E extracta mathematicae Vol. 31, Núm. 1, 25 – 36 (2016)

Adjoints of Generalized Composition Operators
with Linear Fractional Symbol

Aliakbar Salaryan, Hamid Vaezi

Department of Mathematics, Faculty of Mathematical Sciences,
University of Tabriz, P.O. Box: 51666, Tabriz, Iran, a_goshabulaghi@Tabrizu.ac.ir

Department of Mathematics, Faculty of Mathematical Sciences,
University of Tabriz, P.O. Box: 51666,Tabriz, Iran, hvaezi@tabrizu.ac.ir

Presented by Alfonso Montes Received March 30, 2015

Abstract: Given a positive integer n and φ : U → U, an analytic self-map of the open
unit disc in the complex plane, the generalized composition operator C (n)φ is defined by
C

(n)
φ f = f (n) ◦ φ for f belonging to some Hilbert space of analytic functions on U. In this

paper, we investigate some properties of generalized composition operators on the weighted
Hardy spaces. Then we obtain adjoints of generalized composition operators with linear
fractional symbol acting on the Hardy, Bergman and Dirichlet spaces.
Key words: Generalized composition operator, Adjoint, Weighted Hardy space.
AMS Subject Class. (2010): 47B33, 47B38, 47A05.

1. Introduction

Let U denote the open unit disc of the complex plane. For each sequence
β = {βn} of positive numbers, the weighted Hardy space H 2(β) consists of
analytic functions f (z) =

∑∞
n=0 anz

n on U for which the norm

∥f ∥β =
( ∞∑

n=0
|an|2β2n

)1
2

is finite. Notice that the above norm is induced by the following inner product⟨ ∞∑
n=0

anz
n,

∞∑
n=0

bnz
n

⟩
β

=
∞∑

n=0
anbnβ

2
n,

and that the monomials zn form a complete orthogonal system for H 2(β).
Consequently, the polynomials are dense in H 2(β). Observe that particular
instances of the sequence β = {βn} yield well known Hilbert spaces of analytic

25



26 a. salaryan, h. vaezi

functions. Indeed, βn = 1 corresponds to the Hardy space H 2(U). If β0 = 1,
βn = n1/2 for n ≥ 1, the norm obtained is the one in the Dirichlet space
D and if βn = (n + 1)−1/2, we get the Bergman space A2(U). The inner
product of two functions f and g in mentioned spaces may also be computed
by integration. For the Hardy space, H 2(U),

⟨f, g⟩H 2(U) =
∫ 2π

0
f (eiθ)g(eiθ)

dθ

2π
,

where f and g are defined a.e. on ∂U via radial limits (see [4]). In case of the
Bergman space,

⟨f, g⟩A2(U) =
∫
U

f (z)g(z)dA(z),

where dA is the normalized area measure on U and for the Dirichlet space,
the inner product is given by

⟨f, g⟩D = f (0)g(0) +
∫
U

f ′(z)g′(z)dA(z).

If u is analytic on the open unit disc U and φ is an analytic map of
the unit disc into itself, the weighted composition operator on H 2(β) with
symbols u and φ is the operator (Wu,φf )(z) = u(z)f (φ(z)) for f in H 2(β).
When u(z) ≡ 1 we call the operator a composition operator and denote it by
Cφ. The multiplication operator Mu corresponds to the case φ(z) = z and is
given by Muf (z) = u(z)f (z). For general information in this context one can
see the excellent monographs [3], [13] and [15].

In recent years the concept of composition and weighted composition op-
erator has been generalized in the literate. The generalized weighted com-
position operator Dnφ,u for nonnegative integer n, which introduced by Zhu
[16] (see also [14]), is defined by (Dnφ,uf )(z) = u(z)f (n)(φ(z)). We denote the
generalized composition operator by C(n)φ . Motivation to study this type of
operators apart from their own importance is that they appear in the adjoint
of composition operators on the space of analytic functions with derivative in
the Hardy space [12].

