ν(z)|}{1 − |z|}  (2.7)

would be infinite. Since (2.7) is always less than or equal to ∞, we may assume that lim sup_{n→∞} |ν(z_{n})| = 1; passing to a subsequence, we assume further that {ν(z_{n})} converges to a point ω on ∂𝔻. Note that

lim inf_{n→∞} \frac{1 − |μ(ν(z_{n}))|}{1 − |ν(z_{n})|} ≥ lim inf_{z→ω} \frac{1 − |μ(z)|}{1 − |z|},

a quantity which is always greater than 0. Since |ν(z)| < 1 whenever |z| < 1, we may write

\frac{1 − |φ(z)|}{1 − |z|} = \frac{1 − |μ(ν(z))|}{1 − |ν(z)|} · \frac{1 − |ν(z)|}{1 − |z|}

for all z in 𝔻. Therefore

lim inf_{n→∞} \frac{1 − |φ(z_{n})|}{1 − |z_{n}|} ≥ lim inf_{n→∞} \frac{1 − |μ(ν(z_{n}))|}{1 − |ν(z_{n})|} · lim_{n→∞} \frac{1 − |ν(z_{n})|}{1 − |z_{n}|}.

The Julia–Carathéodory theorem shows that

lim inf_{n→∞} \frac{1 − |φ(z_{n})|}{1 − |z_{n}|} = lim_{n→∞} \frac{1 − |φ(z_{n})|}{1 − |z_{n}|} = |φ′(ζ)| < ∞.

Consequently

lim inf_{z→ζ} \frac{1 − |ν(z)|}{1 − |z|} = lim_{n→∞} \frac{1 − |ν(z_{n})|}{1 − |z_{n}|} < ∞,

from which it follows that ν has finite angular derivative at ζ.

Because of its finite angular derivative, ν must have radial limit of modulus 1 at ζ, a point on ∂𝔻 we denote ν(ζ). Our next task is to show that μ has finite angular derivative at ν(ζ). Since both φ and ν have finite angular derivative at ζ, the Julia–Carathéodory theorem dictates that

lim inf_{z→ν(ζ)} \frac{1 − |μ(z)|}{1 − |z|} ≤ lim sup_{r↑1} \frac{1 − |μ(ν(rz))|}{1 − |ν(rz)|}
≤ lim sup_{r↑1} \frac{1 − |φ(rz)|}{1 − |rz|} · lim sup_{r↑1} \frac{1 − |rz|}{1 − |ν(rz)|}
= \frac{|φ′(ζ)|}{|ν′(ζ)|} < ∞.

Hence μ does indeed have finite angular derivative at the point ν(ζ), with

|μ′(ν(ζ))| ≤ \frac{|φ′(ζ)|}{|ν′(ζ)|}.  (2.8)

We now return to the problem at hand. Since ν(0) = 0 and ν is not an automorphism, Lemma 7.33 in [11] shows that

lim inf_{z→ζ} \frac{1 − |ν(z)|}{1 − |z|} > 1.

Appealing once more to the Julia–Carathéodory theorem, we see that that |ν′(ζ)| > 1. Thus line (2.8) shows that

|μ′(ν(ζ))| ≤ \frac{|φ′(ζ)|}{|ν′(ζ)|} < |φ′(ζ)|.

Consequently

|φ′(ζ)|^{−1/2} < |μ′(ν(ζ))|^{−1/2} ≤ lim sup_{|w|↑1} \sqrt{\frac{1 − |w|}{1 − |μ(w)|}}
= lim sup_{|w|↑1} \sqrt{\frac{1 − |w|^{2}}{1 − |μ(w)|^{2}}} = lim sup_{|w|↑1} ∥C_{μ}^{∗}(k_{w})∥_{2} .

In the proof of Proposition 2.10, we noted that

lim sup_{|w|↑1} ∥C_{μ}^{∗}(k_{w})∥_{2} ≤ ∥C_{μ}∥_{e}.

Since ν is an inner function that fixes the origin, Proposition 2.1 dictates that C_{ν} is an isometry of H^{2}. Therefore

∥C_{φ}∥_{e} = ∥C_{μ∘ν}∥_{e} = ∥C_{ν}C_{μ}∥_{e} = ∥C_{μ}∥_{e}.

Combining these observations, we see that

S_{φ}^{∗} < ∥C_{φ}∥_{e} ≤ ∥C_{φ}∥,

as we had hoped to show.

In particular, this result applies in the case where ν(z) = z^{m} for some integer m ≥ 2, so that φ(z) = μ(z^{m}).

It is much more difficult to obtain information about the quantity S_{φ}. Most of our discussion of S_{φ} will be specific to composition operators C_{φ} : H^{2} → H^{2}, where φ : 𝔻 → 𝔻 is a linear fractional map (see Section 3.3). We can, though, obtain one general lemma relating to S_{φ} (Lemma 2.13); our proof depends on Lemma 2.12, which pertains to the lower bounds in expressions (2.1) and (2.2). Lemma 2.12, stated in the context of the Hardy space, appears with a different proof in a current paper of David Pakorny and Jonathan Shapiro [21]. Lemma 2.12 If φ : 𝔻 → 𝔻 is a nonconstant analytic map with φ(0) ≠ 0, then

∥C_{φ} : A_{α}^{2} → A_{α}^{2}∥ > (\frac{1}{1 − |φ(0)|^{2}} )^{(α+2)/2} .

Proof. Our hypotheses guarantee that the function k_{φ(0)} ∘ φ is nonconstant; that is, it is not a scalar multiple of the kernel function K_{0}(z) = 1. Since K_{0} has norm 1, we see that

(\frac{1}{1 − |φ(0)|^{2}} )^{(α+2)/2} = |(k_{φ(0)} ∘ φ) (0)| = ⟨C_{φ} (k_{φ(0)}) , K_{0}⟩_{α}
< ∥C_{φ} (k_{φ(0)})∥_{2,α} ∥K_{0}∥_{2,α} ≤ ∥C_{φ}∥,

30

as we had hoped to show.

Lemma 2.13 Suppose that φ : 𝔻 → 𝔻 is a nonconstant analytic map with φ(0) ≠ 0. If the operator C_{φ} : A_{α}^{2} → A_{α}^{2} attains its norm on a kernel function K_{w}, then |w| > |φ(0)|.

Proof. Suppose that C_{φ} attains its norm on K_{w}; then K_{w} is an eigenfunction for C_{φ}^{∗}C_{φ} corresponding to ∥C_{φ}∥^{2}. Appealing to Observation 2.4, we see that

\frac{1}{1 − \overline{w}φ(0)} = K_{w}(φ(0)) = ∥C_{φ}∥^{2} K_{w}(0) = ∥C_{φ}∥^{2} .

It follows from Lemma 2.12 that

\frac{1}{1 − \overline{w}φ(0)} > \frac{1}{1 − |φ(0)|^{2}} ,

meaning that |w| > |φ(0)|.

We end this section with a restatement of Observation 2.8, valid for α equal to −1, 0, or 1. Recalling equation (2.5) and the univalent version of the essential norm formula (2.3), we obtain the following characterization of the quantities S_{φ} and S_{φ}^{∗}:

Observation 2.14 Consider the composition operator C_{φ} acting on A_{α}^{2}, where α equals −1, 0, or 1. Either S_{φ} = ∥C_{φ}(k_{w})∥_{2,α} for a particular w in 𝔻 or S_{φ} = ∥C_{φ}∥_{e}. Similarly, if φ is univalent, then either S_{φ}^{∗} = ∥C_{φ}^{∗}(k_{w})∥_{2,α} for some w in 𝔻 or S_{φ}^{∗} = ∥C_{φ}∥_{e}.

2.5 Multiplication operators

In accordance with standard usage, we write H^{∞} to denote the Banach space consisting of all bounded analytic function on 𝔻, under the norm

∥f∥_{∞} = sup_{w∈𝔻}{|f(w)|}.

For an element γ of H^{∞}, we define the multiplication operator M_{γ} by the rule

(M_{γ}(f))(z) = γ(z)f(z).

It is not difficult to show that M_{γ} is bounded on each space A_{α}^{2}, with

∥M_{γ} : A_{α}^{2} → A_{α}^{2}∥ = ∥γ∥_{∞}

for every α ≥ −1. Multiplication operators belong to a larger class of operators on A_{α}^{2}, often called the Toeplitz operators.

The basic properties of multiplication operators are easy to determine. For example, if γ and η are both elements of H^{∞}, then

M_{aγ+bη} = aM_{γ} + bM_{η}

for any scalars a and b. The adjoint of a multiplication operator also behaves nicely with respect to the reproducing kernel functions of A_{α}^{2}. Observe that

⟨f, M_{γ}^{∗}K_{w}⟩_{α} = ⟨M_{γ}(f), K_{w}⟩_{α} = γ(w)f(w) = ⟨γ(w)f, K_{w}⟩_{α} = ⟨f, \overline{γ(w)}K_{w}⟩_{α}

for every f in A_{α}^{2}. Therefore M_{γ}^{∗}(K_{w}) = \overline{γ(w)}K_{w} for every kernel function K_{w}.

Abusing notation somewhat, we write M_{z} to denote the operator M_{γ} with γ(z) = z. It is particularly easy to describe the adjoint of the operator M_{z} acting on the Hardy space H^{2}. Let f(z) = ∑_{n=0}^{∞} a_{n}z^{n} be an element of H^{2}; observe that

(M_{z}^{∗}(f))(w) = ⟨M_{z}^{∗}(f), K_{w}⟩ = ⟨f, M_{z}(K_{w})⟩
= ⟨∑_{n=0}^{∞} a_{n}z^{n}, ∑_{n=0}^{∞} \overline{w}^{n}z^{n+1}⟩ = ∑_{n=1}^{∞} a_{n}w^{n−1},

which equals f′(0) when w = 0 and equals

\frac{f(w) − f(0)}{w}

for every w in 𝔻 \ {0}. This fact will prove particularly useful in Chapter 3, when we consider composition operators with linear fractional symbol.

2.6 Functional Banach spaces

Let X be a nontrivial Banach space whose elements are functions on a set U. As one would expect, X is called a functional Banach space if every point-evaluation functional is bounded on X. A Banach space of analytic functions is simply a functional Banach space on an open subset U of ℂ^{n} whose elements are analytic functions. It is perfectly reasonable to consider a composition operator acting on a functional Banach space; the proof of Theorem 1.1 is also valid in this context.

The Hilbert spaces H^{2} and A_{α}^{2}, in fact, belong to a larger class of Banach spaces. For 1 ≤ p < ∞, the Hardy space H^{p} is the Banach space consisting of all analytic functions f on 𝔻 with

∥f∥_{p}^{p} = lim_{r↑1} ∫_{0}^{2π} |f(re^{iθ})|^{p} \frac{dθ}{2π} = ∫_{0}^{2π} |f(e^{iθ})|^{p} \frac{dθ}{2π} < ∞.

Similarly, for α > −1 and 1 ≤ p < ∞, the weighted Bergman space A_{α}^{p} consists of those analytic functions f with

∥f∥_{p,α}^{p} = ∫_{𝔻} |f(z)|^{p} (α + 1) (1 − |z|^{2})^{α} \frac{dA(z)}{π} < ∞.

Each of these spaces is a Banach space of analytic functions on 𝔻. Extending our previous convention, we often write A_{−1}^{p} to denote the Hardy space H^{p}.

Any analytic φ : 𝔻 → 𝔻 induces a bounded composition operator on each weighted Bergman space. In fact,

(\frac{1}{1 − |φ(0)|^{2}} )^{(α+2)/p} ≤ ∥C_{φ} : A_{α}^{p} → A_{α}^{p}∥ ≤ (\frac{1 + |φ(0)|}{1 − |φ(0)|})^{(α+2)/p}

for any α ≥ −1 and 1 ≤ p < ∞. In principle, for a fixed α, the calculation of ∥C_{φ} : A_{α}^{p} → A_{α}^{p}∥ might constitute a substantially different problem for each value of p. When dealing with the Hardy spaces, however, we need only consider C_{φ} acting on the Hilbert space H^{2}. In particular, we have the following result:

Proposition 2.15 Take 1 ≤ p < ∞. For any analytic map φ : 𝔻 → 𝔻,

∥C_{φ} : H^{p} → H^{p}∥^{p} = ∥C_{φ} : H^{2} → H^{2}∥^{2} .

This proposition has been described as a “folk result”; it has been commonly known for several years (see, for example, [16]), but its original source remains obscure.

Proof of Proposition 2.15. Let {f_{n}} be a sequence of unit vectors in H^{p} such that

lim_{n→∞} ∥C_{φ}(f_{n})∥_{p} = ∥C_{φ} : H^{p} → H^{p}∥.

For each n, let B_{n} denote the Blaschke product whose zero-set is identical to that of f_{n}. Every Blaschke product is an isometric zero-divisor for H^{p}; in other words, each function

g_{n} = \frac{f_{n}}{B_{n}}

is also a unit vector in H^{p}. Moreover, since |B_{n}(z)| < 1 for all z in 𝔻,

∥C_{φ}(g_{n})∥_{p} = ∥\frac{f_{n} ∘ φ}{B_{n} ∘ φ}∥_{p} ≥ ∥f_{n} ∘ φ∥_{p} = ∥C_{φ}(f_{n})∥_{p},  (2.9)

which means that

lim_{n→∞} ∥C_{φ}(g_{n})∥_{p} = ∥C_{φ} : H^{p} → H^{p}∥.

Each g_{n}, being a non-vanishing analytic function on 𝔻, has an analytic logarithm. Hence there exist analytic functions (g_{n})^{p/2}, with

∥(g_{n})^{p/2}∥_{2}^{2} = ∫_{0}^{2π} |(g_{n}(e^{iθ}))^{p/2}|^{2} \frac{dθ}{2π} = ∫_{0}^{2π} |g_{n}(e^{iθ})|^{p} \frac{dθ}{2π} = ∥g_{n}∥_{p}^{p} = 1.

