Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
19 (2003), 55–60
www.emis.de/journals
ADJOINTS OF COMPOSITION OPERATORS ON STANDARD
BERGMAN AND DIRICHLET SPACES ON THE UNIT DISK
CARLOS C. PEÑA
Abstract. It is not known a satisfactory way to compute adjoints of composition operators, yet in classical functional Banach spaces (cf. [3]). If K is the
reproducing kernel of a functional Hilbert space H, g ∈ H and the composition
operator Cϕ is bounded then
∗
Cϕ
g (z) = hg(t), K (z, ϕ (t))iH , z ∈ D.
In general, although reproducing kernels might be described in series develop∗ in a closed form. In this article we
ments, it is not possible to determine Cϕ
establish some formulae in order to evaluate adjoints of composition operators
on standard Bergman and Dirichlet spaces.
1. Introduction
Throughout this article we shall consider the standard Bergman and Dirichlet
(Hilbert) spaces on the complex unit disk D defined by
n
o
R
2
2
A2α (D) = f analytic on D : kf kA2 (D) = D |f (z)| dAα (z) < ∞
α
and
R df (z) 2
2
2
Dα (D) = f analytic on D : kf kDα (D) = |f (0)| + D dz dAα (z) < ∞
α
2
respectively, where α > −1, dAα (z) = (1 + α) 1 − |z|
dA(z)/π and dA(z) is
Lebesgue area measure. A composition operator on a Bergman space is bounded
if its symbol is analytic. Indeed, if ϕ is an analytic authomorphism of the disk we
can write ϕ (z) = λ (z + a) / (1 + a z), where |a| < 1, |λ| = 1 and z ∈ D (cf. [5],
Ch. 2, §6, Th. 6.1, page 63). Thus, if f ∈ A2α (D) by the usual change of variables
formula we obtain
Z
2
2
kCϕ f kA2 (D) =
|f (ϕ (z))| dAα (z)
D
Z
2 α
2
=
|f (w)| (1 + α) 1 − ϕ−1 (w)
dA ϕ−1 (w) /π
D
2 !α 2
Z
dϕ−1
1 − ϕ−1 (w)
2
=
|f (w)|
(w) dAα (w)
2
dw
1 − |w|
D
2
2+α Z
|f (w)|
2
= 1 − |a|
2α+4 dAα (w)
D 1 − w aλ
2000 Mathematics Subject Classification. 47B38.
Key words and phrases. Functional Banach space.
kernels.
55
Composition operator.
Reproducing
56
CARLOS C. PEÑA
≤
1 + |a|
1 − |a|
2+α
2
kf kA2α (D) ,
i.e. Cϕ is bounded and kCϕ k ≤ (1 + |ϕ (0)|) / (1 − |ϕ (0)|). Now, if ϕ1 is an analytic
self map of D and ϕ2 = ϕ ◦ ϕ1 then Cϕ2 = Cϕ1 ◦ Cϕ and
Cϕ1 = Cϕ2 ◦ C−1
ϕ = Cϕ2 ◦ Cϕ−1 .
Therefore, Cϕ2 is bounded if and only if Cϕ1 is. Choosing ϕ with ϕ (ϕ1 (0)) = 0
gives ϕ2 (0) = 0 and the claim follows by [2], Th. 3.1, p. 117. In the frame of
Dirichlet spaces boundedness obeys to more restrictive conditions. Indeed, if ϕ is
an analytic self map of D then Cϕ ∈ B [Dα ] if and only if for all ζ ∈ C and all
0 < h < 1 is µα ϕ−1 S (ζ, h) = O hα+2 , where S (ζ, h) is the unit disk intersected
with the disk of radius h centered at ζ and µα is the measure on D defined as
2
dµα = |dϕ/dz| dAα (z) (cf. [4]). Our goal in this article is to develop some formulae
to evaluate adjoints of composition operators on Bergman and Dirichlet spaces. In
general, with the exception of a few particular cases in the context of Hardy spaces,
there is not known a satisfactory way to realize such computations (see [3] and [2],
Ch. 9, §9.1, p. 321). This problem will be considered in Bergman and Dirichlet
contexts in Sections 2 and 3 respectively.