One of the most fundamental questions related to composition and
weighted composition operators and their generalizations is how to obtain
a reasonable representation for their adjoints. The problem of computing
the adjoint of a composition operator induced by linear fractional symbol on
the Hardy space was solved by Cowen [2]. Hurst [8] using an analogous argu-
ment obtained the solution in the weighted Bergman space A2α(U). Mentioned



adjoints of generalized composition operators 27

demonstrations was by composition of another composition operator and two
Toeplitz operators. In 2003, Gallardo and Montes [5] computed the adjoint of
a composition operator with linear fractional symbol acting on the Dirichlet
space by a different method from those used by Cowen and Hurst. Hammond,
Moorhouse and Robbins [7] solved the case for rationally induced composition
operators on the Hardy space, H 2(U). Bourdon and Shapiro [1] obtained the
Hammond-Moorhouse-Robbins formula in a straightforward algebraic fash-
ion. For more investigation we refer the interested reader to [10] and [11].
Goshabulaghi and Vaezi in [6] gave the adjoint formula for rationally induced
composition operators on the Dirichlet and Bergman spaces.

In this paper we investigate some properties of generalized composition
operators on the weighted Hardy spaces. Then we obtain the adjoint of linear
fractionally induced generalized composition operators acting on the Hardy,
Bergman and Dirichlet spaces.

2. Generalized composition operators on H 2(β)

Every weighted Hardy space H 2(β) contains a family of reproducing ker-
nels {Kw : w ∈ U}; that is, ⟨f, Kw⟩ = f (w) for every f ∈ H 2(β). This prop-
erty extends to higher derivatives of elements of H 2(β). Indeed, for any w ∈ U
there exists Kw,n ∈ H 2(β) such that for any f ∈ H 2(β), f (n)(w) = ⟨f, Kw,n⟩β .
We call Kw,n, the generalized reproducing kernel function.

Theorem 2.1. [3, Theorem 2.16] The generalized reproducing kernel
function Kw,n for the weighted Hardy space H 2(β) is given by

Kw,n(z) =
∞∑

m=n

m(m − 1) · · · (m − (n − 1))
β2m

w̄
m−n

zm.

Simple computations based on theorem 2.1 gives the following proposition.

Proposition 2.2. The generalized reproducing kernel function Kw,n at
the point z ∈ U on the Hardy, Bergman and Dirichlet spces is given by

n!zn

(1 − w̄z)n+1
,

(n + 1)!zn

(1 − w̄z)n+2
,

(n − 1)!zn

(1 − w̄z)n
,

respectively.

As a consequence of Littlewood’s Subordination Principle every composi-
tion operator on the Hardy space is bounded. By [9, Proposition 3.4] this fact



28 a. salaryan, h. vaezi

also holds for every composition operator on the Bergman space. The case
for generalized composition operators is rather different. As we will see, if
∥φ∥∞ < 1, then C

(n)
φ even belongs to the class of Hilbert-Schmidt operators

on the aforementioned spaces. In spite of this, there exist examples of φ in
which ∥φ∥∞ = 1 and C

(n)
φ is unbounded.

Proposition 2.3. Let H 2(β) satisfies limm→∞
βm−n

βm
̸= 0 and φ(z) = az

with |a| = 1. Then C(n)φ : H 2(β) → H 2(β) is unbounded.

Proof. For m ≥ n, define fm(z) = 1m(m−1)···(m−(n−1))βm z
m. Then ∥fm∥β =

1
m(m−1)···(m−(n−1)) and so {fm}

∞
m=n converges to zero on H 2(β). But

(C(n)φ fm)(z) = 1βm a
m−nzm−n and accordingly, ∥C(n)φ fm∥ =

βm−n
βm

and hence
C

(n)
φ fm does not converges to zero as we expect. Consequently, C

(n)
φ is un-

bounded on H 2(β).

By Proposition 2.3 for φ(z) = az with |a| = 1, C(n)φ is unbounded on the
Hardy, Bergman and Dirichlet spaces.

Theorem 2.4. Let φ be an analytic self map of U such that for any pos-
itive integer k ≥ n, ∥φk∥β ≤ ck for some constant 0 < c < 1. Then C

(n)
φ is a

Hilbert-Schmidt operator on H 2(β).