Therefore

∥C_{φ} : H^{2} → H^{2}∥^{2} ≥ lim sup_{n→∞} ∥C_{φ}((g_{n})^{p/2})∥_{2}^{2}
= lim sup_{n→_{∞}} ∫_{0}^{2π} |(g_{n}(φ(e^{iθ})))^{p/2}|^{2} \frac{dθ}{2π}
= lim sup_{n→∞} ∫_{0}^{2π} |g_{n}(φ(e^{iθ}))|^{p} \frac{dθ}{2π}
= lim_{n→∞} ∥C_{φ}(g_{n})∥_{p}^{p} = ∥C_{φ} : H^{p} → H^{p}∥^{p}.

Switching the roles of p and 2, we obtain the reverse inequality.

It is unknown whether the analogous result holds for a composition operator acting on the spaces A_{α}^{p} for a fixed α > −1. The proof of Proposition 2.15 relies heavily on the fact that Blaschke products are isometric zero-divisors for H^{p}. It might seem natural, should we attempt to extend this proof to the weighted Bergman spaces, to substitute the canonical divisors of A_{α}^{p} for the Blaschke products; the canonical divisors, however, are often unbounded on 𝔻, which means that line (2.9) is not necessarily valid in this context.

Chapter 3

Composition Operators with

Linear Fractional Symbol

Throughout this chapter, we take

φ(z) = \frac{az + b}{cz + d}  (3.1)

to be a nonconstant linear fractional self-map of 𝔻. We would like to obtain information about the norm of the corresponding composition operator, either on the Hardy space H^{2} or the weighted Bergman spaces A_{α}^{2}. As we have already mentioned, our strategy involves investigating the spectrum of C_{φ}^{∗}C_{φ}. Needless to say, this avenue of inquiry would be useless without a reasonably good representation for the adjoint of C_{φ}. Our results depend on the fact that, for this particular class of maps, we do have an explicit formula for C_{φ}^{∗} (Proposition 3.3). Appealing to this formula, we establish a concrete representation for the operator C_{φ}^{∗}C_{φ} : H^{2} → H^{2}, which we use to determine the conditions under which either S_{φ} or S_{φ}^{∗} equals ∥C_{φ} : H^{2} → H^{2}∥ (Theorem 3.7).

3.1 Cowen’s adjoint formula

In this section, we derive a formula for C_{φ}^{∗}. We begin with the following lemma:

Lemma 3.1 Let φ : 𝔻 → 𝔻 be a nonconstant linear fractional map, as in (3.1); then the map

σ(z) = \frac{\overline{a}z − \overline{c}}{−\overline{b}z + \overline{d}}  (3.2)

is also a self-map of 𝔻; furthermore, both

γ(z) = \frac{1}{−\overline{b}z + \overline{d}}

and

η(z) = cz + d

are bounded, non-vanishing analytic functions on 𝔻.

Proof. For the moment, let us view the nonconstant linear fractional maps as being bijections of the Riemann sphere. Define the set

𝔻̌ = {z : |z| > 1} ∪ {∞}.

Since φ is a self-map of 𝔻, the inverse map φ^{−1} must take 𝔻̌ into 𝔻̌; if this were not the case, then φ^{−1}\binom{}{𝔻̌} would contain a point w in 𝔻, in which case φ(w) would belong to 𝔻̌. A straightforward calculation shows that

σ = ρ ∘ φ^{−1} ∘ ρ,

where ρ(z) = 1/\overline{z}. Note that ρ maps 𝔻 into 𝔻̌, that φ^{−1} maps 𝔻̌ into 𝔻̌, and that ρ maps 𝔻̌ into 𝔻; therefore σ takes 𝔻 into 𝔻, as we had hoped to show.

Because σ is a self-map of 𝔻, it cannot have a singularity in the closed unit disk. Since γ and σ have the same denominator, it follows that γ is bounded on 𝔻; it is evident that γ does not vanish on 𝔻. The function η is bounded on 𝔻; observe that η is the denominator of φ, which implies that η does not vanish on 𝔻.

Since ρ is the identity map on ∂𝔻, we make the following observation, which will prove useful in later sections:

Observation 3.2 Let φ : 𝔻 → 𝔻 be a nonconstant linear fractional map, as in (3.1), and let σ be defined as in (3.2). Suppose that ζ and ω are points on ∂𝔻; then φ(ζ) = ω if and only if σ(ω) = ζ.

Having defined the functions σ, γ, and η, we turn our attention to the Hardy space and the weighted Bergman spaces. Fix a number α ≥ −1. It follows from Lemma 3.1 that the functions

γ_{α}(z) = γ(z)^{α+2} = (\frac{1}{−\overline{b}z + \overline{d}})^{α+2}  (3.3)

and

η_{α}(z) = η(z)^{α+2} = (cz + d)^{α+2}  (3.4)

belong to H^{∞}. We write M_{γ_{α}} and to M_{η_{α}} denote the corresponding multiplication operators on A_{α}^{2}; these operators will appear in our formula for the adjoint of C_{φ}. The following proposition was originally proved by Cowen [9] for composition operators acting on H^{2}; Hurst [15] later extended the result to the weighted Bergman spaces:

Proposition 3.3 (Cowen’s adjoint formula) Take α ≥ −1. Let φ : 𝔻 → 𝔻 be a nonconstant linear fractional map, as in (3.1); let σ, γ_{α}, and η_{α} be defined as in (3.2), (3.3), and (3.4). The adjoint C_{φ}^{∗} on A_{α}^{2} is given by the formula

C_{φ}^{∗} = M_{γ_{α}}C_{σ}M_{η_{α}}^{∗}.

Proof. Recall that the reproducing kernel functions for A_{α}^{2} have the form

K_{w}(z) = (\frac{1}{1 − \overline{w}z} )^{α+2} ;

moreover, C_{φ}^{∗}(K_{w}) = K_{φ(w)} and M_{η_{α}}^{∗}(K_{w}) = \overline{η_{α}(w)}K_{w} for every w in 𝔻. Observe that

((M_{γ_{α}}C_{σ}M_{η_{α}}^{∗}) (K_{w})) (z) = \overline{η_{α}(w)}γ_{α}(z)(C_{σ}(K_{w}))(z)
= (\overline{cw + d})^{α+2} (\frac{1}{−\overline{b}z + \overline{d}})^{α+2} (\frac{1}{1 − \overline{w} (\frac{\overline{a}z−\overline{c}}{−\overline{b}z+\overline{d}})})^{α+2}
= (\frac{\overline{cw} + \overline{d}}{−\overline{b}z + \overline{d} − \overline{wa}z + \overline{wc}})^{α+2}
= (\frac{1}{1 − \overline{φ(w)}z} )^{α+2}
= K_{φ(w)}(z) = (C_{φ}^{∗}(K_{w})) (z).

Therefore the operators C_{φ}^{∗} and M_{γ_{α}}C_{σ}M_{η_{α}}^{∗} agree on every kernel function K_{w}; since the kernel functions span a dense subset of A_{α}^{2}, the two operators must be identical.

3.2 The operator C_{φ}^{∗}C_{φ}

Our goal now is to use Cowen’s adjoint formula to obtain a concrete representation for the operator C_{φ}^{∗}C_{φ}. Our first observation pertains to maps of the form φ(z) = az + b, where |a| + |b| ≤ 1. Such a map, of course, may be written in linear fractional form, as in (3.1), with c = 0 and d = 1. Observe that M_{η_{α}}^{∗} = M_{1}^{∗}; since multiplication by 1 is the identity operator on A_{α}^{2}, so too is its adjoint. Hence Proposition 3.3 dictates that, for any α ≥ −1, the adjoint C_{φ}^{∗} : A_{α}^{2} → A_{α}^{2} may be written as the product M_{γ_{α}}C_{σ}. Thus we obtain the following result:

Observation 3.4 Consider the map φ(z) = az + b where |a| + |b| ≤ 1. For any α ≥ −1, the operator C_{φ}^{∗}C_{φ} : A_{α}^{2} → A_{α}^{2} has the form

((C_{φ}^{∗}C_{φ}) (f)) (z) = (\frac{1}{1 − \overline{b}z} )^{α+2} f(φ(σ(z)))
= (\frac{1}{1 − \overline{b}z} )^{α+2} f(τ(z)),

where τ denotes the composition φ ∘ σ.

We will make use of this representation in Section 4.2.

For the rest of this section, we restrict our attention to composition operators acting on H^{2}. As before, let φ : 𝔻 → 𝔻 be a nonconstant linear fractional map. The only piece of Cowen’s adjoint formula that requires further discussion is the operator M_{η_{α}}^{∗} = M_{η}^{∗}. It follows from our discussion in Section 2.5 that

(M_{η}^{∗}(f)) (z) = (M_{cz+d}^{∗}(f)) (z) = ((\overline{c}M_{z}^{∗} + \overline{d}M_{1}^{∗}) (f)) (z),

for every f in H^{2}, which equals

\overline{c}f′(0) + \overline{d}f(0)

when z = 0 and

\overline{c} (\frac{f(z) − f(0)}{z}) + \overline{d}f(z)

for all z in 𝔻 \ {0}.

This observation allows us to obtain a particularly useful representation for the operator C_{φ}^{∗}C_{φ}. For any f in H^{2}, we see that

((C_{φ}^{∗}C_{φ}) f) (z) = ((M_{γ_{α}}C_{σ}M_{η_{α}}^{∗}C_{φ}) (f)) (z)
= ((M_{γ}C_{σ}M_{η}^{∗}C_{φ}) (f)) (z),

which equals

γ(z) (\overline{c} (\frac{f(φ(σ(z))) − f(φ(0))}{σ(z)}) + \overline{d}f(φ(σ(z))))

= \frac{\overline{c}}{\overline{a}z − \overline{c}} [f(φ(σ(z))) − f(φ(0))] + \frac{\overline{d}}{−\overline{b}z + \overline{d}}f(φ(σ(z)))  (3.5)

for all z in 𝔻 not equal to σ^{−1}(0) = \overline{c}/\overline{a}. Performing the obvious algebraic manipulations, we arrive at the following result:

Observation 3.5 Let φ : 𝔻 → 𝔻 be a nonconstant linear fractional map, as in (3.1). The operator C_{φ}^{∗}C_{φ} : H^{2} → H^{2} may be written

((C_{φ}^{∗}C_{φ}) f) (z) = ψ(z)f(τ(z)) + χ(z)f(φ(0)),  (3.6)

where τ denotes the composition φ ∘ σ and

ψ(z) = \frac{(\overline{ad} − \overline{bc}) z}{(\overline{a}z − \overline{c}) (−\overline{b}z + \overline{d})} and χ(z) = \frac{\overline{c}}{−\overline{a}z + \overline{c}}.

This representation holds for all z in 𝔻 \ {σ^{−1}(0)}.

The point σ^{−1}(0) = \overline{c}/\overline{a} is something of a nuisance, but will not cause us any serious problems. For many choices of φ, this point is totally innocuous; it only lies in 𝔻 if |c| < |a|. In the event that σ^{−1}(0) does belong to 𝔻, we can apply l’Hospital’s rule to expression (3.5) to see that

(C_{φ}^{∗}C_{φ}(f)) (σ^{−1}(0)) = \frac{\overline{c}}{\overline{a}}f′(φ(0))τ′(σ^{−1}(0)) + \frac{\overline{ad}}{\overline{ad} − \overline{bc}}f(φ(0)).  (3.7)

3.3 The quantities S_{φ} and S_{φ}^{∗}

In this section, we restrict our attention to composition operators acting on the Hardy space H^{2}. Applying Observation 3.5, we determine the conditions under which either S_{φ} = ∥C_{φ}∥ or S_{φ}^{∗} = ∥C_{φ}∥ when φ : 𝔻 → 𝔻 is a linear fractional map. The following proposition constitutes the crux of our argument:

Proposition 3.6 Let φ : 𝔻 → 𝔻 be a linear fractional map with φ(0) ≠ 0 and which does not have the form φ(z) = az + b. For any point w in 𝔻,

∥C_{φ}∥ > ∥C_{φ}(k_{w})∥_{2} .

Proof. Suppose, to the contrary, that C_{φ} attains its norm on some normalized kernel function k_{w}; then K_{w} is an eigenfunction for C_{φ}^{∗}C_{φ}. Consequently the subspace K_{{w}} = {αK_{w} : α ∈ ℂ} is invariant under C_{φ}^{∗}C_{φ}. Since C_{φ}^{∗}C_{φ} is self-adjoint, the orthogonal complement K_{{w}}^{⊥} = {f ∈ H^{2} : f(w) = 0} is also invariant under the operator; this observation will give rise to a contradiction.

Lemma 2.13 tells us that w cannot equal 0 or φ(0). Suppose then that w is the problematic point σ^{−1}(0) = \overline{c}/\overline{a}. Referring to line (3.7), we see that

(C_{φ}^{∗}C_{φ}(f)) (σ^{−1}(0)) = \frac{\overline{c}}{\overline{a}}f′(φ(0))τ′(σ^{−1}(0)) + \frac{\overline{ad}}{\overline{ad} − \overline{bc}}f(φ(0)),

which must equal 0 for any f in K_{{w}}^{⊥}. Consider the function f_{1}(z) = (z −φ(0))(z −w) in K_{{w}}^{⊥}. The assumption that φ(z) ≠ az +b guarantees that c ≠ 0; since f_{1}(φ(0)) = 0 and τ = φ ∘ σ is univalent, the term f′_{1}(φ(0)) must equal 0, which is not the case. Therefore w cannot equal σ^{−1}(0). Hence equation (3.6) is valid at w, meaning that

0 = (C_{φ}^{∗}C_{φ}(f)) (w) = ψ(w)f(τ(w)) + χ(w)f(φ(0))

for all f in K_{{w}}^{⊥}.

Again consider the function f_{1}. Observe that f_{1}(φ(0)) = 0; since w ≠ 0, the term ψ(w) is nonzero. Hence f_{1}(τ(w)) = 0, meaning that τ(w) equals either w or φ(0). If τ(w) = φ(0), then w = σ^{−1}(0), which is not the case; therefore τ(w) = w. Now take f_{2}(z) = z − w in K_{\{w\}}^{⊥}. Since f_{2}(τ(w)) = f_{2}(w) = 0 and χ(w) = \overline{c}/(\overline{c} − \overline{a}w) ≠ 0, we see that f_{2}(φ(0)) = 0. Therefore φ(0) must equal w, which is a contradiction.