2. Adjoints on Bergman spaces
1/2
Let α > −1, Ψn (z) = cn z n , where cn = n+1+α
, n ∈ N0 . Then {Ψn }n∈N0
nP
∞
is an orthonormal basis of A2α (D). Let f (z) = n=0 an z n be the Taylor series
on D of a function f ∈ A2α (D), m ∈ N0 and 0 < % < 1. Since the series converges
uniformly on compact subsets of D and D has dAα finite measure is
Z
Z
∞
X
f (z) z m dAα (z) =
an
z n z m dAα (z) = am %2(m+1+α) c−2
(1)
m .
%D
n=0
%D
m
Since |f (z) z | ≤ |f (z)| and by Hölder´s inequality f ∈ L1 (D, dAα ), on letting
% → 1− in (1) we obtain
(2)
am = f (m) (0)/m! = cm hf, Ψm iA2 (D) .
α
Theorem 1. Let ϕ be an analytic self map of the complex unit disk D and Cϕ
be the composition operator
Cϕ g = g ◦ ϕ defined on the standard Bergman space
A2α (D). Then Cϕ ∈ B A2α (D) and
Z
g (t) dAα (t)
∗
2
Cϕ g(z) =
α+2 , g ∈ Aα (D) , z ∈ D.
D 1 − z ϕ (t)
Proof. If ϕ is constant, say ϕ (z) ≡ z0 , then Cϕ is the evaluation functional at z0 .
If f, g ∈ A2α (D), the reproducing formula
Z
f (z)
(3)
f (z0 ) =
2+α dAα (z)
D (1 − z0 z)
holds (cf. [2], p. 27, Ex. 2.1.5 (a)) and
Z
hCϕ f, giA2 (D) = f (z0 ) g(w) dAα (w)
α
D
Z
Z
f (z)
dA
(z)
g(w) dAα (w)
=
α
2+α
D
D (1 − z0 z)
Z
Z
−2−α
g(w) dAα (w) dAα (z) .
=
f (z) (1 − z0 z)
D
D
ADJOINTS OF COMPOSITION OPERATORS. . .
57
−2−α R
Since f is arbitrary we obtain Cϕ∗ g (z) = (1 − z0 z)
g(w) dAα (w) and our
D
claim follows. Let ϕ be non constant, % ∈ (0, 1) , m, n ∈ N0 . Then
Z
∞
X
1
dt
n
n
(4)
ϕ (z) =
an,k z k , with an,k =
ϕ (t)
, z ∈ D,
2πi |t|=%
tk+1
k=0
the series in (4) being uniformly convergent on compact subsets of D. By (2) and (4)
we write
*∞
+
X
hCϕ f, z m iA2 (D) =
hf, Ψn i Cϕ Ψn , z m
α
n=0
(5)
=
∞
X
A2α (D)
n
cn hf, Ψn iA2 (D) hϕ (z) , z m iA2 (D)
α
α
n=0
=
c−2
m
∞
X
n=0
c2n
Z
1
f (z) z dAα (z)
2πi
D
n
Z
n
ϕ (t)
|t|=%
dt
tm+1
.
Now, for each non negative integer n we have
Z
n
(ϕn )(m) (0) 1 Z
2π
dt
Mϕ (%)
n dt n
ϕ (t) m+1 ≤
|ϕ (% exp (it))|
≤
,
=
2πi |t|=%
m!
t
2π%m
%m
0
where Mϕ (%) = max {|ϕ (z)| : |z| ≤ %}. If ϕ is non constant by the maximum
modulus principle Mϕ (%) < 1. So, for p ∈ N and z ∈ D we have
p p (m)
X
n + 1 + α (ϕn )
(0) n |f (z)| X n + 1 + α
n
z ≤ m
Mϕ (%)
f (z)
m!
%
n
n
n=0
n=0
≤%
−m
−2−α
(1 − Mϕ (%))
|f (z)| .
