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

2π

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)

z¯

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