ON THE NORM OF A COMPOSITION OPERATOR WITH LINEAR FRACTIONAL SYMBOL

CHRISTOPHER HAMMOND

Abstract. For any analytic map φ : 𝔻 → 𝔻, the composition operator C_{φ} is bounded on the Hardy space H^{2}, but there is no known procedure for precisely computing its norm. This paper considers the situation where φ is 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 normalized reproducing kernel functions of H^{2}. We also introduce a new set of conditions on φ under which we can calculate ∥C_{φ}∥; moreover, we identify the elements of H^{2} on which such an operator C_{φ} attains its norm. Several specific examples are provided.

1. Introduction

For 1 ≤ p < ∞, the Hardy space H^{p} is the collection of all analytic functions f on 𝔻 = {z ∈ ℂ : |z| < 1} with

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

Under this norm, H^{p} is a Banach space for all such p and a Hilbert space for p = 2. For any analytic map φ : 𝔻 → 𝔻, the composition operator C_{φ} on H^{p} is defined by the rule

C_{φ}(f) = f ∘ φ.

Every composition operator is bounded, with

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

These inequalities 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. In general, though, there is no known procedure for precisely

2000 Mathematics Subject Classification. 47B33. This paper is part of the author’s doctoral thesis, written at the University of Virginia under

the direction of Barbara D. MacCluer.

2 C. HAMMOND

computing the norm of C_{φ}. We see from expression (1.1) that ∥C_{φ}∥ = 1 whenever φ(0) = 0. There are only a few other cases where we can determine the norm exactly; for example

(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.

These results appear in [12], [6], and [7] respectively. Cowen and MacCluer [8] provide a comprehensive treatment of this material, as well as a thorough overview of results relating to composition operators.

A straightforward argument involving Blaschke products shows that the following norm relationship holds for all p ≥ 1:

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

Therefore it suffices to focus our attention on the Hilbert space H^{2}. When studying this space, it is often helpful to consider the reproducing kernel functions {K_{w}}_{w∈𝔻}, defined by the property that ⟨f, K_{w}⟩ = f(w) for all f in H^{2}. These functions have the form K_{w}(z) = (1 − \overline{w}z)^{−1}; hence

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

Throughout this paper, we write k_{w} to denote the normalized kernel function

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

For a subset W of 𝔻, let K_{W} denote the closed linear span of the kernel functions {K_{w}}_{w∈W}. Observe that the orthogonal complement K_{W}^{⊥} is precisely the set of all functions in H^{2} that vanish on W.

The kernel functions provide a valuable tool for the study of composition operators, in part because of the property that C_{φ}^{∗}(K_{w}) = K_{φ(w)} for any adjoint C_{φ}^{∗}. Several authors have explored the connection between the kernel functions and the norm of C_{φ}. In the cases where we know ∥C_{φ}∥, the norm is given by the action of the operator on the set of normalized kernel functions. This situation, however, is not true in general, a fact first proved by Appel, Bourdon, and Thrall [1].

ON THE NORM OF A COMPOSITION OPERATOR 3

The main results of this paper pertain to the situation where φ : 𝔻 → 𝔻 is a linear fractional map. In this case, we determine the conditions under which ∥C_{φ}∥ is given by the action of either C_{φ} or C_{φ}^{∗} on the normalized reproducing kernel functions (Theorem 4.4). We also introduce a new set of conditions on φ under which, at least in principle, we can calculate ∥C_{φ}∥ (Theorem 5.5). For such φ, 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.

2. Preliminaries

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 2.1. 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, or can be deduced from other well-known results (e.g. [10], p. 92). Whenever ∥T (h)∥ = ∥T ∥ ∥h∥ for h ≠ 0, we say that the operator T attains its norm on the element h.

Let ∥ · ∥_{e}, r(·), and r_{e}(·) denote respectively the essential norm, the spectral radius, and the essential spectral radius of an operator. Here the adjective essential signifies that a particular quantity is taken with respect to the Calkin algebra. In light of Proposition 2.1, our next observation follows easily.

Proposition 2.2. 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 the largest element of the spectrum of T^{∗}T does not belong to the essential spectrum, meaning that it is an eigenvalue of finite multiplicity. Consequently T^{∗}T has an eigenvector corresponding to ∥T ∥^{2}, on which the operator T attains its norm. □

4 C. HAMMOND

It is helpful to remember Proposition 2.2 when studying composition operators, especially since we have a formula (due to Joel Shapiro [15]) for the essential norm of C_{φ} on H^{2}. As it happens, in cases (1) and (3) where we know ∥C_{φ}∥, the operators have the property that ∥C_{φ}∥_{e} = ∥C_{φ}∥, a condition sometimes called extremal noncompactness.