Proposition 3.6 allows us easily to prove the main result of this section:

Theorem 3.7 Let φ : 𝔻 → 𝔻 be a linear fractional map with φ(0) ≠ 0 and which does not have the form φ(z) = az + b. Then S_{φ} = ∥C_{φ}∥ if and only if ∥C_{φ}∥_{e} = ∥C_{φ}∥; likewise S_{φ}^{∗} = ∥C_{φ}∥ if and only if ∥C_{φ}∥_{e} = ∥C_{φ}∥.

Proof. Observation 2.14 shows that ∥C_{φ}∥_{e} ≤ S_{φ}^{∗} ≤ S_{φ} ≤ ∥C_{φ}∥ whenever φ is univalent; if ∥C_{φ}∥_{e} = ∥C_{φ}∥, then all of these quantities must be equal. On the other hand, suppose that ∥C_{φ}∥_{e} < ∥C_{φ}∥. Proposition 3.6 dictates that ∥C_{φ}(k_{w})∥_{2} < ∥C_{φ}∥ for all w in 𝔻; it follows from Observation 2.14 that S_{φ} < ∥C_{φ}∥. Since S_{φ}^{∗} ≤ S_{φ}, our claim follows.

As a consequence of this theorem, we see that S_{φ} = ∥C_{φ}∥ if and only if S_{φ}^{∗} = ∥C_{φ}∥. We should mention, though, that there are linear fractional φ such that S_{φ}^{∗} = S_{φ} < ∥C_{φ}∥; for example, Retsek [23] showed that the map φ(z) = 4/(5 − z) has this property.

Theorem 3.7 no longer holds if we eliminate the hypothesis that φ be linear fractional. In light of Proposition 2.10, we see that our assertion for S_{φ}^{∗} holds whenever φ is univalent (an observation also made by Retsek [23]). On the other hand, Bourdon and Retsek [6] proved that S_{φ}^{∗} < ∥C_{φ}∥_{e} = ∥C_{φ}∥ whenever φ is a non-univalent inner function with φ(0) ≠ 0 (see Proposition 2.11 above). Extremal noncompactness implies that S_{φ} = ∥C_{φ}∥ for any φ. It is not difficult, however, to find further examples of analytic φ with φ(0) ≠ 0 and ∥C_{φ}∥_{e} < S_{φ} = ∥C_{φ}∥. To that end, let ν be an inner function that fixes the origin; then (by Proposition 2.1) the composition operator C_{ν} is an isometry of H^{2}. Hence, for any analytic φ : 𝔻 → 𝔻, the operator C_{φ∘ν} = C_{ν}C_{φ} has the same norm as C_{φ}; moreover, S_{φ∘ν} = ∥C_{φ∘ν}∥ if and only if S_{φ} = ∥C_{φ}∥. Consider the map φ(z) = az + b, where both a and b are nonzero and |a| + |b| < 1. We know that S_{φ} = ∥C_{φ}∥, and that both of the operators C_{φ} and C_{φ∘ν} are compact. Hence ∥C_{φ∘ν}∥_{e} = 0 < S_{φ∘ν} = ∥C_{φ∘ν}∥; in particular, this result holds if we take ν(z) = z^{m} for some integer m ≥ 1, so that (φ ∘ ν)(z) = az^{m} + b.

Chapter 4

The Spectrum of a Weighted

Composition Operator

Given an analytic map φ : 𝔻 → 𝔻 and a function ψ in H^{∞}, we define the weighted composition operator W_{φ,ψ} : A_{α}^{2} → A_{α}^{2} by the rule

(W_{φ,ψ}(f)) (z) = ψ(z)f(φ(z))

for all f in A_{α}^{2} and all z in 𝔻. Such an operator, of course, is simply the product of the multiplication operator M_{ψ} with the composition operator C_{φ}. It seems plausible, in view of Cowen’s adjoint formula, that information relating to the spectrum of W_{φ,ψ} might be pertinent to our investigation of the spectrum of C_{φ}^{∗}C_{φ}. Section 4.1 introduces a slightly more general class of operators, one that includes many of the weighted composition operators; we determine the spectrum of any such operator which happens to be compact. Section 4.2 offers a new proof, based on the work of the preceding section, of a previously known norm result for composition operators acting on A_{α}^{2}.

4.1 The spectrum

Let X be a Banach space of analytic functions on 𝔻. Take R_{φ,ψ} to be a bounded operator on X with the following properties:

There is a point w_{0} in 𝔻, an open subset Δ of 𝔻, an analytic map φ : 𝔻 → 𝔻, and an analytic function ψ : Δ → ℂ, such that

• The point w_{0} belongs to Δ,

• The map φ fixes the point w_{0},

• (R_{φ,ψ}(f)) (z) = ψ(z)f(φ(z)) for all f in X and all z in Δ.

To avoid several minor difficulties, we also assume that φ′(w_{0}) ≠ 0.

The operator R_{φ,ψ} behaves like a weighted composition operator in a neighborhood of w_{0}; one could reasonably describe such an operator as locally resembling a weighted composition operator on the set Δ. Having such a representation for R_{φ,ψ}, we write ℳ to denote the set of all non-negative integers m such that there is some function x_{m} in X with a zero of order m at w_{0}. (If 0 belongs to ℳ, then X simply contains a function which does not vanish at w_{0}.) This set plays an important part in describing the spectrum of R_{φ,ψ}. We begin with the following observation: Proposition 4.1 Suppose that φ is not an automorphism of 𝔻 and that ψ(w_{0}) is nonzero. Let m be an element of ℳ; then the point ψ(w_{0})φ′(w_{0})^{m} belongs to the spectrum of R_{φ,ψ}. Proof. Suppose that ψ(w_{0})φ′(w_{0})^{m} belongs to the resolvent of R_{φ,ψ}; in other words, the operator R_{φ,ψ} − ψ(w_{0})φ′(w_{0})^{m} is invertible. Hence there is some f in X such that

R_{φ,ψ}(f) − ψ(w_{0})φ′(w_{0})^{m}f = x_{m},

where x_{m} is an element of X with a zero of order m at w_{0}. In other words,

ψ(z)f(φ(z)) − ψ(w_{0})φ′(w_{0})^{m}f(z) = x_{m}(z)  (4.1)

for all z in Δ.

First of all, suppose that m = 0. Since w_{0} belongs to Δ, we have that

x_{0}(w_{0}) = ψ(w_{0})f(φ(w_{0})) − ψ(w_{0})f(w_{0})
= ψ(w_{0})f(w_{0}) − ψ(w_{0})f(w_{0}) = 0,

which contradicts our assumption that x_{0}(w_{0}) ≠ 0. Therefore ψ(w_{0}) = ψ(w_{0})φ′(w_{0})^{m} must belong to the spectrum of R_{φ,ψ}.

Now suppose that m ≥ 1. We claim that the jth derivative f^{(j)}(w_{0}) must equal 0 for all 0 ≤ j ≤ m − 1, a fact we shall prove by induction. Observe that

0 = x_{m}(w_{0}) = ψ(w_{0})f(φ(w_{0})) − ψ(w_{0})φ′(w_{0})^{m}f(w_{0})
= ψ(w_{0}) (1 − φ′(w_{0})^{m}) f(w_{0}).

Since φ is not an automorphism of 𝔻, it follows from the Schwarz lemma that |φ′(w_{0})| < 1 (see, for example, p. 79 of [27]); therefore, since ψ(w_{0}) ≠ 0, the term f(w_{0}) = f^{(0)}(w_{0}) must equal 0. Now suppose that f^{(k)}(w_{0}) = 0 for all 0 ≤ k ≤ j − 1. Taking the jth derivative of both sides of (4.1), we see that

( \frac{d^{j}}{dz^{j}} [ψ(z)f(φ(z))]|_{z=w_{0}}) − ψ(w_{0})φ′(w_{0})^{n}f^{(j)}(w_{0}) = x_{m}^{(j)}(w_{0}) = 0.

The term

\frac{d^{j}}{dz^{j}} [ψ(z)f(φ(z))]|_{z=w_{0}}

consists of ψ(w_{0})φ′(w_{0})^{j}f^{(j)}(w_{0}), plus the sum of various terms involving lower order derivatives of f at w_{0}. These extraneous terms all equal zero, which means that

0 = x_{m}^{(j)}(w_{0}) = ψ(w_{0})φ′(w_{0})^{j}f^{(j)}(w_{0}) − ψ(w_{0})φ′(w_{0})^{m}f^{(j)}(w_{0})
= ψ(w_{0})φ′(w_{0})^{j} (1 − φ′(w_{0})^{m−j}) f^{(j)}(w_{0}).

Therefore f^{(j)}(w_{0}) = 0 for all 0 ≤ j ≤ m − 1, as we had hoped to show.

Consequently we see that

0 ≠ x_{m}^{(m)}(w_{0}) = ( \frac{d^{m}}{dz^{m}} [ψ(z)f(φ(z))]|_{z=w_{0}}) − ψ(w_{0})φ′(w_{0})^{m}f^{(m)}(w_{0})
= ψ(w_{0})φ′(w_{0})^{m}f^{(m)}(w_{0}) − ψ(w_{0})φ′(w_{0})^{m}f^{(m)}(w_{0}) = 0,

thereby obtaining a contradiction. In other words, ψ(w_{0})φ′(w_{0})^{m} belongs to the spectrum of R_{φ,ψ}.

At this point, we turn our attention specifically to the eigenvalues of R_{φ,ψ}.

Proposition 4.2 Let λ be an eigenvalue of R_{φ,ψ} with a corresponding eigenfunction g; then λ = ψ(w_{0})φ′(w_{0})^{m}, where m denotes the order of the zero of g at the point w_{0}.

Proof. Observe that

λg(z) = ψ(z)g(φ(z))

for all z in Δ. As above, we see that

\frac{d^{m}}{dz^{m}} [ψ(z)g(φ(z))]|_{z=w_{0}}

equals ψ(w_{0})φ′(w_{0})^{m}g^{(m)}(w_{0}), plus terms involving lower order derivatives of g at w_{0}. Therefore

λg^{(m)}(w_{0}) = ψ(w_{0})φ′(w_{0})^{m}g^{(m)}(w_{0}),

48

so that λ = ψ(w_{0})φ′(w_{0})^{m}.

Propositions 4.1 and 4.2, taken together, allow us precisely to determine the spectrum of a compact operator R_{φ,ψ}. Proposition 4.3 Suppose that ℳ is an infinite set. If φ is not an automorphism of 𝔻 and the operator R_{φ,ψ} is compact, then the spectrum of the operator is precisely {0} ∪ {ψ(w_{0})φ′(w_{0})^{m}}_{m∈ℳ} . Proof. Since the functions x_{m} are linearly independent for different values of m, the Banach space X is necessarily infinite dimensional; hence the compact operator R_{φ,ψ} is not invertible, meaning that 0 belongs to its spectrum. Every nonzero element of the spectrum is an eigenvalue and hence must equal ψ(w_{0})φ′(w_{0})^{m}, where m denotes the zero of the corresponding eigenfunction; that is, it must belong to the set {ψ(w_{0})φ′(w_{0})^{m}}_{m∈ℳ} . On the other hand, if ψ(w_{0}) is nonzero, then Proposition 4.1 shows that {ψ(w_{0})φ′(w_{0})^{m}}_{m∈ℳ} belongs to the spectrum of R_{φ,ψ}; if ψ(w_{0}) = 0, then every value ψ(w_{0})φ′(w_{0})^{m} is also 0, which we already know to be in the spectrum.

4.2 An application

Consider the weighted Bergman space A_{α}^{2} for some α ≥ −1 (where A_{−1}^{2}, as always, denotes the Hardy space H^{2}). We now apply the results of Section 4.1 to the operator C_{φ}^{∗}C_{φ} : A_{α}^{2} → A_{α}^{2}, namely to the case where φ has the form φ(z) = az + b. Suppose, first of all, that both a and b are nonzero and that |a| + |b| < 1. Proposition 2.6 guarantees that C_{φ}, and hence C_{φ}^{∗}C_{φ}, is a compact operator on A_{α}^{2}; Observation 3.4 dictates that

(C_{φ}^{∗}C_{φ}(f)) (z) = (\frac{1}{1 − \overline{b}z} )^{α+2} f(τ(z)),

where σ(z) = \overline{a}z/(−\overline{b}z + 1) and

τ(z) = φ(σ(z)) = \frac{(|a|^{2} − |b|^{2}) z + b}{−\overline{b}z + 1}.

In other words, C_{φ}^{∗}C_{φ} is a compact weighted composition operator. By direct calculation, we see that the Denjoy–Wolff point of τ is

w_{0} = \frac{1 − |a|^{2} + |b|^{2} − \sqrt{(1 − |a|^{2} + |b|^{2})^{2} − 4 |b|^{2}}}{2\overline{b}}.

(See Section 5.1 for a brief discussion of the Denjoy–Wolff point.) Since ∥τ∥_{∞} ≤ ∥φ∥_{∞} < 1, the point w_{0} must lie inside 𝔻. Hence C_{φ}^{∗}C_{φ} : A_{α}^{2} → A_{α}^{2} satisfies the hypotheses of Proposition 4.3. Since A_{α}^{2} contains the polynomials, ℳ is equal to the set of non-negative integers; thus the spectrum of C_{φ}^{∗}C_{φ} : A_{α}^{2} → A_{α}^{2} consists of 0, together with all values of the form

(\frac{1}{1 − \overline{b}w_{0}} )^{α+2} τ′(w_{0})^{m} = (\frac{1}{1 − \overline{b}w_{0}} )^{α+2} (\frac{|a|}{1 − \overline{b}w_{0}} )^{2m} = \begin{matrix}|a| & 2m \\ (1 − \overline{b}w_{0}) & α+2+2m \end{matrix}

= |a|^{2m} (1 − \frac{1 − |a|^{2} + |b|^{2} − \sqrt{(1 − |a|^{2} + |b|^{2})^{2} − 4 |b|^{2}}}{2} )^{−(α+2+2m)}
= |a|^{2m} (\frac{2}{1 + |a|^{2} − |b|^{2} + \sqrt{(1 − |a|^{2} + |b|^{2})^{2} − 4 |b|^{2}}} )^{α+2+2m}

for any non-negative integer m.