1
Since f ∈ L (D, dAα ) we apply Lebesgue dominated convergence theorem in (5)
obtaining
"∞
#
Z
Z
X
dt
n
m
−2
2 n 1
(6) hCϕ f, z iA2 (D) = cm
f (z)
cn z
ϕ (t)
dAα (z)
α
2πi |t|=%
tm+1
D
n=0
Moreover, by an analogous argument applied in (6) we have
hCϕ f, z m iA2 (D)
α
"
#
Z
Z
∞ X
1
1
n+1+α
dt
n
= 2
f (z)
(zϕ (t))
dAα (z)
cm D
2πi |t|=% n=0
n
tm+1
"
#
Z
Z
1
1
dt
−2−α
= 2
f (z)
(1 − z ϕ (t) )
dAα (z)
cm D
2πi |t|=%
tm+1
Z
1 ∂m
−2−α
(1 − z ϕ (t) )
|t=0 dAα (z)
=
f (z)
2 ∂tm
m!c
D
m
−2−α
1 ∂m = f (z),
1
−
z
ϕ
(t)
|
.
t=0
m!c2m ∂tm
A2 (D)
α
Since f is arbitrary we have
−2−α
1
∂m 1
−
z
ϕ
(t)
|t=0 .
m! cm ∂tm
On the other hand, if p ∈ N and z, t ∈ D we have
p X
n+1+α
m
n
−2−α
(z ϕ (t)) ≤ (1 − |z|)
t
n
(7)
Cϕ∗ Ψm (z) =
n=0
58
CARLOS C. PEÑA
and by Lebesgue dominated convergence theorem, (2) and (4)
Z
∞ X
Ψm (t) dAα (t)
n+1+α
n
=
z n hϕ (t) , Ψm (t)iA2α (D)
α+2
n
D 1 − z ϕ (t)
n=0
∞ X
n+1+α
= c−1
z n an,m
m
n
n=0
∞ (m)
X
n+1+α
(ϕn )
(0)
−1
= cm
zn
.
n
m!
n=0
As before, if 0 < % < 1 a new application of Lebesgue dominated convergence
theorem and (7) give
Z
Z
∞ X
n
Ψm (t) dAα (t)
n+1+α
1
dt
−1
z n ϕ (t) m+1
α+2 = cm
n
2πi
t
D 1 − z ϕ (t)
|t|=%
n=0
(8)
Z
−2−α dt
1
=
1 − z ϕ (t)
= Cϕ∗ Ψm (z) .
tm+1
2πicm |t|=%
P∞
Finally, we observe that Cϕ∗ g = m=0 hg, Ψm iA2 (D) Cϕ∗ Ψm for all g ∈ A2α (D) and
α
evaluations are bounded linear functionals on A2α (D).1 Thus if z ∈ D by (7) and
(8) is
∞
−2−α
X
1
∂m Cϕ∗ g (z) =
hg, Ψm iA2 (D)
1 − z ϕ (t)
|t=0
m
α
m! cm ∂t
m=0
∞ Z
−2−α
X
1 ∂m =
g (s) sm
1
−
z
ϕ
(t)
|t=0 dAα (s)
m! ∂tm
m=0 D
and the result follows by a last application of Lebesgue dominated convergence
theorem.
Corollary 1. If a ∈ D and ϕ is an analytic self map on D then
h
i
−α−2
−α−2
Cϕ∗ (1 − t a)
(z) = 1 − z ϕ (a)
, z ∈ D.
Remark 1. Observe that if ϕ is an analytic self map of the disk, α > −1 and we
define
Z
g (t) dAα (t)
2
(Hϕ g) (z) =
α+2 , g ∈ Aα (D) , z ∈ D,
D 1 − z ϕ (t)
then Hϕ ∈ B A2α (D) and hCϕ f, giA2 (D) = hf, Hϕ giA2 (D) . Of course, this is conα
α
sequence of Th. 1 and the existence and uniqueness of adjoints. None of the two
assertions are trivial and a direct and perhaps natural point of view is not appropriate nor conducent. For instance, if ϕ is an automorphism on D the usual change
of variable formula gives
2 !α Z
2
1 − ϕ−1 (z)
(1) −1
−1
ϕ
(z)
hCϕ f, giA2 (D) =
f (z) g (ϕ (z))
ϕ
dAα (z) ,
2
α
1 − |z|
D
but it is immediate that in general the function
−1
!α ϕ (z)2
2
1
−
(1) −1
z → g ϕ−1 (z)
ϕ (z) ϕ
2
1 − |z|
1If z ∈ D, f ∈ A2 (D) and δ (z − z ) is the linear evaluation at z by (3) is |hf (z), δ (z − z )i| ≤
0
0
0
0
α
kf kA2 (D) (1 − |z0 |)−2−α , i.e. δ (z − z0 ) is bounded.