The remaining results in this section are specific to composition operators on the Hardy space H^{2}.

Proposition 2.3. 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 of 𝔻.

Proof. Suppose that g(w) = 0 for some w in 𝔻. Then the function

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

belongs to H^{2}, with \left∥{ \atop g}\right∥_{2} = ∥g∥_{2}. Since φ is not an inner function, neither is the composition b_{w} ∘ φ. Therefore

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 2.4. 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. □

We end this section with a straightforward, but remarkably useful observation. Let λ be an eigenvalue for C_{φ}^{∗}C_{φ} with a corresponding eigenfunction g; since C_{φ} fixes the constant function K_{0}(z) = 1, we see that

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

In particular, this result holds for λ = ∥C_{φ}∥^{2} if C_{φ} attains its norm on g.

3. The operator C_{φ}^{∗}C_{φ}

Let

φ(z) = \frac{az + b}{cz + d}

be a nonconstant linear fractional self-map of 𝔻. Cowen [6] proved that the adjoint operator C_{φ}^{∗} may be written T_{γ}C_{σ}T_{η}^{∗}, with

(3.1)  σ(z) = \frac{\overline{a}z − \overline{c}}{−\overline{b}z + \overline{d}},
γ(z) = \frac{1}{−\overline{b}z + \overline{d}},
η(z) = cz + d,

where T_{γ} and T_{η} denote the corresponding Toeplitz operators. Hence (C_{φ}^{∗}C_{φ}) (f) = (T_{γ}C_{σ}T_{η}^{∗}C_{φ}) (f) for any f in H^{2}. Recalling that T_{z}^{∗} is the backward shift on H^{2}, we see that

((C_{φ}^{∗}C_{φ}) f) (z) = γ(z) (\overline{c} (\frac{f(φ(σ(z))) − f(φ(0))}{σ(z)}) + \overline{d}f(φ(σ(z))))
(3.2)  = \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)))

for all z in 𝔻 not equal to σ^{−1}(0) = \frac{\overline{c}}{\overline{a}}. We rewrite this expression simply as

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

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}}.

Equation (3.3) holds for all points except z = σ^{−1}(0), which only belongs to 𝔻 if |c| < |a|. Having such a concrete representation for C_{φ}^{∗}C_{φ} makes it easier to investigate its spectrum.

6 C. HAMMOND

4. The quantities S_{φ} and S_{φ}^{∗}

Let φ be an analytic self-map of 𝔻. Bourdon and Retsek [4] defined the quantities

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

and

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

Among other results, they proved that S_{φ}^{∗} ≤ S_{φ} for all φ and that S_{φ}^{∗} = S_{φ} = ∥C_{φ}∥ whenever φ(0) = 0 or φ has the form φ(z) = az + b; moreover, when φ(0) ≠ 0 and φ(z) ≠ az + b, they showed that S_{φ}^{∗} cannot equal ∥C_{φ}∥ unless ∥C_{φ}∥_{e} = ∥C_{φ}∥. The quantities S_{φ} and S_{φ}^{∗} were also studied, with different notation, by Avramidou and Jafari [2]. In this section, 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.

We begin with a few observations which hold for any analytic φ : 𝔻 → 𝔻. If {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 H^{2}. Therefore, since C_{φ} is a bounded operator, either S_{φ} = ∥C_{φ}(k_{w})∥_{2} for a particular w in 𝔻 or S_{φ} = lim sup_{|w|↑1} ∥C_{φ}(k_{w})∥_{2}. The analogous result holds for S_{φ}^{∗}. Cima and Matheson [5] observed that

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

a fact which follows from the proof of Shapiro’s essential norm formula [15]. In the case where φ is univalent, Shapiro’s formula may be expressed

∥C_{φ}∥_{e} = lim sup_{|w|↑1} \sqrt{\frac{1 − |w|^{2}}{1 − |φ(w)|^{2}}} = lim sup_{|w|↑1} ∥C_{φ}^{∗}(k_{w})∥_{2} .

Therefore S_{φ} ≥ ∥C_{φ}∥_{e} for any φ and S_{φ}^{∗} ≥ ∥C_{φ}∥_{e} whenever φ is univalent.