Having made this observation, we can easily prove the next result:

Proposition 4.4 Take α ≥ −1. Suppose that φ : 𝔻 → 𝔻 has the form φ(z) = az+b, where |a| + |b| ≤ 1; then the operator C_{φ} : A_{α}^{2} → A_{α}^{2} has norm

(\frac{2}{1 + |a|^{2} − |b|^{2} + \sqrt{(1 − |a|^{2} + |b|^{2})^{2} − 4 |b|^{2}}} )^{(α+2)/2} .

Proof. If either a or b equals 0, then our assertion is easy to prove; assume therefore that both terms are nonzero. In view of the preceding comments, the compact case follows immediately: when |a| + |b| < 1, the largest element of the spectrum of C_{φ}^{∗}C_{φ} is

(\frac{2}{1 + |a|^{2} − |b|^{2} + \sqrt{(1 − |a|^{2} + |b|^{2})^{2} − 4 |b|^{2}}} )^{α+2} ,

which equals ∥C_{φ}∥^{2}. Hence we need only consider the situation where |a| + |b| = 1. In this case, our claim simply reduces to

∥C_{φ} : A_{α}^{2} → A_{α}^{2}∥ = |a|^{−(α+2)/2} .

For every 0 < r < 1, we define the map

φ_{r}(z) = rφ(z) = (ra)z + rb.

Since |ra| + |rb| = r(|a| + |b|) < 1, our norm formula is valid for each operator C_{φ_{r}}. For any polynomial f, the functions f ∘ φ_{r} converge to f ∘ φ pointwise on the closed unit disk. Hence Fatou’s lemma dictates that

∥C_{φ}(f)∥_{2,α} ≤ lim inf_{r↑1} ∥C_{φ_{r}}(f)∥_{2,α},

from which we see that

∥C_{φ}∥ ≤ lim inf_{r↑1} ∥C_{φ_{r}}∥
= lim_{r↑1} (\frac{2}{1 + |ra|^{2} − |rb|^{2} + \sqrt{(1 − |ra|^{2} + |rb|^{2})^{2} − 4 |rb|^{2}}} )^{(α+2)/2}
= |a|^{−(α+2)/2} .

We use the same technique as Cowen [9] to obtain the reverse inequality. For 0 < r < 1, consider the point w_{r} = r|a|b/(a|b|) in 𝔻; observe that

∥C_{φ}∥ ≥ lim sup_{r↑1} \begin{matrix}∥C_{φ}^{∗} (K_{w_{r}})∥ & 2,α \\ ∥K_{w_{r}}∥ & 2,α \end{matrix} = lim sup_{r↑1} \begin{matrix}∥K_{φ(w_{r})}∥ & 2,α \\ ∥K_{w_{r}}∥ & 2,α \end{matrix}
= lim sup_{r↑1} (\frac{1 − |w_{r}|^{2}}{1 − |φ(w_{r})|^{2}} )^{(α+2)/2} = lim_{r↑1} (\frac{1 − r^{2}}{1 − (r|a| + |b|)^{2}} )^{(α+2)/2}
= |a|^{−(α+2)/2} .

Therefore our formula is valid for every map φ(z) = az + b with |a| + |b| ≤ 1.

As we mentioned in Chapter 2, this result was originally established by Cowen [9] for composition operators on H^{2}; his proof was later extended to the weighted Bergman spaces by Hurst [15]. While there are obvious differences between our proof and Cowen’s, it is impossible not to notice certain similarities as well. Both arguments, for example, require the calculation of the Denjoy–Wolff point of τ = φ∘σ. Approaching Cowen’s work with the correct mindset, one can interpret a significant portion of his proof as being a statement about the spectrum of C_{φ}^{∗}C_{φ}.

Chapter 5

Norms of Composition Operators

with Linear Fractional Symbol

Let φ : 𝔻 → 𝔻 be a nonconstant linear fractional map, as in (3.1):

φ(z) = \frac{az + b}{cz + d}.

Let σ be defined as in (3.2):

σ(z) = \frac{\overline{a}z − \overline{c}}{−\overline{b}z + \overline{d}}.

Our goal now is to find a set of conditions, formulated in terms of the map τ = φ ∘ σ, under which we can determine ∥C_{φ}∥. We introduce these conditions in Section 5.1, establishing a representation for the norm of a corresponding composition operator (Theorem 5.11). Our technique also allows us to determine the elements on which such an operator attains its norm, a question we discuss further in Section 5.2. Section 5.3 provides several specific examples of linear fractional φ that satisfy these conditions. We conclude the chapter with a few related results, including the statement of a theorem recently obtained by the author in collaboration with three other mathematicians.

The principal results of this section also appear, in somewhat abbreviated form, in a current paper of the author [14].

5.1 The spectrum of C_{φ}^{∗}C_{φ}

For a non-negative integer j, let τ_{j} denote the jth iterate of τ = φ ∘ σ; that is, τ_{0} is the identity map on 𝔻 and τ_{j+1} = τ ∘ τ_{j}. Throughout the next two sections, we make the following assumption:

There is some integer n ≥ 0 such that τ_{n}(φ(0)) = 0.  (5.1)

In effect, this condition is a generalization of the case where φ(0) = 0. To avoid a triviality, we also assume that φ does not have the form φ(z) = az.

We begin with several basic observations about φ:

Lemma 5.1 Suppose that τ_{n}(φ(0)) = 0 and that φ(z) ≠ az; then the map φ cannot be an automorphism of 𝔻.

Proof. Suppose that φ is an automorphism; in other words, φ has the form

φ(z) = s (\frac{w − z}{1 − \overline{w}z} ) = \frac{sz − sw}{\overline{w}z − 1},  (5.2)

where w is a point in 𝔻 and s is a constant of modulus 1. The corresponding map σ may be written

σ(z) = \frac{\overline{s}z − w}{\overline{sw}z − 1}.

By direct calculation, we see that

τ(z) = φ(σ(z)) = \frac{|s|^{2} (1 − |w|^{2}) z}{(1 − |w|^{2})} = z.

Therefore, since τ_{n}(φ(0)) = 0, the point φ(0) must equal 0. Considering line (5.2), we see that φ must have the form

φ(z) = −sz,

which is excluded by our initial assumptions.

Since φ is not an automorphism, neither is the map τ = φ ∘ σ. Hence τ has a Denjoy–Wolff point, that is, a unique fixed point w_{0} in the closed unit disk such that the iterates τ_{j} converge to w_{0} uniformly on compact subsets of 𝔻. (See Section 2.3 of [11] for a detailed discussion of this topic.) This fact allows us to prove the next three lemmas:

Lemma 5.2 Suppose that τ_{n}(φ(0)) = 0 and that φ(z) ≠ az; then the map φ cannot have the form φ(z) = az + b.

Proof. Suppose that φ(z) = az + b; then

σ(z) = \frac{\overline{a}z}{−\overline{b}z + 1}.

Observe that σ(0) = 0, which means that

0 = τ_{n}(φ(0)) = τ_{n}(φ(σ(0))) = τ_{n+1}(0).

Consequently τ_{k(n+1)}(0) = 0 for any positive integer k. Since the iterates τ_{j}(0) converge to w_{0}, we conclude that w_{0} = 0. Hence τ(0) = 0, so

0 = τ(0) = φ(σ(0)) = φ(0),

which implies that φ has the form φ(z) = az.

Lemma 5.3 Suppose that τ_{n}(φ(0)) = 0 and that φ(z) ≠ az; then the point τ_{j}(φ(0)) never equals σ^{−1}(0). Proof. Suppose that τ_{m}(φ(0)) = σ^{−1}(0) for some integer m ≥ 0. In this case,

τ_{m+1}(φ(0)) = τ(τ_{m}(φ(0))) = τ(σ^{−1}(0)) = φ(0).

Hence τ_{k(m+1)}(φ(0)) = φ(0) for any positive integer k, which implies that w_{0} = φ(0). Consequently τ(φ(0)) = φ(0), from which we see that

φ(0) = τ_{n}(φ(0)) = 0.

Since τ fixes φ(0) = 0, we see that τ(0) = 0 as well; it follows that σ(0) = 0. In other words, the map

φ(z) = \frac{az + b}{cz + d}

has both b = 0 and c = 0, which means that it has the forbidden form φ(z) = (a/d)z. Lemma 5.4 Suppose that τ_{n}(φ(0)) = 0 and that φ(z) ≠ az; then the point τ_{j}(φ(0)) does not equal τ_{m}(φ(0)) if j ≠ m. Proof. Suppose that τ_{j}(φ(0)) = τ_{m}(φ(0)) for j < m. In this case,

τ_{m−j}(φ(0)) = φ(0),

from which it follows that

τ(τ_{m−j−1}(φ(0))) = φ(0),

or

τ_{m−j−1}(φ(0)) = σ^{−1}(0).

Lemma 5.3 tells us that this situation is impossible.

In particular, Lemma 5.4 shows that τ_{j}(φ(0)) ≠ 0 for j ≠ n; thus there is no ambiguity associated with the possibility that multiple integers satisfy property (5.1). While various indices appear in different contexts during the course of this argument, we always reserve n for the unique integer such that τ_{n}(φ(0)) = 0.

Let W denote the set of points {τ_{j}(φ(0))}_{j=0}^{n}. Lemma 5.3 guarantees that equation (3.6), our usual representation for C_{φ}^{∗}C_{φ}, is valid at each point in W. Consider K_{W}^{⊥}, the subspace of H^{2} consisting of all functions that vanish on W. The following observation is crucial to the results of this section: Proposition 5.5 The subspace K_{W}^{⊥} is invariant under the operator C_{φ}^{∗}C_{φ}. Proof. Suppose that f belongs to K_{W}^{⊥}; it follows from equation (3.6) that

(C_{φ}^{∗}C_{φ}(f)) (τ_{j}(φ(0))) = ψ(τ_{j}(φ(0)))f(τ_{j+1}(φ(0))) + χ(τ_{j}(φ(0)))f(φ(0))
= ψ(τ_{j}(φ(0)))f(τ_{j+1}(φ(0))).

For 0 ≤ j ≤ n − 1, the term f(τ_{j+1}(φ(0))) equals 0; for j = n, the term ψ(τ_{j}(φ(0))) = ψ(0) vanishes. Therefore (C_{φ}^{∗}C_{φ}) (f) also belongs to K_{W}^{⊥}.

It is not difficult to determine the spectrum of the operator C_{φ}^{∗}C_{φ} : K_{W}^{⊥} → K_{W}^{⊥}, at least whenever C_{φ} is compact: Proposition 5.6 Suppose that the operator C_{φ} : H^{2} → H^{2} is compact; then the spectrum of C_{φ}^{∗}C_{φ} : K_{W}^{⊥} → K_{W}^{⊥} is precisely 0∪{ψ(w_{0})τ′(w_{0})^{m}}_{m=0}^{∞}, where w_{0} denotes the Denjoy–Wolff point of τ = φ ∘ σ. Proof. Since any function in K_{W}^{⊥} vanishes at φ(0), the operator C_{φ}^{∗}C_{φ} : K_{W}^{⊥} → K_{W}^{⊥} may be written

(C_{φ}^{∗}C_{φ}(f)) (z) = ψ(z)f(τ(z)).

for all z in the open set 𝔻 \ {σ^{−1}(0)}. The compactness of C_{φ} implies that ∥τ∥_{∞} ≤ ∥φ∥_{∞} < 1; hence w_{0}, the Denjoy–Wolff point of τ, lies in 𝔻. If w_{0} = σ^{−1}(0), then

σ^{−1}(0) = τ(σ^{−1}(0)) = φ(0),

which contradicts Lemma 5.3; consequently w_{0} belongs to 𝔻 \ {σ^{−1}(0)}. In other words, the operator C_{φ}^{∗}C_{φ} : K_{W}^{⊥} → K_{W}^{⊥} locally resembles a weighted composition operator on the set 𝔻 \ {σ^{−1}(0)}, in the sense we defined in Section 4.1.

Lemma 5.4 implies that τ does not fix any point τ_{j}(φ(0)), which means that w_{0} does not belong to the set W = {τ_{j}(φ(0))}_{j=0}^{n}. Therefore, for any integer m ≥ 0, the function

x_{m}(z) = (z − w_{0})^{m} ∏_{j=0}^{n}(z − τ_{j}(φ(0)))

belongs to K_{W}^{⊥} and has a zero of order m at w_{0}. Consequently every non-negative integer belongs to the set ℳ, so our claim follows from Proposition 4.3.

This fact, although interesting, does not particularly help us to determine ∥C_{φ}∥. Proposition 5.6 does, however, prove to be relevant to our work in Section 6.3

We now turn our attention to K_{W}, the span of the kernel functions {K_{τ_{j}(φ(0))}}_{j=0}^{n}; observe that it has dimension n + 1. Since C_{φ}^{∗}C_{φ} : H^{2} → H^{2} is self-adjoint, the subspace K_{W} is also invariant under the operator. Our strategy for finding ∥C_{φ}∥ centers around determining the spectrum, namely the eigenvalues, of the operator C_{φ}^{∗}C_{φ} : K_{W} → K_{W}.

The next several results pertain to the eigenvalues and eigenfunctions of C_{φ}^{∗}C_{φ}. The following proposition serves as a generalization of Observation 2.4:

Proposition 5.7 Let λ be an eigenvalue of C_{φ}^{∗}C_{φ} : H^{2} → H^{2} with a corresponding eigenfunction g. For every integer j ≥ 0, the following relationship holds:

λ^{j+1}g(0) = [∏_{m=0}^{j−1} ψ(τ_{m}(φ(0)))] g(τ_{j}(φ(0)))

+ ∑_{k=0}^{j−1} χ(τ_{k}(φ(0))) [∏_{m=0}^{k−1} ψ(τ_{m}(φ(0)))] λ^{j−k}g(0),

where we take ∏_{m=0}^{−1}(·) to equal 1 and ∑_{k=0}^{−1}(·) to equal 0.