α
ADJOINTS OF COMPOSITION OPERATORS. . .
59
does not belong to A2α (D).
Example 1. If a, b ∈ C are such that 0 < |a| ≤ |a| + |b| ≤ 1 the function
ϕ (z) = az + b is an analytic self map of D and
az
1
2
Cϕ∗ g(z) = g
α+2 , g ∈ Aα (D) , z ∈ D.
1−b z
1−b z
3. Adjoints on Dirichlet spaces
Results of Section 3 are outlined, their proofs follow similarly as in Section 2.
Let Λ0 (z) = 1 and Λn (z) = cn−1 z n /n, n ∈ N. Now {Λn }n∈N0 is an orthonormal
P∞
basis of Dα (D). Let f (z) = n=0 an z n be the Taylor series on D of a function
f ∈ Dα (D) and m ∈ N0 .
(
hf, Λ0 iDα (D)
if m = 0,
f (m) (0)
=
(9)
am =
m!
cm−1 /m hf, Λm iDα (D) if m > 0.
Moreover, let us consider the function KDα on D × D defined as
Z w
−α−1
(1 − t z)
−1
KDα (z, w) = 1 +
dt.
(α + 1) t
0
So, for z ∈ D fixed the function w → KDα (z, w) (= KDα (w, z) ) belongs to Dα ,
and f (z) = hf (w) , KDα (z, w)iDα if f ∈ Dα (cf. [1]). Let ϕ be an analytic self map
of D such that Cϕ ∈ B [Dα (D)]. Then
hCϕ f, Λ0 i = f (ϕ (0)) = hf (z) , KDα (ϕ (0) , z)iDα , f ∈ Dα ,
i.e.
Cϕ∗ Λ0
(z) = KDα (ϕ (0) , z) if z ∈ D. Moreover, for m ∈ N is
Cϕ∗ Λm (z) =
Therefore, if f (z) =
Cϕ∗ f
=
P∞
m=0
∂m
1
[KDα (ϕ (t) , z)] |t=0 .
(m − 1)! cm−1 ∂tm
hf, Λm i Λm (z) in Dα we write
(z)
∞
X
hf, Λm i Cϕ∗ Λm (z)
m=0
∞ Z
X
df sm−1
∂m
dAα (s)
[KDα (ϕ (t) , z)] |t=0
ds (m − 1)!
∂tm
m=0 D
Z
df ∂
= f (0) KDα (z, ϕ (0)) +
KDα (z, ϕ (s)) dAα (s) .
D ds ∂s
= f (0) KDα (ϕ (0) , z) +
Theorem 2. Let ϕ be an analytic self map of the complex unit disk D such that
Cϕ ∈ B [Dα (D)]. Then
*
+
Z ϕ(t)
−1−α
(1 − z s)
−1
∗
Cϕ g (z) = g(t), 1 +
ds
, g ∈ Dα , z ∈ D.
(1 + α) s
0
Dα
References
[1] A.L. Barrenechea and C.C. Peña. On Hadamard – Dirichlet algebras. Acta Math. Universitatis
Comenianae., LXXI:9–17, 2002.
[2] C. Cowen and B. MacCluer. Composition operators on spaces of analytic functions. Studies
in Advances Math. CRC Press, Boca Raton - N.Y. - London - Tokyo, 1995.
[3] C. Cowen and B. MacCluer. Some problems on composition operators. Contemporary Mathematics, volume 213, pages 15–25. 1998.
[4] B. McCluer and J. Shapiro. Angular derivatives and compact composition operators on the
hardy and berman spaces. Canadian J. Math., 38:878–906, 1996. MR 87h#47048.
60
CARLOS C. PEÑA
[5] W. Veech. A second course in complex analysis. W. A. Benjamin, Inc., N.Y., Amsterdam,
1967.
Received June 05, 2002.
Departamento de Matemáticas,
Universidad Nacional del Centro de la Provincia de Buenos Aires
NuCoMPA, Argentina
E-mail address: ccpenia@exa.unicen.edu.ar