THE NORM OF A COMPOSITION OPERATOR WITH LINEAR SYMBOL ACTING ON THE DIRICHLET SPACE

CHRISTOPHER HAMMOND

Abstract. We obtain a representation for the norm of a composition operator on the Dirichlet space induced by a map of the form φ(z) = az+b. We compare this result to an upper bound for ∥C_{φ}∥ that is valid whenever φ is univalent. Our work relies heavily on an adjoint formula recently discovered by Gallardo-Gutiérrez and Montes-Rodríguez.

1. Introduction

Let 𝔻 = {z ∈ ℂ : |z| < 1} and let dA denote normalized area measure on 𝔻. The Dirichlet space D is the set of all analytic functions f on 𝔻 for which

∥f∥_{D}^{2} := |f(0)|^{2} + ∫_{𝔻} |f′(z)|^{2} dA(z)

is finite. The Dirichlet space is a Hilbert space under the obvious inner product. Moreover, D is a reproducing kernel Hilbert space; that is, for any point w in 𝔻, there is a corresponding K_{w} in D such that ⟨f, K_{w}⟩_{D} = f(w) for all f in D. In the case of the Dirichlet space, these functions have the form K_{w}(z) = 1 + log(1/(1 − \overline{w}z)). Note that

∥K_{w}∥_{D}^{2} = ⟨K_{w}, K_{w}⟩_{D} = K_{w}(w) = 1 + log(1/(1 − |w|^{2})).

Given an analytic map φ : 𝔻 → 𝔻, we define the composition operator C_{φ} on D by the rule

C_{φ}(f) = f ∘ φ.

It is certainly not obvious that every C_{φ} should take D into itself, and there are actually many examples for which it does not (see Proposition 3.12 in [9]). Nevertheless, any univalent φ (or even any φ with bounded valence) is guaranteed to induce a bounded composition operator on D. This paper is only concerned with

2000 Mathematics Subject Classification. 47B33.

Keywords: Composition operator, Dirichlet space.

2 C. HAMMOND

univalent φ, so the problem of unbounded composition operators will not arise. There are, of course, many other Hilbert spaces on which one can consider the action of a composition operator. We will make mention of several results pertaining to the Hardy space H^{2} and the weighted Bergman spaces A_{α}^{2}. Cowen and MacCluer’s book [4] serves as a standard reference for such topics.

One of the major impediments to the study of (bounded) composition operators is the lack of a reasonable representation for the adjoint C_{φ}^{∗}. It is well known, and is easy to prove, that C_{φ}^{∗}(K_{w}) = K_{φ(w)} for any reproducing kernel function K_{w}. Beyond this fact, not much is known about the adjoints of composition operators. Working on the Hardy space H^{2}, Cowen [3] obtained an explicit representation for C_{φ}^{∗} in the case where φ is a linear fractional map; his result was later extended to the weighted Bergman spaces A_{α}^{2} by Hurst [8]. Their arguments rely heavily on the particular form of the reproducing kernel functions for those spaces, and hence cannot be adapted to the Dirichlet space. Gallardo-Gutiérrez and Montes-Rodríguez [5], however, have recently discovered a representation for C_{φ}^{∗} : D → D when φ is linear fractional. Their adjoint formula (which appears in Section 3 below) provides the foundation for the results of this paper.

One of Cowen’s original applications for his adjoint formula was to determine the norm of an operator C_{φ} : H^{2} → H^{2} when φ has the form φ(z) = az + b. Similarly, Hurst was able to calculate the norms of the analogous operators on A_{α}^{2}. (See also [12] and [13].) Now that we have an adjoint formula that is valid for the Dirichlet space, it seems reasonable to consider the same problem in this context. The actual result we obtain (Theorem 4.2) has a rather different appearance from its counterparts in the Hardy and weighted Bergman spaces.

The question of actually calculating the norm of a composition operator is not a trivial one. Aside from the aforementioned results of Cowen [3] and Hurst [8], there are not many instances for which we know the exact value of the norm. Even the case where φ is linear fractional has proved quite difficult. Bourdon, Fry, Spofford, and the author [1] (and the author individually [6]) have considered this question in the context of the Hardy space. Gallardo-Gutiérrez and Montes-Rodríguez [5], as a consequence of their work with adjoints, were able to determine the norm of such an operator acting on the subspace D_{0} = {f ∈ D : f(0) = 0}. As we stated in the last

THE NORM OF A COMPOSITION OPERATOR 3

paragraph, our goal here is much more modest: simply to obtain a representation for the norm of C_{φ} : D → D when φ has the form φ(z) = az + b. As it turns out, the techniques we employ will be similar to those used to deal with the linear fractional case on the Hardy space.