Proof (by induction). Since λg(0) = g(φ(0)), the claim holds for j = 0. For any j ≥ 0, equation (3.6) dictates that

λg(τ_{j}(φ(0))) = ((C_{φ}^{∗}C_{φ}) g) (τ_{j}(φ(0)))
= ψ(τ_{j}(φ(0)))g(τ_{j+1}(φ(0))) + χ(τ_{j}(φ(0)))λg(0).  (5.3)

Now assume that our claim holds for the index j. Multiplying both sides of the consequent equation by λ and substituting expression (5.3) for λg(τ_{j}(φ(0))), we obtain

λ^{j+2}g(0) = [∏_{m=0}^{j−1} ψ(τ_{m}(φ(0)))] [ψ(τ_{j}(φ(0)))g(τ_{j+1}(φ(0))) + χ(τ_{j}(φ(0)))λg(0)]

+ ∑_{k=0}^{j−1} χ(τ_{k}(φ(0))) [∏_{m=0}^{k−1} ψ(τ_{m}(φ(0)))] λ^{j+1−k}g(0)

= [∏_{m=0}^{j} ψ(τ_{m}(φ(0)))] g(τ_{j+1}(φ(0)))

+ ∑_{k=0}^{j} χ(τ_{k}(φ(0))) [∏_{m=0}^{k−1} ψ(τ_{m}(φ(0)))] λ^{j+1−k}g(0).

Hence our claim also holds for the index j + 1.

Since both K_{W} and K_{W}^{⊥} are invariant under C_{φ}^{∗}C_{φ} : H^{2} → H^{2}, each eigenvalue λ of C_{φ}^{∗}C_{φ} has an eigenfunction belonging to one of the two subspaces. The next proposition provides a distinguishing characteristic for eigenfunctions in K_{W}^{⊥}.

Proposition 5.8 Let g be an eigenfunction for C_{φ}^{∗}C_{φ} : H^{2} → H^{2}; then g belongs to K_{W}^{⊥} if and only if g(0) = 0.

Proof. If g belongs to K_{W}^{⊥}, then by definition g(0) = g(τ_{n}(φ(0))) equals 0. Conversely, suppose that g is an eigenfunction for C_{φ}^{∗}C_{φ} with g(0) = 0. In this case, Proposition 5.7 dictates that

0 = λ^{j+1}g(0) = [∏_{m=0}^{j−1} ψ(τ_{m}(φ(0)))] g(τ_{j}(φ(0)))

for all j ≥ 0. Since ψ(τ_{m}(φ(0))) is nonzero for 0 ≤ m ≤ n − 1, the function g must vanish on the entire set {τ_{j}(φ(0))}_{j=0}^{n}. In other words, g belongs to the subspace K_{W}^{⊥}.

Corollary 5.9 Suppose that g_{1} and g_{2} are eigenfunctions for C_{φ}^{∗}C_{φ} which belong to K_{W}; if they correspond to the same eigenvalue, then one is a scalar multiple of the other.

Proof. We appeal to the proof of Corollary 2.3, bearing in mind that no eigenfunction in K_{W} can vanish at 0.

Consequently every eigenspace of C_{φ}^{∗}C_{φ} : K_{W} → K_{W} has dimension 1. Since C_{φ}^{∗}C_{φ} : K_{W} → K_{W} is a self-adjoint operator on a finite dimensional space, we know that K_{W} is spanned by eigenfunctions of C_{φ}^{∗}C_{φ}. Since K_{W} has dimension n + 1, the operator C_{φ}^{∗}C_{φ} : K_{W} → K_{W} must have n + 1 distinct eigenvalues.

We return to the result of Proposition 5.7. Taking j = n and observing that χ(τ_{n}(φ(0))) = χ(0) = 1, we obtain the expression

λ^{n+1}g(0) = ∑_{k=0}^{n} χ(τ_{k}(φ(0))) [∏_{m=0}^{k−1} ψ(τ_{m}(φ(0)))] λ^{n−k}g(0).

Suppose that the eigenfunction g belongs to K_{W}; then g(0) ≠ 0 and

λ^{n+1} = ∑_{k=0}^{n} χ(τ_{k}(φ(0))) [∏_{m=0}^{k−1} ψ(τ_{m}(φ(0)))] λ^{n−k}.

In other words, any eigenvalue λ of C_{φ}^{∗}C_{φ} : K_{W} → K_{W} is a solution to this polynomial equation. Since there are n + 1 distinct eigenvalues and the equation has no more than n + 1 roots, we conclude that every solution is an eigenvalue. In other words,

p(λ) = λ^{n+1} − ∑_{k=0}^{n} χ(τ_{k}(φ(0))) [∏_{m=0}^{k−1} ψ(τ_{m}(φ(0)))] λ^{n−k}  (5.4)

is the characteristic polynomial of the operator C_{φ}^{∗}C_{φ} : K_{W} → K_{W}.

Finally, we make an observation regarding the essential norm of C_{φ}. (The author is indebted to Paul Bourdon for suggesting the proof of this proposition.) Proposition 5.10 Under the assumptions of this section, ∥C_{φ}∥_{e} < 1. Proof. If ∥φ∥_{∞} < 1, then C_{φ} is compact, so our claim holds. Suppose then that ∥φ∥_{∞} = 1; since φ is not an automorphism, there is precisely one pair of points ζ and ω on ∂𝔻 with φ(ζ) = ω. Observation 3.2 shows that σ(ω) = ζ; Bourdon, Levi, Narayan, and Shapiro [5] proved in general that σ′(ω) = (φ′(ζ))^{−1}. Therefore τ(ω) = ω and τ′(ω) = 1. Consider the map τ_{n} ∘ φ. The assumptions of this section guarantee that τ_{n} ∘ φ fixes the origin; since φ is not surjective, τ_{n} ∘ φ cannot be an automorphism of 𝔻. Hence Lemma 7.33 in [11] shows that

lim inf_{z→ζ} \frac{1 − |(τ_{n} ∘ φ)(z)|}{1 − |z|} > 1.

Since τ_{n} ∘ φ is analytic on a neighborhood of the closed unit disk and (τ_{n} ∘ φ)(ζ) = τ_{n}(ω) = ω, the Julia–Carathéodory theorem (see Theorem 2.44 in [11]) dictates that |(τ_{n} ∘ φ)′(ζ))| > 1. Consequently

1 < |(τ_{n})′(φ(ζ)) · φ′(ζ)| = |(τ_{n})′(ω) · φ′(ζ)| = |φ′(ζ)|.

Appealing to expression (2.4), we see that

∥C_{φ}∥_{e}^{2} = max {|φ′(w)|^{−1} : |w| = |φ(w)| = 1} = |φ′(ζ)|^{−1} < 1,

as we had hoped to show.

Since ∥C_{φ}∥ ≥ 1, Proposition 1.3 dictates that C_{φ} : H^{2} → H^{2} attains its norm on an element of H^{2}; that is, ∥C_{φ}∥^{2} is an eigenvalue of C_{φ}^{∗}C_{φ} : H^{2} → H^{2}. Proposition 2.2 guarantees that any corresponding eigenfunction must belong to K_{W}. In other words, ∥C_{φ}∥^{2} is the largest eigenvalue of C_{φ}^{∗}C_{φ} : K_{W} → K_{W}, meaning that it is the largest zero of the polynomial p. Hence we have proved the following result:

Theorem 5.11 Let φ : 𝔻 → 𝔻 be a linear fractional map, with φ(z) ≠ az. Suppose that τ_{n}(φ(0)) = 0 for some integer n ≥ 0; then ∥C_{φ}∥^{2} is the largest zero of the polynomial p in equation (5.4), and the elements on which C_{φ} attains its norm are linear combinations of the kernel functions {K_{τ_{j}(φ(0))}}_{j=0}^{n}.

Whenever n ≥ 1, Theorem 3.7 dictates that S_{φ} < ∥C_{φ}∥. Assuming that we can find examples of such φ, this would appear to be the first case of a composition operator whose norm we can calculate, for which the norm is not given by the action of C_{φ} on the reproducing kernel functions of H^{2}.

5.2 The eigenfunctions of C_{φ}^{∗}C_{φ}

Having determined a particular eigenvalue λ of C_{φ}^{∗}C_{φ} : K_{W} → K_{W}, it is possible to find the corresponding eigenfunctions. In particular, considering Theorem 5.11, we can identify the functions on which the operator C_{φ} attains its norm. Let λ be such an eigenvalue and g be its unique eigenfunction in K_{W} with g(0) = g(τ_{n}(φ(0))) = 1. We write

g(z) = ∑_{i=0}^{n} \frac{α_{i}}{1 − \overline{τ_{i}(φ(0))}z},

where we hope to determine the coefficients α_{i}. For any index 0 ≤ j ≤ n − 1, we may appeal to Proposition 5.7 to find g(τ_{j}(φ(0))) explicitly in terms of λ:

g(τ_{j}(φ(0))) = \frac{λ^{j+1} − ∑_{k=0}^{j−1} χ(τ_{k}(φ(0))) [∏_{m=0}^{k−1} ψ(τ_{m}(φ(0)))] λ^{j−k}}{∏_{m=0}^{j−1} ψ(τ_{m}(φ(0)))} .

Therefore we obtain the matrix equation

[\frac{1}{1 − \overline{τ_{i}(φ(0))}τ_{j}(φ(0))}]_{0≤j,i≤n} [α_{i}]_{0≤i≤n} = [g(τ_{j}(φ(0)))]_{0≤j≤n}.

The (n + 1) × (n + 1) matrix is simply the Gram matrix of the vectors {K_{τ_{i}(φ(0))}}_{i=0}^{n}, whose determinant is positive since the kernel functions are linearly independent (see Theorem XX.1 in [19]). Hence we can use Cramer’s rule to solve explicitly for the coefficients.

For example, take n = 1. Then

[ \begin{matrix}\frac{1}{1−|φ(0)|^{2}} & 1 \\ 1 & 1 \end{matrix} ] [ \begin{matrix}α_{0} \\ α_{1} \end{matrix} ] = [ \begin{matrix}g(φ(0)) \\ g(0) \end{matrix} ] = [ \begin{matrix}λ \\ 1 \end{matrix} ],

so

α_{0} = \begin{matrix}| & \begin{matrix}λ & 1 \\ 1 & 1 \end{matrix} & | \\ | & \begin{matrix}\frac{1}{1−|φ(0)|^{2}} & 1 \\ 1 & 1 \end{matrix} & | \end{matrix} = \frac{λ − 1}{\frac{1}{1−|φ(0)|^{2}} − 1} and α_{1} = \begin{matrix}| & \begin{matrix}\frac{1}{1−|φ(0)|^{2}} & λ \\ 1 & 1 \end{matrix} & | \\ | & \begin{matrix}\frac{1}{1−|φ(0)|^{2}} & 1 \\ 1 & 1 \end{matrix} & | \end{matrix} = \frac{\frac{1}{1−|φ(0)|^{2}} − λ}{\frac{1}{1−|φ(0)|^{2}} − 1}.

5.3 Examples

It is not difficult to find examples of φ with the property that τ(φ(0)) = 0. In terms of its coefficients, a linear fractional map φ : 𝔻 → 𝔻 (which does not have the form φ(z) = az) satisfies this condition if and only if

|d|^{2} − |b|^{2} = \frac{a}{b} (\overline{c}d − \overline{a}b) .

In such a case, the polynomial p in (5.4) may be written

p(λ) = λ^{2} − χ(φ(0))λ − ψ(φ(0)).

Hence any eigenvalue λ of C_{φ}^{∗}C_{φ} : K_{W} → K_{W} has the form

λ = \frac{χ(φ(0)) ± \sqrt{χ(φ(0))^{2} + 4ψ(φ(0))}}{2} = \frac{\frac{a\overline{c}d}{b} ± \sqrt{(\frac{a\overline{c}d}{b})^{2} − 4 (\overline{ad} − \overline{bc}) ad}}{2 (|d|^{2} − |b|^{2})}.

In particular, ∥C_{φ}∥^{2} is the larger of these two values.

For example, take

φ(z) = \frac{16z + 8}{19z + 32}.

Since ∥φ∥_{∞} < 1, the operator C_{φ} is compact. Observe that τ(φ(0)) = 0, which means that

∥C_{φ}∥^{2} = \frac{19 + \sqrt{181}}{30} ≈ 1.081787468.

Moreover, the operator attains its norm on the element (−11 + \sqrt{181}) K_{1/4} + 13 − \sqrt{181}.

At this point, we turn our attention to a larger class of examples. Let n be a positive integer and r a real number greater than n. Define

φ(z) = \frac{rz − n}{−(n + 1)z + (r + 1)}.

In order to show that φ is a self-map of 𝔻, we shift our attention momentarily to the right half-plane {z : Re(z) > 0}. Consider the map

L(z) = \frac{1 + z}{1 − z},

a bijection that takes 𝔻 onto the right half-plane. A simple calculation shows that

(L ∘ φ ∘ L^{−1}) (z) = (\frac{r − n}{r + n + 1}) z + \frac{1}{r + n + 1}.

Since both (r − n)/(r + n + 1) and 1/(r + n + 1) are positive numbers, the map L ∘ φ ∘ L^{−1} takes the right half-plane into itself. Consequently φ takes 𝔻 into 𝔻. Furthermore, since

|φ(1)| = \frac{r − n}{r − n} = 1

and

|φ(−1)| = \frac{r + n}{r + n + 2} < 1,

we see that ∂φ(𝔻) ∩ ∂𝔻 = {1}.

The essential norm of the corresponding composition operator is easy to calculate:

∥C_{φ}∥_{e}^{2} = max {|φ′(w)|^{−1} : |w| = |φ(w)| = 1}
= |φ′(1)|^{−1} = \frac{(r − n)^{2}}{r(r + 1) − n(n + 1)} = \frac{r − n}{r + n + 1}.