Before proving our results for linear fractional φ, we need the following pair of general lemmas. The first, which pertains to the lower bound in expression (1.1), appears with a different proof in a current paper of David Pakorny and Jonathan Shapiro [13].

Lemma 4.1. If φ : 𝔻 → 𝔻 is a nonconstant analytic map with φ(0) ≠ 0, then

∥C_{φ}∥ > \sqrt{\frac{1}{1 − |φ(0)|^{2}}} .

Proof. The Hardy space H^{2} has the property that |f(0)| < ∥f∥_{2} for any nonconstant element f. Observe that the function k_{φ(0)} ∘ φ is nonconstant; therefore

∥C_{φ}∥ ≥ ∥C_{φ} (k_{φ(0)})∥_{2} > |(k_{φ(0)} ∘ φ) (0)| = \sqrt{\frac{1}{1 − |φ(0)|^{2}}} ,

as we had hoped to show. □

Lemma 4.2. Suppose that φ : 𝔻 → 𝔻 is a nonconstant analytic map with φ(0) ≠ 0. If the operator C_{φ} attains its norm on a normalized 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 equation (2.1), 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 4.1 that

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

meaning that |w| > |φ(0)|. □

Now we turn our attention to the situation where φ is a linear fractional map.

Proposition 4.3. 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}. Hence 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 4.2 tells us that w cannot equal 0 or φ(0). Suppose then that w is the problematic point σ^{−1}(0) = \frac{\overline{c}}{\overline{a}}. Applying L’Hôpital’s rule to expression (3.2), we obtain

(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

8 C. HAMMOND

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 (3.3) 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) = \frac{\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. □

We now state main result of this section:

Theorem 4.4. 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. Recall that ∥C_{φ}∥_{e} ≤ S_{φ}^{∗} ≤ S_{φ} ≤ ∥C_{φ}∥ for any univalent φ; if ∥C_{φ}∥_{e} = ∥C_{φ}∥, then all of these quantities are equal. On the other hand, suppose that ∥C_{φ}∥_{e} < ∥C_{φ}∥. Since ∥C_{φ}(k_{w})∥_{2} < ∥C_{φ}∥ for all w in 𝔻, it follows from our characterization of S_{φ} that S_{φ} < ∥C_{φ}∥. Since S_{φ}^{∗} ≤ S_{φ}, our result 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 [14] showed that the map φ(z) = \frac{4}{5−z} has this property.

Theorem 4.4 no longer holds if we eliminate the hypothesis that φ be linear fractional. In light of the aforementioned results of Bourdon and Retsek, we see that our assertion for S_{φ}^{∗} holds whenever φ is univalent (an observation also made by Retsek [14]). On the other hand, Bourdon and Retsek [4] proved that S_{φ}^{∗} < ∥C_{φ}∥_{e} = ∥C_{φ}∥ whenever φ is a non-univalent inner function with φ(0) ≠ 0. 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 (as shown by Nordgren [12]) the

ON THE NORM OF A COMPOSITION OPERATOR 9

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.

5. The spectrum of C_{φ}^{∗}C_{φ}

Let φ : 𝔻 → 𝔻 be a nonconstant linear fractional map, as discussed in Section 3; let σ be defined as in (3.1). Our goal now is to find a set of conditions under which we can determine ∥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.

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. These assumptions guarantee that τ_{j}(φ(0)) never equals σ^{−1}(0) and that τ_{j}(φ(0)) ≠ 0 for j ≠ n, as we can see from arguments involving fixed points. Furthermore, these conditions exclude all disk automorphisms and all maps of the form φ(z) = az + b.

Let W denote the set of points {τ_{j}(φ(0))}_{j=0}^{n}; recall that K_{W}^{⊥} is the subspace of H^{2} consisting of all functions that vanish on W. We claim that K_{W}^{⊥} is invariant under the operator C_{φ}^{∗}C_{φ}. Suppose that f belongs to K_{W}^{⊥}; it follows from equation (3.3) 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}^{⊥}.

Since C_{φ}^{∗}C_{φ} : K_{W}^{⊥} → K_{W}^{⊥} “looks like” a weighted composition operator, we can deduce a good deal of information about its spectrum. For example, if C_{φ} : H^{2} → H^{2} is compact, then the spectrum of C_{φ}^{∗}C_{φ} : K_{W}^{⊥} → K_{W}^{⊥} is precisely

10 C. HAMMOND

0 ∪ {ψ(w_{0})(τ′(w_{0}))^{j}}_{j=0}^{∞}, where w_{0} denotes the Denjoy-Wolff point of τ. This fact, however, does not help us to determine ∥C_{φ}∥ and is not proved here; details appear in the author’s thesis [9].