In spite of the difficulties associated with computing the norm exactly, it is often possible to find sharp estimates for ∥C_{φ}∥ in terms of the value |φ(0)|. Such results are well known for operators acting on the Hardy and weighted Bergman spaces (see Section 3.1 of [4]). Martín and Vukotić [10] recently obtained an upper bound for the norm of C_{φ} : D → D when φ is univalent. We will discuss this result, and its relationship to our norm representation, in Section 5.

2. Preliminaries

Let T be a bounded operator on a Hilbert space H, with T^{∗} denoting its adjoint operator. Since the spectral radius of T^{∗}T equals ∥T^{∗}T ∥ = ∥T ∥^{2}, it seems reasonable to study the spectrum of T^{∗}T when trying to determine ∥T ∥. This point of view has served as the basis for much of the recent work relating to the norms of composition operators (e.g. [1] and [6]). Even the results of Cowen [3] and Hurst [8], when viewed in a particular manner, can be seen as statements pertaining to the spectrum of C_{φ}^{∗}C_{φ} (see Chapter 4 of [7]). The following proposition, which can be proved with a straightforward Hilbert space argument (see Proposition 1.2 in [7]), further emphasizes the connection between ∥T ∥ and the spectrum of T^{∗}T.

Proposition 2.1. Let h be an element of H; then ∥T (h)∥ = ∥T ∥ ∥h∥ if and only if (T^{∗}T)(h) = ∥T ∥^{2} h.

Whenever there is a nonzero h such that ∥T (h)∥ = ∥T ∥ ∥h∥, we say that the operator T is norm-attaining. Proposition 2.1 tells us that an operator T has this property if and only if ∥T ∥^{2} is an eigenvalue for T^{∗}T. The next result, whose proof appears in [6], provides further insight into this situation. Recall that the essential norm ∥T ∥_{e} of an operator T : H → H is simply the norm of its equivalence class in the Calkin algebra; that is,

∥T ∥_{e} := inf_{K} ∥T − K∥,

the infimum being taken over all compact operators K : H → H.

Proposition 2.2. If ∥T ∥_{e} < ∥T ∥, then the operator T is norm-attaining.

The object of this section is to show that the composition operators we are currently studying have the property that ∥C_{φ}∥_{e} < ∥C_{φ}∥, from which it will follow that they are norm-attaining. We begin with the following (widely known) result.

Proposition 2.3. Let φ : 𝔻 → 𝔻 be an analytic map that induces a bounded composition operator on D; then ∥C_{φ}∥ ≥ \sqrt{1 + log(1/(1 − |φ(0)|^{2}))}.

Proof. Observe that the kernel function K_{0}(z) = 1 is a unit vector and that

∥C_{φ}^{∗}(K_{0})∥_{D}^{2} = ∥K_{φ(0)}∥_{D}^{2} = 1 + log(1/(1 − |φ(0)|^{2})).

Our claim follows immediately, since ∥C_{φ}∥ = ∥C_{φ}^{∗}∥. □

Our next step is to obtain an estimate for the essential norm of C_{φ}.

Proposition 2.4. Let φ : 𝔻 → 𝔻 be a univalent map; then the essential norm of the operator C_{φ} : D → D is no greater than 1.

Proof. Let P denote the orthogonal projection from D onto the subspace D_{0}; that is, (P f)(z) = f(z) − f(0). Let Q = I − P, where I denotes the identity map on D. Since Q is a compact operator, it follows that

∥C_{φ}∥_{e} = ∥P C_{φ} + QC_{φ}∥_{e} = ∥P C_{φ}∥_{e} ≤ ∥P C_{φ}∥.