The following observation will allow us to determine ∥C_{φ}∥ as well:

Lemma 5.12 Each iterate τ_{j} has the form

τ_{j}(z) = \frac{(r + n − j + 1) z + j}{−jz + (r + n + j + 1)}.

Proof (by induction). Observe that

τ_{0}(z) = z = \frac{(r + n − 0 + 1) z + 0}{−0z + (r + n + 0 + 1)},

so the claim holds for j = 0. Assume then that the assertion holds for a particular index j. Since

σ(z) = \frac{rz + (n + 1)}{nz + (r + 1)},

the map τ = φ ∘ σ may be written

τ(z) = \frac{(r^{2} − n^{2}) z + (r − n)}{(−r + n)z + (r^{2} − n^{2} + 2r − 2n)} = \frac{(r + n)z + 1}{−z + (r + n + 2)}.

Note that

τ_{j+1}(z) = (τ ∘ τ_{j})(z),

which we may write as

\frac{az + b}{cz + d}

where

a = (r + n)(r + n − j + 1) + (1)(−j),
b = (r + n)(j) + (1)(r + n + j + 1),
c = (−1)(r + n − j + 1) + (r + n + 2)(−j),
d = (−1)(j) + (r + n + 2)(r + n + j + 1).

Performing routine manipulations, we see that

τ_{j+1}(z) = \frac{(r + n + 1)(r + n − j)z + (r + n + 1)(j + 1)}{(r + n + 1)(−j − 1)z + (r + n + 1)(r + n + j + 2)}
= \frac{(r + n − j)z + (j + 1)}{(−j − 1)z + (r + n + j + 2)}.

Therefore the claim also holds for the index j + 1.

Lemma 5.12 dictates that

τ_{j}(φ(0)) = \frac{(r + n − j + 1) (−\frac{n}{r+1}) + j}{−j (−\frac{n}{r+1}) + (r + n + j + 1)} = \frac{j − n}{r + j + 1}.

In particular τ_{n}(φ(0)) = 0, so Theorem 5.11 applies for any such φ. Observe that

ψ(τ_{j}(φ(0))) = \begin{matrix}(r (r + 1) − n (n + 1)) & (\frac{j−n}{r+j+1}) \\ (r (\frac{j−n}{r+j+1}) + n + 1) & (n (\frac{j−n}{r+j+1}) + r + 1) \end{matrix}
= \begin{matrix}(r − n) & (j − n) & (r + j + 1) \\ (j + 1) & (r + n + 1) & (r + j − n + 1) \end{matrix}

and

χ(τ_{j}(φ(0))) = \frac{n + 1}{r (\frac{j−n}{r+j+1}) + n + 1} = \begin{matrix}(n + 1) & (r + j + 1) \\ (j + 1) & (r + n + 1) \end{matrix}.

Hence the characteristic polynomial of C_{φ}^{∗}C_{φ} : K_{W} → K_{W} may be written

p(λ) = λ^{n+1}−∑_{k=0}^{n} \begin{matrix}(n + 1) & (r + k + 1) \\ (k + 1) & (r + n + 1) \end{matrix} [∏_{m=0}^{k−1} \begin{matrix}(r − n) & (m − n) & (r + m + 1) \\ (m + 1) & (r + n + 1) & (r + m − n + 1) \end{matrix}] λ^{n−k},

and ∥C_{φ}∥^{2} is the largest zero of this polynomial.

In particular, if n = 1 then

∥C_{φ}∥^{2} = \frac{r + 1}{r + 2} + \frac{1}{r + 2}\sqrt{\frac{2(r + 1)}{r}}.

For n = 2, we solve the resulting cubic equation to obtain

∥C_{φ}∥^{2} = \frac{r + 1}{r + 3} + \frac{2}{r + 3}\sqrt[3]{\frac{3 (r + 1)}{r (r − 1)}} Re (\sqrt[3]{(r + 4) + i (r − 2) \sqrt{\frac{2(r + 2)}{r − 1}}}),

where we take the principal branch of the cube root function. For example, if

φ(z) = \frac{4z − 2}{−3z + 5}

then

∥C_{φ}∥^{2} = \frac{5}{7} + \frac{2 Re \sqrt[3]{10 + 5i}}{7} = \frac{5}{7} + \frac{2\sqrt{5}}{7} cos (\frac{arctan (\frac{1}{2})}{3}) ≈ 1.345547525.

5.4 Further results

The strategy we used to prove Theorem 5.11 cannot be employed, at least without substantial modification, to determine ∥C_{φ}∥ for any other linear fractional φ. For example, it is generally impossible to decompose H^{2} into the direct sum of two nontrivial subspaces K_{W} and K_{W}^{⊥}, each invariant under the operator C_{φ}^{∗}C_{φ}. To be more precise, we have the following result: Proposition 5.13 Let φ : 𝔻 → 𝔻 be a non-automorphic linear fractional map which does not have the form φ(z) = az + b. Let W be a collection of points in 𝔻 such that both K_{W} and K_{W}^{⊥} are nontrivial subspaces of H^{2}. Suppose that K_{W} is invariant under the operator C_{φ}^{∗}C_{φ} : H^{2} → H^{2}; then τ_{n}(φ(0)) = 0 for a unique integer n ≥ 0, there are n + 1 points in W, and W = {τ_{j}(φ(0))}_{j=0}^{n} . Taking W to be a singleton set, we obtain a variation of the statement of Proposition 3.6. Not surprisingly, the proofs of these two propositions involve the same basic idea.

Proof of Proposition 5.13. The orthogonal complement K_{W}^{⊥} is also invariant under the self-adjoint operator C_{φ}^{∗}C_{φ}. We use this fact to determine the structure of W. To make our argument more comprehensible, we break the proof into several smaller claims: Claim 5.14 The point σ^{−1}(0) does not belong to W. Proof. Suppose that σ^{−1}(0) = \overline{c}/\overline{a} does belong to W. Line (3.7) dictates that

(C_{φ}^{∗}C_{φ}(f)) (σ^{−1}(0)) = \frac{\overline{c}}{\overline{a}}f′(φ(0))τ′(σ^{−1}(0)) + \frac{\overline{ad}}{\overline{ad} − \overline{bc}}f(φ(0)),

which must equal 0 for any f in K_{W}^{⊥}. Let f_{1} be a function in K_{W}^{⊥} which has a zero of order 1 at each point in W, a zero of order 1 at φ(0), and is nonzero at every other point in 𝔻. (It is possible, of course, that φ(0) belongs to W, in which case f_{1} simply has a zero of order 1 at each point in W.) The assumption that φ(z) ≠ az + b guarantees that c ≠ 0; since f_{1}(φ(0)) = 0 and τ = φ∘σ is univalent, the term f′_{1}(φ(0)) must equal 0, which is not the case. ♢

It follows that equation (3.6) is valid at each point y in W, which means that

0 = (C_{φ}^{∗}C_{φ}(f)) (y) = ψ(y)f(τ(y)) + χ(y)f(φ(0))

for all f in K_{W}^{⊥}.

Claim 5.15 If y is a nonzero element of W, then τ(y) also belongs to W.

Proof. Again consider the function f_{1} in K_{W}^{⊥}, as defined in the proof of Claim 5.14. Since f_{1}(φ(0)) = 0 and ψ(y) ≠ 0, it follows that f_{1}(τ(y)) must also vanish. Therefore τ(y) belongs to the set W ∪ {φ(0)}. If τ(y) = φ(0), then y would equal σ^{−1}(0), contradicting Claim 5.14. Therefore τ(y) must belong to W. ♢

Claim 5.16 The point φ(0) belongs to W.

Proof. Suppose first that 0 is the only point in W. In this case, the span of the constant function K_{0}(z) = 1 is invariant under C_{φ}^{∗}C_{φ}. Therefore

(C_{φ}^{∗}C_{φ}) (K_{0}) = C_{φ}^{∗}(K_{0}) = K_{φ(0)}

is a constant function; hence φ(0) equals 0, which belongs to W. Now assume that W contains a nonzero element y; suppose that φ(0) does not belong to W. Consider a function f_{2} in K_{W}^{⊥} which does not vanish at any point in 𝔻 \ W. Claim 5.15 shows that τ(y) also belongs to W, which means that

0 = (C_{φ}^{∗}C_{φ}(f_{2})) (y) = ψ(y)f_{2}(τ(y)) + χ(y)f_{2}(φ(0)) = χ(y)f_{2}(φ(0)).

Since χ(y) ≠ 0 and f_{2}(φ(0)) ≠ 0, we obtain a contradiction. Therefore φ(0) must belong to W. ♢

Claim 5.17 There is a unique integer n ≥ 0 such that τ_{n}(φ(0)) = 0; moreover, every point in the set {τ_{j}(φ(0))}_{j=0}^{n} belongs to W.

Proof. If φ(0) = 0, then this fact is apparent; thus we consider the situation where φ(0) ≠ 0. Suppose that τ_{j}(φ(0)) ≠ 0 for all j. Since φ(0) belongs to W, Claim 5.15 shows that every point τ_{j}(φ(0)) belongs to W. The sequence {τ_{j}(φ(0))}_{j=0}^{∞} converges to w_{0}, the Denjoy–Wolff point of τ. In order for K_{W}^{⊥} to be nontrivial, the set W must have no limit point in 𝔻; suppose then that w_{0} lies on ∂𝔻. In this case, Lemma 3.3 in [4] dictates that there is some positive constant C such that

1 − |τ_{j}(φ(0))| ≥ Cj^{−1}

for all j ≥ 1. Therefore {τ_{j}(φ(0))}_{j=0}^{∞} is not a Blaschke sequence, which means that there is no element of H^{2}, other than the zero function, which vanishes on the set (see Theorem 2.3 in [13]). Hence the subspace K_{W}^{⊥} must be trivial, which contradicts our initial assumptions.

In other words, there is some integer n ≥ 0 such that τ_{n}(φ(0)) = 0. Since φ does not have the form φ(z) = az, Lemma 5.4 guarantees that n is the unique integer with this property. Since τ_{j}(φ(0)) ≠ 0 for 0 ≤ j ≤ n − 1, Claim 5.15 shows that every point in the set {τ_{j}(φ(0))}_{j=0}^{n} also belongs to W. ♢

Claim 5.18 Every point in W belongs to the set {τ_{j}(φ(0))}_{j=0}^{n}.

Proof. Let y be an element of W. Repeating the argument of Claim 5.17, we see that the points τ_{j}(y) cannot all be nonzero; let m be the smallest integer such that τ_{m}(y) = 0. Suppose that m > n. Observe that τ_{m−n}(y) = φ(0), or

τ_{m−n−1}(y) = σ^{−1}(0).

If m − n − 1 = 0, then y = σ^{−1}(0); if m − n − 1 > 0, then Claim 5.15 shows that τ_{m−n−1}(y) = σ^{−1}(0) belongs to W. In either case, we obtain a contradiction to Claim 5.14. Consequently m ≤ n, which means that y = τ_{n−m}(φ(0)). ♢ Combining these observations, we obtain the desired conclusion.

While we cannot decompose H^{2} as we did in Section 5.1, it is possible to adapt the basic techniques of that argument to obtain an “n = ∞” version of Theorem 5.11. We sketch several results along these lines, all of which appear in a current paper of Paul Bourdon, Erin Fry, Christina Spofford, and the author [4]. The following theorem may be considered our main result: Theorem 5.19 Let φ be a linear fractional self-map of 𝔻 that fixes the point 1. If ∥C_{φ}∥ > ∥C_{φ}∥_{e}, then ∥C_{φ}∥^{2} is the largest value of λ which satisfies the equation

∑_{k=0}^{∞} χ(τ_{k}(φ(0))) [∏_{m=0}^{k−1} ψ(τ_{m}(φ(0)))] (\frac{1}{λ})^{k+1} = 1.  (5.5)

We provide a rudimentary outline of the proof, omitting several important details; the complete argument appears in [4]. Outline of the proof of Theorem 5.19. Let λ > ∥C_{φ}∥_{e}^{2} be an eigenvalue for C_{φ}^{∗}C_{φ} which has a corresponding eigenfunction g that does vanish at 0. Checking the necessary details, we obtain the same functional equation as in Section 5.1:

λ^{j+1}g(0) = [∏_{m=0}^{j−1} ψ(τ_{m}(φ(0)))] g(τ_{j}(φ(0)))

+ ∑_{k=0}^{j−1} χ(τ_{k}(φ(0))) [∏_{m=0}^{k−1} ψ(τ_{m}(φ(0)))] λ^{j−k}g(0).

Dividing both sides by λ^{j+1}, we see that

g(0) = [∏_{m=0}^{j−1} ψ(τ_{m}(φ(0)))] (\frac{1}{λ})^{j+1} g(τ_{j}(φ(0)))

+ ∑_{k=0}^{j−1} χ(τ_{k}(φ(0))) [∏_{m=0}^{k−1} ψ(τ_{m}(φ(0)))] (\frac{1}{λ})^{k+1} g(0).

We can show that the term

[∏_{m=0}^{j−1} ψ(τ_{m}(φ(0)))] (\frac{1}{λ})^{j+1} g(τ_{j}(φ(0)))

tends to 0 as j goes to infinity. Therefore, since g(0) ≠ 0, the eigenvalue λ satisfies equation (5.5).

On the other hand, if λ > ∥C_{φ}∥_{e}^{2} is a solution to (5.5), we can prove that it must be an eigenvalue for C_{φ}^{∗}C_{φ}. If λ were not an eigenvalue, then it would not belong to the spectrum of C_{φ}^{∗}C_{φ}; in other words, the operator C_{φ}^{∗}C_{φ} − λ would be invertible. Using this fact, we can demonstrate that

∑_{k=0}^{∞} h(τ_{k}(φ(0))) [∏_{m=0}^{k−1} ψ(τ_{m}(φ(0)))] (\frac{1}{λ})^{k+1} = 0

for all h in H^{2}. We can show that this equation contradicts (5.5).