Proof. Let {em}∞m=0 be defined by em(z) =
1

βm
zm. Then {em}∞m=0 forms

an orthonormal basis for H 2(β) and for each m ≥ n,

e(n)m (z) =
m(m − 1) · · · (m − (n − 1))

βm
zm−n.

Accordingly,
∞∑

m=0
∥C(n)φ em∥

2
β =

∞∑
m=0

∥e(n)m ◦ φ∥
2
β

=
∞∑

m=n

m2(m − 1)2 · · · (m − (n − 1))2

β2m
∥φm−n∥2β

≤
∞∑

m=n

m2(m − 1)2 · · · (m − (n − 1))2

β2m
c2(m−n) = ∥Kc,n∥2β < ∞,

which leads to C(n)φ being Hilbert-Schmidt on H 2(β).



adjoints of generalized composition operators 29

In view of Proposition 2.3, for φ(z) = az with |a| = 1, C(n)φ is unbounded
on the Hardy and Bergman spaces. If φ is an analytic self map of U and
∥φ∥∞ < 1, the situation is very different than when ∥φ∥∞ = 1, as the following
corollary shows.

Corollary 2.5. Let φ be an analytic self map of U and ∥φ∥∞ < 1. Then
C

(n)
φ is Hilbert-Schmidt on the Hardy and Bergman spaces.

Theorem 2.6. Let φ be an analytic self map of U such that ∥φ∥∞ < 1 .
Then C(n)φ is Hilbert-Schmidt on the Dirichlet space D.

Proof. Let {em}∞m=0 be defined by em(z) =
1√
m

zm. Then {em}∞m=0 forms
an orthonormal basis for D and for each m ≥ n + 1,

e(n)m (z) =
m(m − 1) · · · (m − (n − 1))

√
m

zm−n.

It is clear that φm−n ∈ D. Moreover, for each m ≥ n + 1 we have

∥φ′φm−n−1∥2A2 ≤ ∥φ
′∥2A2 ∥φ∥

2(m−n−1)
∞ ≤ ∥φ∥

2
D∥φ∥

2(m−n−1)
∞ ,

and hence

∥φm−n∥2D = |φ(0)|
2(m−n) + (m − n)2∥φ′φm−n−1∥2A2

≤ |φ(0)|2(m−n) + (m − n)2∥φ∥2D∥φ∥
2(m−n−1)
∞ .

Accordingly, using the root test we see that
∞∑

m=n+1
∥C(n)φ em∥

2
D =

∞∑
m=n+1

∥e(n)m ◦ φ∥
2
D

=
∞∑

m=n+1

m2(m − 1)2 · · · (m − (n − 1))2

m
∥φm−n∥2D

≤
∞∑

m=n+1
m(m − 1)2 · · · (m − (n − 1))2|φ(0)|2(m−n)

+
∞∑

m=n+1
m(m − 1)2 · · · (m − n)2∥φ∥2D∥φ∥

2(m−n−1)
∞ < ∞,

which leads to C(n)φ being Hilbert-Schmidt on D.



30 a. salaryan, h. vaezi

3. Adjoints of generalized composition operator with
linear fractional symbol

Corollary 2.5 and Theorem 2.6 of Section 2 guaranties that a wide class
of generalized composition operators are bounded on the Hardy, Bergman
and Dirichlet spaces. Specially, there always exist bounded linear fractionally
induced generalized composition operators on the mentioned spaces and there-
fore computing the adjoint of generalized composition operators with linear
fractional symbol makes sense. Now let φ(z) = az+b

cz+d be a linear fractional self
map of U. It was shown by Cowen [3] that corresponding to φ, σ(z) = āz−c̄−b̄z+d̄
is a self map of U. In the sequel, for n ≥ 1, we compute the adjoint of lin-
ear fractionally induced generalized composition operator C(n)φ on the Hardy,
Bergman and Dirichlet spaces.

Theorem 3.1. Let C(n)φ be a bounded generalized composition operator
on H 2(U) with linear fractional symbol. Then for each f ∈ H 2(U) and z ∈ U,

C(n)
∗

φ f (z) = f (0)g0(z) + D
n
σ,uMvf (z),

where u(z) = z
n

(−b̄z+d̄)n+1 , v(z) =
(d̄z+c̄)n+1

z
and

g0(z) = −
zn

(−b̄z + d̄)n+1

(
(d̄z + c̄)n+1

z

)(n)
(σ(z)) +

n!d̄
n+1

zn

(−b̄z + d̄)n+1
.

