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Signaler un problème

J. Math. Anal. Appl. 341 (2008) 626–639
www.elsevier.com/locate/jmaa
Adjoints of composition operators with rational symbol
a,∗ b cChristopher Hammond , Jennifer Moorhouse , Marian E. Robbins
a Department of Mathematics and Computer Science, Connecticut College, Box 5384, 270 Mohegan Avenue, New London, CT 06320, USA
b Department of Mathematics, Colgate University, 13 Oak Drive, Hamilton, NY 13346, USA
c Mathematics Department, California Polytechnic State University, San Luis Obispo, CA 93407, USA
Received 15 May 2007
Available online 28 October 2007
Submitted by J.H. Shapiro
Abstract
Building on techniques developed by Cowen and Gallardo-Gutiérrez, we find a concrete formula for the adjoint of a composition
2operator with rational symbol acting on the Hardy space H . We consider some specific examples, comparing our formula with
several results that were previously known.
© 2007 Elsevier Inc. All rights reserved.
Keywords: Composition operator; Adjoint; Hardy space
1. Preliminaries
2Let D denote the open unit disk in the complex plane. The Hardy space H is the Hilbert space consisting of all

nanalytic functions f(z) = a z on D such thatn


2f = |a | < ∞.2 n
n=0

n n 2If f(z) = a z and g(z) = b z belong to H , the inner product f,g can be written in several ways. Forn n
example,
2π∞ dθiθ iθf,g = a b = lim f re g re .n n
r↑1 2π
n=0 0
* Corresponding author.
E-mail addresses: cnham@conncoll.edu (C. Hammond), jmoorhouse@colgate.edu (J. Moorhouse), mrobbins@calpoly.edu (M.E. Robbins).
0022-247X/$ – see front matter © 2007 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmaa.2007.10.039C. Hammond et al. / J. Math. Anal. Appl. 341 (2008) 626–639 627
2Any function f in H can be extended to the boundary of D by means of radial limits; in particular, f(ζ) =
lim f(rζ) exists for almost all ζ in ∂D. (See Theorem 2.2 in [5].) Furthermore, we can writer↑1

dθ 1 dζiθ iθf,g = f e g e = f(ζ)g(ζ) .
2π 2πi ζ
0 ∂D
2 2 n ∞ 2It is often helpful to think of H as a subspace of L (∂D). Taking the basis {z } for L (∂D), we can identifyn=−∞
the Hardy space with the collection of functions whose Fourier coefficients vanish for n −1.
2One important property of H is that it is a reproducing kernel Hilbert space. In other words, for any point w in D
2 2there is some function K in H (known as a repr kernel function) such that f,K = f(w) for all f in H .w w
In the case of the Hardy space, it is easy to see that K (z) = 1/(1−¯ wz).w
At this point, we will introduce our principal object of study. Let ϕ be an analytic map that takes D into itself. The
2composition operator C on H is defined by the ruleϕ
C (f ) = f ◦ ϕ.ϕ
2It follows from Littlewood’s Subordination Theorem (see Theorem 2.22 in [4]) that every such operator takes H into
itself. These operators have received a good deal of attention in recent years. Both [4] and [10] provide an overview
of many of the results that are known.
2. Adjoints
One of the most fundamental questions relating to composition operators is how to obtain a reasonable representa-
∗tion for their adjoints. It is difficult to find a useful description for C , apart from the elementary identityϕ
∗C (K ) = K . (1)w ϕ(w)ϕ
(See Theorem 1.4 in [4].) In 1988, Cowen [2] used this fact to establish the first major result pertaining to the adjoints
of composition operators:
Theorem1 (Cowen). Let
az + b
ϕ(z) =
cz + d
∗ ∗be a nonconstant linear fractional map that takes D into itself. The adjoint C can be written T C T ,forg σϕ h
1 az¯ −¯ c
g(z) = ,σ(z) = , and h(z) = cz +d,
¯ ¯ ¯ ¯−bz + d −bz + d
where T and T denote the Toeplitz operators with symbols g and h, respectively.g h
While Cowen only stated this result for nonconstant ϕ, it is easy to see that the formula also holds for constant
maps, provided that ϕ is written in the form
b 0z + b
ϕ(z) = = ,
d 0z + d
so that σ(z) = 0. In that case, C and C can simply be considered point-evaluation functionals.ϕ σ
∗It is sometimes helpful to have a more concrete version of Cowen’s adjoint formula. Recalling that T is thez
2backward shift on H , we see that