Since ∥C_{φ}∥_{e} < ∥C_{φ}∥, Proposition 1.3 dictates that ∥C_{φ}∥^{2} is the largest eigenvalue for C_{φ}^{∗}C_{φ}; Proposition 2.2 guarantees that the corresponding eigenfunctions do not vanish at 0. Consequently ∥C_{φ}∥^{2} satisfies equation (5.5) and is greater than every other solution.

In the case where τ_{n}(φ(0)) = 0 for some integer n ≥ 0, line (5.5) reduces to the polynomial equation from Theorem 5.11. Theorems 5.11 and 5.19 are similar in spirit: they both describe ∥C_{φ}∥ in terms of the zeros of a particular analytic function.

Theorem 5.19 has several interesting consequences. For example, let

φ(z) = \frac{α − 1}{α − z}

for some α > 1; it is not difficult to show that ∥C_{φ}∥ > ∥C_{φ}∥_{e}. For such a φ, equation (5.5) may be rewritten

∑_{k=0}^{∞} \frac{(α − 1)^{k+1}}{(k + 1)α − 1} (\frac{1}{λ})^{k+1} = 1.  (5.6)

This representation for ∥C_{φ}∥ allows us to prove an astonishing fact. Take α = 2, so that φ(z) = 1/(2 − z); in this case, line (5.6) reduces to

∑_{k=0}^{∞} \frac{1}{2k + 1} (\frac{1}{λ})^{k+1} = 1.

Summing the series and taking λ = ∥C_{φ}∥^{2}, we see that

\frac{1}{2∥C_{φ}∥} log (\frac{∥C_{φ}∥ + 1}{∥C_{φ}∥ − 1}) = 1.  (5.7)

If ∥C_{φ}∥ happened to be an algebraic number, then Baker’s theorem [3] would dictate that the left-hand side of (5.7) must be transcendental; since 1 is algebraic, we conclude that the norm of C_{φ} is, in fact, a transcendental number. This example shows that there can be no algebraic formula, involving only the coefficients of φ, which gives ∥C_{φ}∥ for every linear fractional φ. In particular, Proposition 4.4 cannot be extended in an obvious manner to include all linear fractional maps.

The second half of the proof of Theorem 5.19 yields a helpful test for determining the relationship between ∥C_{φ}∥ and ∥C_{φ}∥_{e}: Corollary 5.20 Let φ be a non-automorphic linear fractional self-map of 𝔻 that fixes the point 1. If there is a number Λ > ∥C_{φ}∥_{e}^{2} with

∑_{k=0}^{∞} χ(τ_{k}(φ(0))) [∏_{m=0}^{k−1} ψ(τ_{m}(φ(0)))] (\frac{1}{Λ})^{k+1} = 1,

then Λ is an eigenvalue for C_{φ}^{∗}C_{φ}; in particular, ∥C_{φ}∥^{2} ≥ Λ > ∥C_{φ}∥_{e}^{2}.

For example, we can apply this test to a map of the form

φ(z) = \frac{(r + s)z + (1 − s)}{r(1 − s)z + (1 + sr)},

where 0 < s < 1 and −1 < r < 0, to show that ∥C_{φ}∥_{e} is strictly less than ∥C_{φ}∥. For any such φ, we know that the spectral radius of C_{φ} is equal to ∥C_{φ}∥_{e} (see Theorem 3.9 in [11]). Consequently C_{φ}^{∗} cannot be a hyponormal operator, which answers a question posed by Cowen and MacCluer [12].

One can also prove a version of Theorem 5.19 which holds for certain compact composition operators. In this case, the criterion guaranteeing the validity of our representation for ∥C_{φ}∥ is no longer that ∥C_{φ}∥_{e} < ∥C_{φ}∥ (which would always hold, since ∥C_{φ}∥_{e} = 0); instead the condition is that |ψ(w_{0})| < ∥C_{φ}∥^{2}, where w_{0} denotes the Denjoy–Wolff point of τ.

Chapter 6

Norms of Composition Operators

on Subspaces of H^{2}

Consider the Hardy space H^{2}. For an integer n ≥ 0, we write H_{(n)}^{2} to denote the subspace of H^{2} consisting of all functions with a zero of order at least n at the origin. Throughout this chapter, we take φ : 𝔻 → 𝔻 to be a nonconstant analytic map with φ(0) = 0. Observe that each subspace H_{(n)}^{2} is invariant under the corresponding composition operator C_{φ}. While it is easy to determine the norm of C_{φ} on H^{2} (namely ∥C_{φ}∥ = 1), there is no obvious method for finding the norm of C_{φ} on H_{(n)}^{2}. In this chapter, we approach this question using the techniques we have already developed. Not surprisingly, we obtain particularly interesting results in the case where φ is a linear fractional map.

6.1 The essential norm of C_{φ} on H_{(n)}^{2}

As it turns out, it is not difficult to determine the essential norm of C_{φ} on the subspace H_{(n)}^{2}. We state the following proposition without proof; this fact, which appears as Proposition 5.1 in [26], serves as an important step in the proof of Shapiro’s essential norm formula: Proposition 6.1 Let T be a bounded operator on a Hilbert space H. Let {K_{m}} be a sequence of compact self-adjoint operators on H; write P_{m} = I − K_{m}, where I denotes the identity operator on H. Suppose that ∥P_{m}∥ = 1 for each m and that lim_{m→∞} ∥P_{m}(h)∥ = 0 for each h in H; then ∥T ∥_{e} = lim_{m→∞} ∥T P_{m}∥.

In particular, take H to be H_{(n)}^{2}. For m ≥ n + 1, let K_{m} denote the orthogonal projection of H_{(n)}^{2} onto the subspace spanned by the monomials {z^{k}}_{k=n}^{m−1}. Each K_{m}, having finite dimensional range, is compact. Observe that P_{m} = I − K_{m} is the projection of H_{(n)}^{2} onto H_{(m)}^{2}; therefore ∥P_{m}∥ = 1 for every m. For any function f(z) = ∑_{k=n}^{∞} a_{k}z^{k} in H_{(n)}^{2}, we see that

∥P_{m}(f)∥_{2}^{2} = ∑_{k=m}^{∞} |a_{k}|^{2}

for all m ≥ n + 1; hence lim_{m→∞} ∥P_{m}(f)∥_{2} = 0. In other words, the operators K_{m} and P_{m} satisfy the hypotheses of Proposition 6.1.

Since (C_{φ}P_{m}) (f) = C_{φ}(f) for every f in H_{(m)}^{2} and ∥P_{m}∥ = 1, we note that

∥C_{φ}P_{m} : H_{(n)}^{2} → H_{(n)}^{2}∥ = ∥C_{φ} : H_{(m)}^{2} → H_{(m)}^{2}∥

for any m ≥ n+1. Considering Proposition 6.1, we arrive at the following conclusion: Observation 6.2 Let φ be an analytic self-map of 𝔻 with φ(0) = 0; then

∥C_{φ} : H_{(n)}^{2} → H_{(n)}^{2}∥_{e} = lim_{m→∞} ∥C_{φ} : H_{(m)}^{2} → H_{(m)}^{2}∥.

Since this observation holds for any value of n, the next result follows immediately: Observation 6.3 Let φ be an analytic self-map of 𝔻 with φ(0) = 0. For any integer n ≥ 0, the essential norm of the operator C_{φ} : H_{(n)}^{2} → H_{(n)}^{2} is equal to the essential norm of C_{φ} : H^{2} → H^{2}.

For the rest of this chapter, without risk of ambiguity, we write ∥C_{φ}∥_{e} to denote the essential norm of the operator C_{φ} acting on any of the subspaces H_{(n)}^{2}.

6.2 The norm of C_{φ} on H_{(n)}^{2}

At this point, we commence our investigation of the norm of C_{φ} : H_{(n)}^{2} → H_{(n)}^{2}. We begin by introducing more manageable notation. Suppose that H_{(n)}^{2} is invariant under some operator T : H^{2} → H^{2}; we write ∥T ∥_{(n)} to denote the quantity

∥T : H_{(n)}^{2} → H_{(n)}^{2}∥.

Given a map φ, we are interested in describing the behavior of ∥C_{φ}∥_{(n)} as n varies. If m ≤ n, then H_{(n)}^{2} is contained in H_{(m)}^{2}, so ∥C_{φ}∥_{(n)} is certainly less than or equal to ∥C_{φ}∥_{(m)}. In many cases, we can say a good deal more. For example, suppose that φ is an inner function; then (by Proposition 2.1) the operator C_{φ} is an isometry of H^{2}, which means that ∥C_{φ}∥_{(n)} = 1 for every n ≥ 0. Applying Observation 6.2, we obtain the following result: Observation 6.4 Suppose that φ : 𝔻 → 𝔻 is an inner function with φ(0) = 0; then ∥C_{φ}∥_{(n)} = ∥C_{φ}∥_{e} = 1 for every n ≥ 0.

This situation, however, is somewhat atypical. When m < n, it is quite often the case that ∥C_{φ}∥_{(n)} < ∥C_{φ}∥_{(m)}. The next proposition allows us to discuss this issue more precisely:

Proposition 6.5 Let φ : 𝔻 → 𝔻 be an analytic map with φ(0) = 0. Suppose, for some n ≥ 1, that ∥C_{φ}∥_{(n)} is strictly greater than ∥C_{φ}∥_{e}; then

∥C_{φ}∥_{(n)} < ∥C_{φ}∥_{(n−1)} .

Proof. Since C_{φ} : H_{(n)}^{2} → H_{(n)}^{2} is not extremally noncompact, Proposition 1.3 dictates that the operator attains its norm on some unit vector g in H_{(n)}^{2}. Note that the function g̃(z) = g(z)/z is a unit vector in the subspace H_{(n−1)}^{2}. In light of our hypotheses, Observation 6.4 shows that φ cannot be an inner function; hence

lim_{r↑1} |g(φ(re^{iθ}))| < lim_{r↑1} |\frac{g(φ(re^{iθ}))}{φ(re^{iθ})}|

for θ in a set of positive measure. Therefore

∥C_{φ}∥_{(n)} = ∥C_{φ}(g)∥_{2} < ∥C_{φ}(g̃)∥_{2} ≤ ∥C_{φ}∥_{(n−1)} ,

as we had hoped to show.

Appealing to this proposition, we obtain a new proof of a related result, originally due to Joel Shapiro [28]: Proposition 6.6 Suppose that φ : 𝔻 → 𝔻 is analytic with φ(0) = 0; then φ is inner if and only if ∥C_{φ}∥_{(1)} = 1. Proof. Observation 6.4 shows that ∥C_{φ}∥_{(1)} = 1 whenever φ is inner. Suppose then that φ is not inner; Proposition 4.1 in [28] dictates that

∥C_{φ}∥_{e} < \sqrt{\frac{1 + |φ(0)|}{1 − |φ(0)|}} = 1.

Since the norm of an operator is never less than its essential norm, ∥C_{φ}∥_{(1)} ≥ ∥C_{φ}∥_{e}. If ∥C_{φ}∥_{(1)} = ∥C_{φ}∥_{e}, then ∥C_{φ}∥_{(1)} < 1. If ∥C_{φ}∥_{(1)} > ∥C_{φ}∥_{e}, then Proposition 6.5 shows that ∥C_{φ}∥_{(1)} < ∥C_{φ}∥_{(0)} = 1.

In other words, whenever φ is not inner, the norm of C_{φ} acting on H_{(1)}^{2} is strictly less than its norm on H_{(0)}^{2} = H^{2}. As there is no obvious characteristic distinguishing this pair of subspaces from any other, one might expect to find an analogous result for other values of n; for example, perhaps ∥C_{φ}∥_{(1)} > ∥C_{φ}∥_{(2)} whenever φ is not inner. We shall see, however, that this phenomenon does not actually occur; Section 6.3 will provide specific counterexamples.

In light of Observation 6.2, we see that the values ∥C_{φ}∥_{(n)} form a non-increasing sequence converging to ∥C_{φ}∥_{e}. Therefore Proposition 6.5 shows that one of the following two situations must hold:

(1) The sequence ∥C_{φ}∥_{(n)} is strictly decreasing, with ∥C_{φ}∥_{(n)} > ∥C_{φ}∥_{e} for each n and lim_{n→∞} ∥C_{φ}∥_{(n)} = ∥C_{φ}∥_{e}.

(2) The sequence ∥C_{φ}∥_{(n)} is strictly decreasing up to some integer N ≥ 0,

with ∥C_{φ}∥_{(n)} = ∥C_{φ}∥_{(N)} = ∥C_{φ}∥_{e} for all n ≥ N. As shown by Observation 6.4, any inner function φ provides a rather trivial example of situation (2). If φ is not inner, then either situation can manifest itself. For example, if the operator C_{φ} : H^{2} → H^{2} is compact, then situation (1) must hold, since ∥C_{φ}∥_{(n)} never equals 0. In the next section, we shall encounter non-inner φ which serve as further examples of situation (2).

6.3 The linear fractional case

As our final topic, we consider a composition operator C_{φ} whose symbol

φ(z) = \frac{az}{cz + 1}  (6.1)

is a nonconstant linear fractional self-map of 𝔻 with φ(0) = 0; without loss of generality, we have taken the coefficient d to equal 1. As in Section 3.1, we define the map

σ(z) = \overline{a}z − \overline{c}.

Recall that φ is a self-map of 𝔻 if and only if σ is as well; that is, if and only if |a| + |c| ≤ 1. The operator C_{φ} is noncompact if and only if ∥φ∥_{∞} = 1; that is, if and only if there are points ζ and ω on ∂𝔻 such that φ(ζ) = ω. Observation 3.2 shows that this situation occurs if and only if there are ζ and ω on ∂𝔻 such that σ(ω) = ζ, which occurs precisely when |a| + |c| = 1.

Whenever C_{φ} is noncompact, it is easy to determine its essential norm: Lemma 6.7 Let φ : 𝔻 → 𝔻 be a nonconstant linear fractional self-map of 𝔻 with φ(0) = 0, as in (6.1). Suppose that |a| + |c| = 1, so that the operator C_{φ} is noncompact; then ∥C_{φ}∥_{e}^{2} = |a|. Proof. If |a| = 1 and c = 0, then φ is inner, so ∥C_{φ}∥_{e}^{2} = 1 = |a|. If c is nonzero, then there is exactly one point ζ on ∂𝔻 such that

1 = |φ(ζ)| = |\frac{aζ}{cζ + 1}|.