Proof. A simple computation shows that

1 − φ(w)z = (−b̄z + d̄)
(

1 − σ(z)w̄
c̄w̄ + d̄

)
. (3.1)

Moreover for every g ∈ H 2(U), Proposition 2.2 implies that

g(n)(w) = ⟨g, Kw,n⟩ =
∫ 2π

0

n!g(eiθ)e−inθ

(1 − we−iθ)n+1
dθ

2π
.

Hence for f ∈ H 2(U) with f (0) = 0 and z ̸= 0 we have



adjoints of generalized composition operators 31

C(n)
∗

φ f (z) = ⟨C
(n)∗
φ f, Kz ⟩ = ⟨f, C

(n)
φ Kz ⟩ = ⟨f, K

(n)
z ◦ φ⟩

= ⟨f, n!z̄
n
(Kz ◦ φ)n+1⟩

= n!zn
∫ 2π

0

f (eiθ)
(1 − φ(eiθ)z)n+1

dθ

2π

= n!zn
∫ 2π

0

f (eiθ)

(−b̄z + d̄)n+1
(

1−σ(z)e−iθ
c̄e−iθ +d̄

)n+1 dθ2π
=

n!zn

(−b̄z + d̄)n+1

∫ 2π
0

(c̄e−iθ + d̄)n+1f (eiθ)
(1 − σ(z)e−iθ)n+1

dθ

2π

=
zn

(−b̄z + d̄)n+1

∫ 2π
0

n! (d̄e
iθ +c̄)n+1

eiθ
f (eiθ)e−inθ

(1 − σ(z)e−iθ)n+1
dθ

2π

=
zn

(−b̄z + d̄)n+1
⟨h, Kσ(z),n⟩ =

zn

(−b̄z + d̄)n+1
h(n)(σ(z)),

(3.2)

where h(z) = (d̄z+c̄)
n+1

z
f (z) for z ̸= 0 and h(0) = c̄n+1f ′(0). Notice that

letting g1(z) = (d̄z + c̄)n+1 we have g1 ∈ H ∞(U) and hence h ∈ H 2(U). Now
let f ∈ H 2(U) be arbitrary. Then for z ̸= 0,

(C(n)
∗

φ f (0))(z) = ⟨C
(n)∗
φ f (0), Kz ⟩ = ⟨f (0), C

(n)
φ Kz ⟩ = ⟨f (0), K

(n)
z ◦ φ⟩

= ⟨f (0), n!z̄
n
(Kz ◦ φ)n+1⟩ = n!f (0)zn(Kz ◦ φ)(0)

n+1

= n!f (0)zn(Kφ(0)(z))
n+1 =

n!f (0)zn

(1 − φ(0)z)n+1

=
n!d̄

n+1
f (0)zn

(−b̄z + d̄)n+1
.

(3.3)

On the other hand, by (3.2)

(C(n)
∗

φ (f − f (0)))(z) =
zn

(−b̄z + d̄)n+1
h

(n)
0 (σ(z)), (3.4)

where h0(z) =
(d̄z+c̄)n+1

z
(f − f (0)))(z). Combining equalities (3.3) and (3.4)



32 a. salaryan, h. vaezi

results

C(n)
∗

φ f (z) = f (0)


 n!d̄n+1 zn

(−b̄z + d̄)n+1
−

zn

(−b̄z + d̄)n+1

(
(d̄z + c̄)n+1

z

)(n)
(σ(z))




+
zn

(−b̄z + d̄)n+1

(
(d̄z + c̄)n+1

z
f (z)

)(n)
(σ(z))

= f (0)g0(z) + Dnσ,uMvf (z).

Now, by analyticity the statement of theorem holds for arbitrary z ∈ U.