It is not difficult to see that the operator P C_{φ} is a contraction on D. Because φ is univalent, the standard change-of-variables formula simply reduces to w = φ(z) and dA(w) = |φ′(z)|^{2} dA(z); therefore

∥(P C_{φ})(f)∥_{D}^{2} = ∫_{𝔻} |f′(φ(z))|^{2} |φ′(z)|^{2} dA(z) = ∫_{φ(𝔻)} |f′(w)|^{2} dA(w) ≤ ∥f∥_{D}^{2}

for all f in D. Consequently ∥C_{φ}∥_{e} ≤ 1, as we had hoped to show. □

Whenever φ : 𝔻 → 𝔻 is a univalent map with φ(0) ≠ 0, Propositions 2.3 and 2.4 combine to show us that ∥C_{φ}∥_{e} < ∥C_{φ}∥. Therefore Proposition 2.2 dictates that such an operator C_{φ} : D → D is norm-attaining, a fact which will prove useful in Section 4.

THE NORM OF A COMPOSITION OPERATOR 5

3. Operators with linear fractional symbol

Our immediate goal is to obtain a functional equation that relates an eigenvalue of C_{φ}^{∗}C_{φ} to the values of its eigenfunctions at particular points in the disk. We will perform our preliminary work in the more general setting where φ is a linear fractional map. To that end, let

(1)  φ(z) = \frac{az + b}{cz + d}

be a nonconstant linear fractional self-map of 𝔻. As in the case of the Hardy and weighted Bergman spaces, the adjoint C_{φ}^{∗} : D → D can be written in terms of

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

The map σ takes 𝔻 into itself whenever φ has the same property (see Lemma 9.1 in [4]). Gallardo-Gutiérrez and Montes-Rodríguez [5] showed that the adjoint C_{φ}^{∗} on D can be written

(C_{φ}^{∗}f) (z) = f(σ(z)) + f(0)K_{φ(0)}(z) − f(σ(0))

for any f in D. Following the convention established in [1] and [6], we write τ to denote the map

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

(Gallardo-Gutiérrez and Montes-Rodríguez use ϕ to signify the same map.) It follows that the operator C_{φ}^{∗}C_{φ} : D → D can be written

(C_{φ}^{∗}C_{φ}f) (z) = f(τ(z)) + f(φ(0))K_{φ(0)}(z) − f(τ(0))

for all z in 𝔻, for any f in D. In particular, if g is an eigenfunction for C_{φ}^{∗}C_{φ} corresponding to an eigenvalue λ, then

(4)  λg(z) = g(τ(z)) + g(φ(0))K_{φ(0)}(z) − g(τ(0)).

We would like to exploit equation (4) to obtain information about the possible values of λ, and hence about the norm of C_{φ}. Our strategy is based largely on the methods developed in [1] and [6].

Throughout this paper, we write τ_{j} to denote the jth iterate of τ; that is, τ_{0} is the identity map on 𝔻 and τ_{j+1} = τ ∘τ_{j}. Our next result is analogous to Proposition 5.1 in [6].

Proposition 3.1. Let g be an eigenfunction for C_{φ}^{∗}C_{φ} corresponding to an eigenvalue λ. For any natural number n, the equation

λ^{n}g(z) = g(τ_{n}(z)) + ∑_{j=1}^{n} λ^{n−j}[g(φ(0))K_{φ(0)}(τ_{j−1}(z)) − g(τ(0))]

holds for all z in 𝔻.

Proof. This assertion follows from an elementary induction argument, the base case and the induction step both coming as consequences of equation (4). □

Since ∥C_{φ}∥^{2} > 1 whenever φ(0) ≠ 0, it seems fair to restrict our attention to the eigenvalues of C_{φ}^{∗}C_{φ} which are larger than 1. For such values of λ, we are able to obtain an “n = ∞” version of Proposition 3.1.

Proposition 3.2. Let g be an eigenfunction for C_{φ}^{∗}C_{φ} corresponding to an eigenvalue λ > 1; then

(5)  g(z) = ∑_{j=1}^{∞} (\frac{1}{λ})^{j} [g(φ(0))K_{φ(0)}(τ_{j−1}(z)) − g(τ(0))]

for all z in 𝔻.