In this case, expression (2.4) shows that

∥C_{φ}∥_{e}^{2} = max {|φ′(w)|^{−1} : |w| = |φ(w)| = 1}
= |φ′(ζ)|^{−1} = \frac{|cζ + 1|^{2}}{|a|} = |\frac{cζ + 1}{aζ}|^{2} |aζ^{2}| = |a|,

which proves our assertion.

We are interested in determining ∥C_{φ}∥_{(n)} for various values of n, in particular n = 1. As usual, our strategy involves considering the spectrum of the operator C_{φ}^{∗}C_{φ}, which we write in terms of the functions

τ(z) = φ(σ(z)),
ψ(z) = \frac{\overline{a}z}{\overline{a}z − \overline{c}},
χ(z) = \frac{\overline{c}}{−\overline{a}z + \overline{c}}.

We need to be somewhat careful, though, about the precise definition of C_{φ}^{∗}. Even though each subspace H_{(n)}^{2} is invariant under C_{φ}, the same is not necessarily true for C_{φ}^{∗}. The adjoint of C_{φ} on H_{(n)}^{2} is, in fact, the restriction of the operator P_{n}C_{φ}^{∗} to H_{(n)}^{2}, where C_{φ}^{∗} denotes the adjoint of C_{φ} on H^{2} and P_{n} is the orthogonal projection of H^{2} onto H_{(n)}^{2}; we write C_{φ}^{(∗,n)} to denote this operator.

For one particular set of maps, C_{φ}^{(∗,n)} actually does equal the restriction of C_{φ}^{∗} to H_{(n)}^{2}, a fact which allows us to determine ∥C_{φ}∥_{(n)}:

Proposition 6.8 If φ has the form φ(z) = az, then ∥C_{φ}∥_{(n)} = |a|^{n} for any n.

Proof. If |a| = 1, then C_{φ} is an inner function, which guarantees that ∥C_{φ}∥_{(n)} = 1 for any n. Now consider the case where 0 < |a| < 1, for which the operator C_{φ} is compact. Cowen’s adjoint formula (Proposition 3.3) dictates C_{φ}^{∗}, the adjoint of C_{φ} on H^{2}, may be written C_{σ}, where σ(z) = \overline{a}z. Note that each subspace H_{(n)}^{2} is invariant under C_{φ}^{∗}, which means that C_{φ}^{(∗,n)} is simply the restriction of C_{φ}^{∗} = C_{σ}. Hence the operator C_{φ}^{(∗,n)}C_{φ} : H_{(n)}^{2} → H_{(n)}^{2} may be written C_{τ}, where τ(z) = |a|^{2} z.

Applying the results of Chapter 4, we can determine the spectrum of C_{φ}^{(∗,n)}C_{φ} = C_{τ} precisely. Observe that 0 is the unique fixed point of the map τ : 𝔻 → 𝔻. For any m ≥ n, the subspace H_{(n)}^{2} contains the monomial z^{m}, a function with a zero of order m at the origin; by definition, H_{(n)}^{2} does not contain any function with a zero of order less than n at 0. Hence the set ℳ consists exactly of the integers which are greater than or equal to n. Proposition 4.3 shows that the spectrum of C_{τ} on H_{(n)}^{2} is precisely 0 ∪ {|a|^{2m}}_{m=n}^{∞}, which means that ∥C_{φ}∥_{(n)}^{2} = ∥C_{τ}∥_{(n)} = |a|^{2n}.

For any f in H^{2}, observe that

(C_{φ}^{∗}(f)) (0) = ⟨C_{φ}^{∗}(f), K_{0}⟩ = ⟨f, C_{φ}(K_{0})⟩ = ⟨f, K_{0}⟩ = f(0).

Consequently the subspace H_{(1)}^{2} is invariant under C_{φ}^{∗}, so C_{φ}^{(∗,1)} is actually just the restriction of C_{φ}^{∗} to H_{(1)}^{2}. This fact, which is specific to this particular subspace, is crucial to our next result: Proposition 6.9 Let φ : 𝔻 → 𝔻 be a nonconstant linear fractional map with φ(0) = 0, as in (6.1), which does not have the form φ(z) = az. If the operator C_{φ} : H^{2} → H^{2} is compact, then ∥C_{φ}∥_{(1)}^{2} = ψ(w_{0}), where w_{0} denotes the Denjoy– Wolff point of τ = φ ∘ σ. Proof. Since τ_{0}(φ(0)) = φ(0) = 0, the map φ is an example of the class discussed in Section 5.1. In this instance, the subspace K_{W}^{⊥} = K_{{0}}^{⊥} is identical to H_{(1)}^{2}. Therefore Proposition 5.6 shows that the spectrum of C_{φ}^{∗}C_{φ} : K_{W}^{⊥} → K_{W}^{⊥}, in other words C_{φ}^{(∗,1)}C_{φ} : H_{(1)}^{2} → H_{(1)}^{2}, is precisely {0} ∪ {ψ(w_{0})τ′(w_{0})^{m}}_{m=0}^{∞}. The largest element of the spectrum is ψ(w_{0}), which equals ∥C_{φ}∥_{(1)}^{2}.

Proposition 6.9 allows us to determine ∥C_{φ}∥_{(1)} for any linear fractional φ. Theorem 6.10 Let

φ(z) = \frac{az}{cz + 1}

be a nonconstant linear fractional self-map of 𝔻 with φ(0) = 0; then

∥C_{φ}∥_{(1)}^{2} = \frac{2 |a|^{2}}{1 + |a|^{2} − |c|^{2} + \sqrt{(1 − |a|^{2} + |c|^{2})^{2} − 4 |c|^{2}}} .

Proof. If the coefficient c equals 0, then φ has the form φ(z) = az; in this case, our claim reduces to saying that ∥C_{φ}∥_{(1)}^{2} = |a|^{2}, which follows from Proposition 6.8. For the rest of the proof, we assume that c ≠ 0.

Suppose first of all that C_{φ} is compact; that is, |a| + |c| < 1. Proposition 6.9 tells us that ∥C_{φ}∥_{(1)}^{2} = ψ(w_{0}), where w_{0} denotes the Denjoy–Wolff point of

τ(z) = \frac{|a|^{2} z − a\overline{c}}{(\overline{a}c)z + (1 − |c|^{2})}.

By direct calculation, we see that

w_{0} = \frac{−1 + |a|^{2} + |c|^{2} + \sqrt{(1 − |a|^{2} − |c|^{2})^{2} − 4 |ac|^{2}}}{2\overline{a}c}.

Therefore

∥C_{φ}∥_{(1)}^{2} = ψ(w_{0}) = \frac{\overline{a}w_{0}}{\overline{a}w_{0} − \overline{c}} = \frac{−2\overline{a}cw_{0}}{−2\overline{a}cw_{0} + 2 |c|^{2}}
= \frac{1 − |a|^{2} − |c|^{2} − \sqrt{(1 − |a|^{2} − |c|^{2})^{2} − 4 |ac|^{2}}}{1 − |a|^{2} + |c|^{2} − \sqrt{(1 − |a|^{2} − |c|^{2})^{2} − 4 |ac|^{2}}} .  (6.2)

Performing elementary manipulations, we see that line (6.2) equals

\frac{2 |a|^{2}}{1 + |a|^{2} − |c|^{2} + \sqrt{(1 − |a|^{2} − |c|^{2})^{2} − 4 |ac|^{2}}}

= \frac{2 |a|^{2}}{1 + |a|^{2} − |c|^{2} + \sqrt{(1 − |a|^{2} + |c|^{2})^{2} − 4 |c|^{2}}} ,

thus proving our result for compact C_{φ}.

We now consider a map that induces a noncompact composition operator; that is, φ with |a| + |c| = 1. In this case, our claim simply asserts that ∥C_{φ}∥_{(1)}^{2} = |a|. For every 0 < r < 1, we define the map

φ_{r}(z) = rφ(z) = \frac{(ra)z}{cz + 1}.

Since |ra| + |c| < |a| + |c| = 1, our norm formula is valid for each operator C_{φ_{r}}. For any polynomial f, the functions f ∘ φ_{r} converge to f ∘ φ pointwise on the closed unit disk. Hence Fatou’s lemma dictates that

∥C_{φ}(f)∥_{2} ≤ lim inf_{r↑1} ∥C_{φ_{r}}(f)∥_{2} ,

from which we see that

∥C_{φ}∥_{(1)}^{2} ≤ lim inf_{r↑1} ∥C_{φ_{r}}∥_{(1)}^{2}
= lim_{r↑1} \frac{2 |ra|^{2}}{1 + |ra|^{2} − |c|^{2} + \sqrt{(1 − |ra|^{2} + |c|^{2})^{2} − 4 |c|^{2}}}
= |a|.

We appeal to Lemma 6.7 to establish the reverse inequality:

∥C_{φ}∥_{(1)}^{2} ≥ ∥C_{φ}∥_{e}^{2} = |a|.

Therefore our formula is valid for every linear fractional φ : 𝔻 → 𝔻 with φ(0) = 0.

Combining the results of Lemma 6.7 and Theorem 6.10, we make the following observation:

Observation 6.11 Let φ : 𝔻 → 𝔻 be a nonconstant linear fractional self-map of 𝔻 with φ(0) = 0, as in (6.1). Suppose that |a| + |c| = 1, so that the operator C_{φ} is noncompact; then ∥C_{φ}∥_{(1)}^{2} = ∥C_{φ}∥_{e}^{2} = |a|.

In light of this fact, our next result follows immediately:

Theorem 6.12 Let φ : 𝔻 → 𝔻 be a nonconstant linear fractional self-map of 𝔻 with φ(0) = 0, as in (6.1). Suppose that |a| + |c| = 1, so that the operator C_{φ} is noncompact; then ∥C_{φ}∥_{(n)}^{2} = ∥C_{φ}∥_{e}^{2} = |a| for all n ≥ 1.

Proof. We know that ∥C_{φ}∥_{e} ≤ ∥C_{φ}∥_{(n)} ≤ ∥C_{φ}∥_{(1)} for any integer n ≥ 1. Since ∥C_{φ}∥_{(1)} = ∥C_{φ}∥_{e}, all of these quantities must be equal.

In other words, these maps provide a nontrivial example of situation (2), as described at the end of Section 6.2. They also serve as a counterexample to the hypothesis that ∥C_{φ}∥_{(1)} > ∥C_{φ}∥_{(2)} for any non-inner φ.

The astute reader will notice a definite resemblance between Theorem 6.10 and Proposition 4.4; the methods of proof and the actual norm formulae are strikingly similar. This correspondence is not accidental. An alternate proof of Theorem 6.10 helps to shed light on this situation:

Alternate proof of Theorem 6.10. In his proof of Theorem 5 in [9], Cowen showed that the adjoint C_{φ}^{(∗,1)} : H_{(1)}^{2} → H_{(1)}^{2} is unitarily equivalent to the operator \overline{a}C_{σ} : H^{2} → H^{2}, where σ(z) = \overline{a}z − \overline{c}. Hence Proposition 4.4 dictates that

∥C_{φ}∥_{(1)}^{2} = ∥C_{φ}^{(∗,1)}∥_{(1)}^{2} = |a|^{2} ∥C_{σ}∥_{(0)}^{2}
= \frac{2 |a|^{2}}{1 + |a|^{2} − |c|^{2} + \sqrt{(1 − |a|^{2} + |c|^{2})^{2} − 4 |c|^{2}}}

for any such φ.

This point of view also provides further insight into Observation 6.11. Reworded slightly, that result simply states that the operator C_{φ} : H_{(1)}^{2} → H_{(1)}^{2} is extremally noncompact. This fact makes more sense once we recognize that \overline{a}C_{σ} : H^{2} → H^{2}, which is unitarily equivalent to C_{φ}^{(∗,1)} : H_{(1)}^{2} → H_{(1)}^{2}, possesses the same property.

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List of Symbols

\begin{matrix}K_{w} & 4 \\ k_{w} & 4 \\ K_{W} & 4 \\ K_{W}^{⊥} & 4 \\ C_{φ} & 4 \\ ∥ · ∥_{e} & 7 \\ r_{e}(·) & 7 \\ r(·) & 7 \\ S_{φ}^{∗} & 9 \\ S_{φ} & 9 \\ 𝔻 & 11 \\ H^{2} & 12 \\ ∥ · ∥_{2} & 12 \\ A_{α}^{2} & 16 \\ ∥ · ∥_{2,α} & 16 \end{matrix}  \begin{matrix}⟨·,·⟩_{α} & 16 \\ A^{2} & 17 \\ A_{−1}^{2} & 17 \\ H^{∞} & 31 \\ ∥ · ∥_{∞} & 31 \\ M_{γ} & 31 \\ H^{p} & 32 \\ ∥ · ∥_{p} & 32 \\ A_{α}^{p} & 32 \\ R_{φ,ψ} & 45 \\ ℳ & 45 \\ τ_{j} & 53 \\ H_{(n)}^{2} & 74 \\ ∥ · ∥_{(n)} & 76 \\ C_{φ}^{(∗,n)} & 80 \end{matrix}

Index of Terminology

Automorphism, 18 Banach space of analytic functions, 32 Bergman space (unweighted), 17 Calkin algebra, 7 Composition operator, 4 Cowen’s adjoint formula, 37 Denjoy–Wolff point, 54 Essential norm, 7 Essential spectral radius, 7 Essential spectrum, 7 Extremal noncompactness, 8 Functional Banach space, 32 Functional Hilbert space, 3 Hardy space, 12, 32 Hilbert space of analytic functions, 4 Inner function, 13 Iterate, 53 Local resemblance (to a weighted

composition operator), 45 Multiplication operator, 31 Nevanlinna-type counting function, 20 Norm attainment, 7 Reproducing kernel function, 4 Reproducing kernel Hilbert space, 4 Self-map, 4 Symbol (of a composition operator), 5 Toeplitz operator, 31 Weighted Bergman space, 16, 32 Weighted composition operator, 44