1 f(σ(z)) −f(0)∗ ¯C f (z) = c¯ +df σ(z)ϕ ¯ ¯ σ(z)−bz + d


¯ 1 c¯ +dσ(z) cf¯ (0)
= f σ(z) −
¯ ¯ σ(z) σ(z)−bz + d
(ad − bc)z cf¯ (0)
= f σ(z) + . (2)
¯ ¯ c¯−¯ az(az¯ −¯ c)(−bz +d)
A similar calculation appears in [7].


628 C. Hammond et al. / J. Math. Anal. Appl. 341 (2008) 626–639
In recent years, numerous authors have made the observation that
2π iθ f(e ) dθ∗C f (w)= f,K ◦ ϕ = . (3)wϕ iθ 2π1 − ϕ(e )w
0
This fact seems particularly helpful when considering composition operators induced by rational maps. In an unpub-
∗lished manuscript, Bourdon [1] uses it to find a representation for C when ϕ belongs to a certain class of “quadraticϕ
fractional” maps. It is the principal tool used by Effinger-Dean, Johnson, Reed, and Shapiro [6] to calculate C ϕ
when ϕ is a rational map satisfying a particular finiteness condition. Equation (3) is also the starting point from which
both Martín and Vukotic´ [8] and Cowen and Gallardo-Gutiérrez [3] attempt to describe the adjoints of all compo-
sition operators with rational symbol. It is the content of this last paper that serves as the catalyst for our current
discussion.
The results of Cowen and Gallardo-Gutiérrez are stated in terms of multiple-valued weighted composition oper-
ators. Suppose that ψ and σ are a compatible pair of multiple-valued analytic maps on D (in a sense the authors
describe in their paper), with σ(D) ⊆ D. The operator W is defined by the ruleψ,σ

(W f )(z) = ψ(z)f σ(z) ,ψ,σ
the sum being taken over all branches of the pair ψ and σ . Whenever we encounter such an operator in this paper, the
function ψ will actually be defined in terms of σ .
Before considering their adjoint theorem, we need to remind the reader of a particular piece of notation. If f is a
˜(possibly multiple-valued) function acting on a subset U of the Riemann sphere, we define the function f on the set
{z ∈ C∪{∞}:1/z¯ ∈ U} by the rule


1
˜f(z) = f . (4)

Cowen and Gallardo-Gutiérrez state their adjoint formula in terms of this notation:
Theorem2 (Cowen and Gallardo-Gutiérrez). Let ϕ be a nonconstant rational map that takes D into itself. The adjoint
∗C can be written BW , where B denotes the backward shift operator and W is the multiple-valued weightedψ,σ ψ,σϕ
−1 −1 −1composition operator induced by σ = 1/ϕ and ψ = (ϕ ) /ϕ .
−1 −1 Note that the function (ϕ ) in the numerator of ψ represents the “tilde transform” of (ϕ ) , as defined in line (4),
−1rather than the derivative of ϕ . It is clear from the context of this theorem that the authors consider both B and Wψ,σ
2to be operators from H into itself.
As we shall see, Theorem 2 is not actually correct in all cases. We will begin by considering whether this result is
valid for linear fractional maps.
3. Linearfractionalexamples
If Theorem 2 were to hold in general, it would certainly have to agree with Theorem 1 in the case of linear fractional
maps. We shall show that these two theorems rarely yield the same result. Let
az + b
ϕ(z) =
cz + d
be a nonconstant map that takes D into itself. Note that
dz − b−1ϕ (z) = .
−cz + a
Using the notation of Theorem 2, we can write
¯ ¯−bz + d−1 −1ϕ (z) = ϕ (1/z)¯ = ,
az¯ −¯ c