Proof. Our result will follow directly from Proposition 3.1, if only we can show that

lim_{n→∞} \frac{g(τ_{n}(z))}{λ^{n}} = 0

for any z in 𝔻. There are several cases to consider. First of all, if φ happens to be an automorphism, then one can easily see that σ = φ^{−1}; hence τ is simply the identity map on 𝔻, from which our claim follows. If the Denjoy–Wolff point w_{0} of τ lies inside 𝔻, then the terms g(τ_{n}(z)) are converging pointwise to g(w_{0}), in which case our claim also holds. Suppose then that φ is not an automorphism and that w_{0} lies on the unit circle ∂𝔻. In this case, the map τ must be of parabolic type; that is τ′(w_{0}) = 1. This fact can be deduced directly from Lemma 5.1 in [2] (or rather the remark immediately following its proof), or as a consequence of Theorem 4.1 in [5]. Hence we can appeal to the argument used to prove Lemma 3.3 in [1] to see that, for every z in 𝔻, there is a constant C such that (1 − |τ_{n}(z)|)^{−1} ≤ Cn.

Thus it follows that

|g(τ_{n}(z))| ≤ ∥g∥_{D} ∥K_{τ_{n}(z)}∥_{D}
= ∥g∥_{D} \sqrt{1 + log(1/(1 − |τ_{n}(z)|^{2}))}
≤ ∥g∥_{D} \sqrt{1 + log(Cn)},

from which we obtain the desired result. □

Unfortunately, Proposition 3.2 (which is based on the proof of Theorem 3.5 in [1]) does not allow us readily to determine ∥C_{φ}∥. The fact that expression (5) involves both g(φ(0)) and g(τ(0)) prevents us from obtaining an equation solely in terms of λ. The maps in which we are most interested, though, are precisely those for which φ(0) = τ(0). In that case, as we shall see in the next section, Proposition 3.2 will allow us to find a series representation for ∥C_{φ}∥^{2}.

4. Operators with linear symbol

For the remainder of our discussion, we will restrict our attention to the setting where φ has the form φ(z) = az + b, with a and b both nonzero and |a| + |b| ≤ 1. (Note that b = 0 implies ∥C_{φ}∥ = 1.) We can, of course, view φ as being a linear fractional map, as in (1), with c = 0 and d = 1. Define the maps σ and τ as in (2) and (3). In this case, we see that σ(0) = 0, which means that τ(0) = φ(0). Hence Proposition 3.2 becomes a more manageable result; equation (5) can be rewritten

\begin{aligned}g(z) & = g(τ(0)) ∑_{j=1}^{∞} (\frac{1}{λ})^{j} [K_{τ(0)}(τ_{j−1}(z)) − 1] \\ & = g(τ(0)) ∑_{j=1}^{∞} (\frac{1}{λ})^{j} log(1/(1 − \overline{b}τ_{j−1}(z))). \end{aligned} ; (6)

If g(τ(0)) = 0, then equation (6) would dictate that the function g(z) is identically 0. Thus any eigenfunction g must have the property that g(τ(0)) ≠ 0. Therefore, taking z = τ(0), we see that any eigenvalue λ of C_{φ}^{∗}C_{φ} (with λ > 1) must satisfy the condition

(7)  1 = ∑_{j=1}^{∞} (\frac{1}{λ})^{j} log(\frac{1}{1 − \overline{b}τ_{j}(0)}).

Our main result will be stated in terms of this equation.

At this point, we will briefly discuss the possible solutions to equation (7). The following lemma will greatly simplify the situation.

Lemma 4.1. For any j ≥ 1, the point \overline{b}τ_{j}(0) is a positive real number; moreover, \overline{b}τ_{j}(0) < \overline{b}w_{0}, where w_{0} denotes the Denjoy–Wolff point of τ.

Proof. In the case we are considering, the map τ has the form

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

A simple calculation shows that

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

The numerator of this expression is a real number greater than 2 |b|^{2}, which means that \overline{b}w_{0} is a positive real number and that |w_{0}| > |b|. Thus our claim is equivalent to saying that every point τ_{j}(0) belongs to the set S = \{tb : 0 < t < \overline{b}w_{0}/ |b|^{2}\}, the line segment connecting the points 0 and w_{0}. Consider the image of S under the linear fractional map τ. The image must be an open-ended line segment inside 𝔻. (The image cannot be an arc, since the point τ^{−1}(∞) = b/ |b|^{2} lies on the line containing S.) Since τ(w_{0}) = w_{0}, one endpoint of the image segment must be w_{0}; the other endpoint is τ(0) = b. Since b itself belongs to S, we conclude that the image of S under τ must actually be a subsegment of S. Thus, by induction, we see that each point τ_{j}(0) does indeed lie on the segment S. □

Consider the analytic function

(8)  F (z) = ∑_{j=1}^{∞} log(\frac{1}{1 − \overline{b}τ_{j}(0)}) z^{j}.