Theorem 3.2. Let C(n)φ and Cn+1σ be bounded generalized composition
operators on A2(U) with linear fractional symbol. Then for each f ∈ A2(U)
and z ∈ U,

C(n)
∗

φ f (z) = f (0)g0(z) + D
n+1
σ,u MvQf (z),

where

g0(z) =
zn

(−b̄z + d̄)n+2


(n + 1)!d̄n+2 −

(
(d̄z + c̄)n+2

z

)(n+1)
(σ(z))


 ,

u(z) = z
n

(−b̄z+d̄)n+2 , v(z) =
(d̄z+c̄)n+2

z2
and Qf = F is the antiderivative of f

with F (0) = 0.

Proof. For arbitrary g ∈ A2(U), Proposition 2.2 implies that

g(n)(w) = ⟨g, Kw,n⟩ =
∫
U

(n + 1)!g(t)t̄
n

(1 − t̄w)n+2
dA(t).

Furthermore Kz (t) = 1(1−z̄t)2 and hence K
(n)
z (t) = (n+1)!z̄

n

(1−z̄t)n+2 . Therefore

(K(n)z ◦ φ)(t) =
(n + 1)!z̄

n

(1 − z̄φ(t))n+2
.

Let g(w) = (cw+d)
n+2

(1−σ(z)w)n+2
. Then g ∈ H 2(U) and by Lemma 2 of [10] for any f

contained in the Dirichlet space, and hence for any polynomial f ,

⟨f ′, g⟩A2 = ⟨f, ιg⟩H 2 , (3.5)



adjoints of generalized composition operators 33

where ι is the identity map ι(z) = z. Now, using (3.1) and (3.5), for any
polynomial f with f (0) = 0 we have

C(n)
∗

φ f (z) = ⟨C
(n)∗
φ f, Kz ⟩ = ⟨f, C

(n)
φ Kz ⟩ = ⟨f, K

(n)
z ◦ φ⟩

= ⟨f, (n + 1)!z̄
n
(Kz ◦ φ)n+2⟩ =

∫
U

(n + 1)!f (w)zn

(1 − φ(w)z)n+2
dA(w)

= (n + 1)!zn
∫
U

f (w)

(−b̄z + d̄)n+2
(

1−σ(z)w̄
c̄w̄+d̄

)n+2 dA(w)
=

(n + 1)!zn

(−b̄z + d̄)n+2

∫
U

(c̄w̄ + d̄)n+2f (w)
(1 − σ(z)w̄)n+2

dA(w)

=
(n + 1)!zn

(−b̄z + d̄)n+2
⟨f, g⟩A2 =

(n + 1)!zn

(−b̄z + d̄)n+2
⟨F, ιg⟩H 2

=
(n + 1)!zn

(−b̄z + d̄)n+2

∫ 2π
0

F (eiθ)e−iθ(c̄e−iθ + d̄)n+2

(1 − σ(z)e−iθ)n+2
dθ

2π

=
(n + 1)!zn

(−b̄z + d̄)n+2

∫ 2π
0

F (eiθ) (d̄e
iθ +c̄)n+2
e2iθ

e−i(n+1)θ

(1 − σ(z)e−iθ)n+2
dθ

2π

=
zn

(−b̄z + d̄)n+2

(
(d̄z + c̄)n+2F (z)

z2

)(n+1)
(σ(z)).

(3.6)

The last equality follows from generalized reproducing kernel property of
H 2(U). Let A20(U) = {f ∈ A

2(U) : f (0) = 0} and define the operator
T0 : A20(U) → A

2
0(U) by T f (z) =

F (z)
z2

for z ̸= 0 and T f (0) = 12 f
′(0). Then T0

is bounded on A20(U). Therefor, by continuity (3.6) holds for any f ∈ A
2(U)

with f (0) = 0. Moreover for arbitrary f ∈ A2(U),

(C(n)
∗

φ f (0))(z) = ⟨f (0), K
(n)
z ◦ φ⟩ = f (0)(K

(n)
z ◦ φ)(0) =

f (0)(n + 1)!zn

(1 − φ(0)z)n+2

=
f (0)(n + 1)!d̄

n+2
zn

(−b̄z + d̄)n+2
. (3.7)

It is clear that the antiderivative of f − f (0) at z is F (z) − f (0)z, hence by



34 a. salaryan, h. vaezi

(3.6),

(C(n)
∗

φ (f − f (0)))(z)

=
zn

(−b̄z + d̄)n+2

(
(d̄z + c̄)n+2

z2
(F (z) − f (0)z)

)(n+1)
(σ(z)).