Now consider K_{W}, the span of the kernel functions {K_{τ_{j}(φ(0))}}_{j=0}^{n}; observe that it has dimension n + 1. The subspace K_{W} is also invariant under the self-adjoint operator C_{φ}^{∗}C_{φ} : H^{2} → H^{2}. 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 equation (2.1).

Proposition 5.1. 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 (3.3) dictates that

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

Now assume that our claim holds for the index j. Multiplying the consequent equation by λ and substituting expression (5.1) 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. □

ON THE NORM OF A COMPOSITION OPERATOR 11

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 5.2. 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 5.1 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 5.3. 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 2.4, 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 5.1. 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}.

12 C. HAMMOND

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,

(5.2)  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}

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 5.4. 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. Bourdon, Levi, Narayan, and Shapiro [3] proved in general that σ(ω) = ζ and σ′(ω) = (φ′(ζ))^{−1}; hence τ(ω) = ω and τ′(ω) = 1. Since the map τ_{n} ∘ φ fixes the origin and (τ_{n} ∘ φ)(ζ) = ω, it follows from Lemma 7.33 in [8], together with the Julia-Carathéodory theorem, that |(τ_{n} ∘ φ)′(ζ))| > 1. Therefore

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

Since φ is univalent on a neighborhood of the closed unit disk, Shapiro’s essential norm formula [15] yields

∥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 2.2 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 2.3 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 5.5. 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 (5.2), 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 4.4 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 normalized reproducing kernel functions of H^{2}.

6. 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 5.5, 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 5.1 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 [11], p. 595). 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}.

7. Examples

It is not difficult to find examples of linear fractional φ : 𝔻 → 𝔻 with τ(φ(0)) = 0. In terms of the coefficients of φ, this condition is equivalent to

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

In this case, considering the polynomial p in equation (5.2), we see that 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.

We now 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)}.

It is easy to show that φ is a self-map of 𝔻 and that ∂φ(𝔻) ∩ ∂𝔻 = {1}. Note that

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

A straightforward induction argument shows that each iterate τ_{j} has the form

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

ON THE NORM OF A COMPOSITION OPERATOR 15

Consequently

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},

from which we see that τ_{n}(φ(0)) = 0. 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 for 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.

References

[1] M. J. Appel, P. S. Bourdon and J. J. Thrall, Norms of composition operators on the Hardy space, Experiment. Math., 5 (1996), 111-117.

[2] P. Avramidou and F. Jafari, On norms of composition operators on Hardy spaces, Function Spaces (Edwardsville, 1998), 47-54, Contemp. Math., 232, Amer. Math. Soc., Providence, 1999.

[3] P. S. Bourdon, D. Levi, S. Narayan and J. H. Shapiro, Which linear-fractional composition operators are essentially normal?, preprint, 2002.

[4] P. S. Bourdon and D. Q. Retsek, Reproducing kernels and norms of composition operators, Acta Sci. Math. (Szeged), 67 (2001), 387-394.

[5] J. A. Cima and A. L. Matheson, Essential norms of composition operators and Aleksandrov measures, Pacific J. Math., 179 (1997), 59-64.

[6] C. C. Cowen, Linear fractional composition operators on H^{2}, Integral Equations Operator Theory, 11 (1988), 151-160.

[7] C. C. Cowen and T. L. Kriete, Subnormality and composition operators on H^{2}, J. Funct. Anal., 81 (1988), 298-319.

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

[9] C. Hammond, On the Norm of a Composition Operator, Thesis, University of Virginia, 2003.

[10] A. A. Kirillov and A. D. Gvishiani, Theorems and Problems in Functional Analysis, translated by H. H. McFaden, Springer-Verlag, New York, 1982.

[11] D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht/Boston/London, 1993.

[12] E. A. Nordgren, Composition operators, Canadian J. Math., 20 (1968), 442-449.

[13] D. B. Pakorny and J. E. Shapiro, Continuity of the norm of a composition operator, Integral Equations Operator Theory, to appear.

[14] D. Q. Retsek, The Kernel Supremum Property and Norms of Composition Operators, Thesis, Washington University, 2001.

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

C. Hammond, Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA

E-mail address: cnh5u@virginia.edu