Lemma 4.1 guarantees that each of the coefficients log(1/(1 − \overline{b}τ_{j}(0))) is a positive real number. Therefore, since the points τ_{j}(0) converge to w_{0}, we make the following elementary observations:

• The power series that defines F (z) has radius of convergence 1.

• F (x) is a non-negative real number for all x in the interval [0, 1).

• F (0) = 0 and the series F (1) diverges to infinity.

• F′(x) > 0 for all x in the interval (0, 1).

THE NORM OF A COMPOSITION OPERATOR 9

Based on this information, we conclude that there is exactly one ξ in the interval (0, 1) such that F (ξ) = 1. In other words, there is exactly one number λ > 1 that satisfies equation (7).

Combining the results of the last three sections, we obtain the following characterization of ∥C_{φ}∥.

Theorem 4.2. Let φ(z) = az +b, where a and b are both nonzero and |a|+|b| ≤ 1, and consider the operator C_{φ} : D → D. Then λ = ∥C_{φ}∥^{2} is the unique positive real solution to the equation

1 = ∑_{j=1}^{∞} (\frac{1}{λ})^{j} log(\frac{1}{1 − \overline{b}τ_{j}(0)}),

where τ_{j} denotes the jth iterate of the map

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

Proof. The arguments set forth in Section 2 show that ∥C_{φ}∥^{2} is an eigenvalue for C_{φ}^{∗}C_{φ}, with ∥C_{φ}∥^{2} > 1. Hence λ = ∥C_{φ}∥^{2} is a positive real solution to equation (7). We have just observed that only one such solution exists. □

Remark. In certain instances, our norm representation takes on a somewhat more tractable form. In particular, consider the maps φ(z) = az + b with 0 < |b| < 1 and |a| + |b| = 1. For such φ, an induction argument shows that

τ_{j}(z) = \frac{(1 − (j + 1) |b|) z + jb}{−j\overline{b}z + 1 + (j − 1) |b|}

for all j ≥ 1. Hence equation (7) can be rewritten

(9)  1 = ∑_{j=1}^{∞} (\frac{1}{λ})^{j} log(\frac{1 + (j − 1) |b|}{(1 − |b|) (1 + j |b|)}).

For example, take φ(z) = (1/2) z + 1/2; in this case, equation (9) simply becomes

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

Using numerical methods, we see that ∥C_{φ}∥ = \sqrt{λ} ≈ 1.195830076.

10 C. HAMMOND

5. Operators with maximal norm

As we mentioned in the introduction, Martín and Vukotić [10] recently obtained an upper bound for the norm of C_{φ} : D → D when φ is univalent; in particular,

(10)  ∥C_{φ}∥ ≤ \sqrt{\frac{2 + L + \sqrt{L(4 + L)}}{2}},

where L = log(1/(1 − |φ(0)|^{2})). Moreover, they showed that equality occurs whenever φ is a full map, that is, the area of 𝔻 \ φ(𝔻) equals 0. Whenever there is equality in (10) for some univalent φ, we say that the operator C_{φ} has maximal norm on D. In light of the work done by Joel Shapiro [11] on the Hardy space, it seems reasonable to try to determine which composition operators possess this property. We have already noted that any univalent full map induces a composition operator with maximal norm. It also follows from the work of Martín and Vukotić (in particular, Theorem 2 in [10]) that, whenever φ(0) ≠ 0, the operator C_{φ} cannot have maximal norm unless ∥φ∥_{∞} = sup{|φ(z)| : z ∈ 𝔻} = 1. We can actually use our own norm representation to contribute a small piece of information to this line of inquiry: namely that no map of the form φ(z) = az + b, with b ≠ 0, induces a composition operator with maximal norm on D. The substance of this assertion lies in the following estimate. (The reader will notice a certain symmetry between this result and Proposition 2.3.)