(3.8)

Combine (3.7) and (3.8) and obtain

C(n)
∗

φ f (z) =
f (0)zn

(−b̄z + d̄)n+2

(
(n + 1)!d̄

n+2
− (

(d̄z + c̄)n+2

z
)(n+1)(σ(z))

)

+
zn

(−b̄z + d̄)n+2

(
(d̄z + c̄)n+2

z2
F (z)

)(n+1)
(σ(z))

= f (0)g0(z) + Dn+1σ,u MvQf (z).

Now, by analyticity the statement of theorem holds for arbitrary z ∈ U.

Theorem 3.3. Let C(n)φ be a bounded generalized composition operator
on D with linear fractional symbol. Then for f ∈ D and z ∈ U,

C(n)
∗

φ f (z) = f (0)g0(z) + D
n
σ,uMvf (z),

where g0(z) =
(n−1)!d̄

n

zn

(−b̄z+d̄)n , u(z) =
(ād̄−b̄c̄)zn+1
(−b̄z+d̄)n+1 and v(z) = (d̄z + c̄)

n−1.

Proof. On the Dirichlet space D, Kz (t) = 1 + ln 11−z̄t and hence K
(n)
z (t) =

(n−1)!z̄
n

(1−z̄t)n . Therefore using (3.1) we have

(K(n)z ◦ φ)(w) =
(n − 1)!z̄

n

(1 − z̄φ(w))n
=

(n − 1)!z̄
n

(−bz̄ + d)n

(
cw + d

1 − σ(z)w

)n
,

which implies

(K(n)z ◦ φ)
′(w) =

n!(ad − bc)z̄
n+1

(−bz̄ + d)n+1
.

(cw + d)n−1

(1 − σ(z)w)n+1
. (3.9)



adjoints of generalized composition operators 35

Now, abusing of notation, generalized reproducing kernel property of H 2(U)
implies that

⟨f (w),
w(cw + d)n−1

(1 − σ(z)w)n+1
⟩H 2 =

∫ 2π
0

f (eiθ)e−iθ(c̄e−iθ + d̄)n−1

(1 − σ(z)e−iθ)n+1
dθ

2π

=
∫ 2π

0

f (eiθ)(d̄eiθ + c̄)n−1e−inθ

(1 − σ(z)e−iθ)n+1
dθ

2π

=

(
(d̄z + c̄)n−1f (z)

)(n)
(σ(z))

n!
,

since (K(n)z ◦ φ)′ ∈ H 2(U), using (3.9) we see that

C(n)
∗

φ f (z) = ⟨C
(n)∗
φ f, Kz ⟩D = ⟨f, C

(n)
φ Kz ⟩D = ⟨f, K

(n)
z ◦ φ⟩D

= f (0)K(n)z (φ(0)) + ⟨f ′, (K(n)z ◦ φ)
′⟩A2

= f (0)K(n)z (φ(0)) + ⟨f (w), w(K(n)z ◦ φ)
′(w)⟩H 2

= f (0)
(n − 1)!d̄

n

zn

(−b̄z + d̄)n
+

n!(ād̄ − b̄c̄)zn+1

(−b̄z + d̄)n+1
⟨f (w),

w(cw + d)n−1

(1 − σ(z)w)n+1
⟩H 2

= f (0)
(n − 1)!d̄

n

zn

(−b̄z + d̄)n

+
n!(ād̄ − b̄c̄)zn+1

(−b̄z + d̄)n+1
.

(
(d̄z + c̄)n−1f (z)

)(n)
(σ(z))

n!

= f (0)
(n − 1)!d̄

n

zn

(−b̄z + d̄)n
+

(ād̄ − b̄c̄)zn+1

(−b̄z + d̄)n+1
(
(d̄z + c̄)n−1f (z)

)(n)
(σ(z))

= f (0)g0(z) + Dnσ,uMvf (z).

Finally, by analyticity the statement of theorem holds for arbitrary z ∈ U.

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36 a. salaryan, h. vaezi

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