Proposition 5.1. Suppose that φ(z) = az + b, where a and b are both nonzero and |a| + |b| ≤ 1; then

(11)  ∥C_{φ}∥ ≤ \sqrt{1 + log(\frac{1}{1 − \overline{b}w_{0}} )},

where w_{0} denotes the Denjoy–Wolff point of τ.

Proof. Consider the analytic function F (z), as defined in line (8). Bearing in mind the result of Lemma 4.1, we see that

log(\frac{1}{1 − \overline{b}τ_{j}(0)}) ≤ log(\frac{1}{1 − \overline{b}w_{0}} )

for all j ≥ 1. Hence, for any x in the interval [0, 1), it follows that

(12)  F (x) ≤ ∑_{j=1}^{∞} log(\frac{1}{1 − \overline{b}w_{0}} ) x^{j} = \frac{x}{1 − x} log(\frac{1}{1 − \overline{b}w_{0}} ).

THE NORM OF A COMPOSITION OPERATOR 11

We have already noted that there is a unique ξ in (0, 1) such that F (ξ) = 1; furthermore, we know that F′(x) > 0 on (0, 1). Hence (12) shows that ξ is greater than or equal to any x satisfying the equation

\frac{x}{1 − x} log(\frac{1}{1 − \overline{b}w_{0}} ) = 1.

In other words,

ξ ≥ (1 + log(\frac{1}{1 − \overline{b}w_{0}} ))^{−1} .

Our claim follows from Theorem 4.2, which dictates that ∥C_{φ}∥ = \sqrt{1/ξ}. □

In the case where |a| + |b| < 1, Proposition 5.1 has an interesting geometrical interpretation. Appealing to Lemma 4 in [12], we see that line (11) can be rewritten

∥C_{φ}∥ ≤ \sqrt{1 + log(\frac{r}{R})},

where R denotes the Euclidean radius of the disk φ(𝔻) and r its pseudohyperbolic radius. When |a| + |b| = 1, on the other hand, our estimate simply becomes

∥C_{φ}∥ ≤ \sqrt{1 + log(\frac{1}{1 − |b|})}.

This last statement will allow us to obtain our final result.

Theorem 5.2. Let φ(z) = az +b, where a and b are both nonzero and |a|+|b| ≤ 1. The operator C_{φ} : D → D does not have maximal norm.

Proof. If |a| + |b| < 1, then ∥φ∥_{∞} < 1. As we have already mentioned, Theorem 2 in [10] guarantees that no such map induces a composition operator with maximal norm. Suppose then that |a| + |b| = 1. In view of Proposition 5.1, we simply need to show that, for any b = φ(0) ≠ 0, the quantity \sqrt{1 + log(1/(1 − |b|))} is strictly less than the term on right-hand side of (10). After some manipulation, this claim reduces to the easily verifiable inequality

log(1 − |b|) + log(1 + |b|) < log(1 − |b|) log(1 + |b|).

Therefore the upper bound stated in Proposition 5.1 is strictly less than the bound obtained by Martín and Vukotić [10]. In other words, none of the maps we are considering induces a composition operator with maximal norm. □

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

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[7] C. Hammond, On the norm of a composition operator, Ph.D. thesis, University of Virginia, 2003.

[8] P. R. Hurst, Relating composition operators on different weighted Hardy spaces, Arch. Math. (Basel) 68 (1997), 503–513.

[9] B. D. MacCluer and J. H. Shapiro, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math. 38 (1986), 878–906.

[10] M. J. Martín and D. Vukotić, Norms and spectral radii of composition operators acting on the Dirichlet space, J. Math. Anal. Appl., in press.

[11] J. H. Shapiro, What do composition operators know about inner functions?, Monatsh. Math. 130 (2000), 57–70.

[12] D. Vukotić, On norms of composition operators acting on Bergman spaces, J. Math. Anal. Appl. 291 (2004), 189–202.

[13] D. Vukotić, Erratum to “On norms of composition operators acting on Bergman spaces” [J. Math. Anal. Appl. 291 (2004), 189–202], in press.

Department of Mathematics and Computer Science, Connecticut College, Box 5384, 270 Mohegan Avenue, New London, CT 06320, USA

   E-mail address: cnham@conncoll.edu