CUBO A Mathematical Journal
Vol.15, No¯ 02, (43–52). June 2013

Isometric weighted composition operators on weighted
Banach spaces of holomorphic functions defined on the unit

ball of a complex Banach space

Elke Wolf

University of Paderborn,

Mathematical Institute,

D-33095 Paderborn, Germany.

lichte@math.uni-paderborn.de

ABSTRACT

Let X and Y be complex Banach spaces and BX resp. BY the closed unit ball. Analytic

maps φ : BY → BX and ψ : BX → C induce the weighted composition operator:

Cφ,ψ : H(BY) → H(BX), f 7→ ψ(f ◦ φ),

where H(BY) resp. H(BX) denotes the collection of all analytic functions f : BX(resp.BY) →
C. We study when such operators acting between weighted spaces of analytic functions

are isometric.

RESUMEN

Sea X y Y espacios de Banach complejos, BX y BY las bolas unitarias cerradas cor-

respondientes. Las aplicaciones anaĺıticas φ : BY → BX y ψ : BX → C inducen el
operador de composición con pesos:

Cφ,ψ : H(BY) → H(BX), f 7→ ψ(f ◦ φ),

donde H(BY) y H(BX) denotan la colección de todas las funciones anaĺıticas f : BX(resp.BY) →
C. Estudiamos cuándo dichos operadores que actúan entre los espacios con peso de fun-

ciones anaĺıticas son isométricas.

Keywords and Phrases: weighted composition operators, weighted spaces of holomorphic func-

tions on the unit ball of a complex Banach space.

2010 AMS Mathematics Subject Classification: 47B38, 47B33.



44 Elke Wolf CUBO
15, 2 (2013)

1 Introduction

Let D denote the open unit disk in the complex plane and H(D) the collection of all analytic func-

tions on D. Then, an analytic self-map φ of D induces through composition a linear composition

operator

Cφ : H(D) → H(D), f 7→ f ◦ φ.

Since such operators appear naturally in a variety of problems and since they link - in the classical

setting of the Hardy space H2 (see [10] and [24]) - operator theoretical questions with classical

results in complex analysis their study has a long and rich history. Now, let ψ ∈ H(D). The next

step is to combine the composition operator Cφ with a multiplication operator Mψ : H(D) →
H(D), f 7→ ψf to obtain the so-called weighted composition operator

Cφ,ψ := MψCφ : H(D) → H(D), f 7→ ψ(f ◦ φ).

For a bounded and continuous function (weight) v : D → (0,∞) we consider

H∞v := {f ∈ H(D); ‖f‖v := sup
z∈D

v(z)|f(z)| < ∞}.

Endowed with norm ‖.‖v, these spaces are Banach spaces and in the sequel we refer to them as

weighted Banach spaces of holomorphic functions. Such spaces arise in functional analysis, partial

differential equations and convolution equations as well as in distribution theory. They have been

studied intensively in several articles, see e.g. [1], [2], [3], [4], [18], [19].

In [6] Bonet, Domański, Lindström and Taskinen characterized boundedness and compactness of

operators

Cφ : H
∞

v → H
∞

w , f 7→ f ◦ φ

in terms of the inducing symbol φ as well as the involved weights v and w. The same properties of

the weighted composition operator Cφ,ψ : H
∞

v → H
∞

w were analyzed independently by Contreras

and Hernández-Dı́az as well as Montes-Rodŕıguez. In [8] we investigated under which conditions

the weighted composition operator Cφ,ψ acting on H
∞

v is isometric. The work of Bonet, Domański,

Lindström and Taskinen motivated Garcia, Maestre and Sevilla-Peris to study boundedness and

compactness of composition operators in the following setting.

Let X be a complex Banach space, BX its open unit ball and H(BX) the collection of all holomorphic

functions f : BX → C. Moreover, we consider continuous and bounded functions v : BX → (0,∞).
Such a map is called a weight. A weight v induces the space

Hv(BX) :=

{
f ∈ H(BX); ‖f‖v = sup

x∈BX

v(x)|f(x)| < ∞
}

which, endowed with the weighted sup-norm ‖.‖v is a Banach space as in the onedimensional case.

Now, an analytic map φ : BY → BX induces an operator

Cφ : H(BY) → H(BX), f 7→ f ◦ φ.



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Isometric weighted composition operators on weighted Banach ... 45

Garcia, Maestre and Sevilla-Peris, characterized when an operator

Cφ : Hv(BY) → Hw(BX), f 7→ f ◦ φ

is bounded and compact, i.e. they gave sufficient and necessary conditions in terms of the inducing

map φ as well as of the involved weights v and w for a composition operator to be bounded resp.

compact.

In this article we are interested in weighted composition operators

Cφ,ψ : Hv(BY) → Hw(BY), f 7→ ψ(f ◦ φ).

Motivated by [14] we will investigate when such an operator is bounded. A full characterization

when such an operator is bounded follows easily with a similar proof as given in [14]. The more

interesting question (motivated by [8]) is the following: When is a bounded operator Cφ,ψ acting

on Hv(BX) an isometry.

2 Basics on weights and weighted spaces

This section is devoted to collect some basic facts on weights and weighted spaces in the setting

of a complex Banach space X and its open unit ball BX. These can be found in [13] and [14]. We

say that a set A ⊂ BX is BX-bounded if there exists 0 < r < 1 such that A ⊂ rBX. We write

Hb(BX) = {f ∈ H(BX); f bounded on the BX-bounded sets } .

We consider

Hv(BX) =

{
f ∈ H(BX); ‖f‖v := sup

x∈BX

v(x)|f(x)| < ∞
}
.

With the norm ‖.‖v, the space Hv(BX) is a Banach space. A weight v is radial if v(λx) = v(x) for

every λ ∈ C with |λ| = 1 and every x ∈ BX.

A weight v satisfies Condition I if infx∈rBX v(x) > 0 for every 0 < r < 1. If v satisfies Condition

I, then Hv(BX) ⊂ Hb(BX). If X is finite-dimensional, then all weights on BX enjoy Condition I. In

the sequel we will assume that each weight v satisfies the Condition I.

Given any weight v we consider

ṽ(z) =
1

sup{|f(z)|; ‖f‖v ≤ 1}
.

By [14] Proposition 1.1 the following hold:

(1) 0 < v ≤ ṽ and ṽ is bounded and continuous, i.e. ṽ is a weight.

(2) ṽ is radial and decreasing whenever v is so.

(3) ‖f‖v ≤ 1 ⇐⇒ ‖f‖ṽ ≤ 1.



46 Elke Wolf CUBO
15, 2 (2013)

(4) For every x ∈ BX there is fx ∈ H
∞

v with ‖f‖v ≤ 1 such that ṽ(x) = |fx(x)|.

We say that a weight v is norm-radial if v(x) = v(y) for every x,y with ‖x‖ = ‖y‖. We need

some extra condition on the weight -which in a sense - is an analogon to the Lusky condition (L1)

which appeared during his studies on the isomorphism classes of H∞v , see [18]. Let v be a norm-

radial weight that is continuously differentiable w.r.t. x. Then we say that v satisfies condition (B)

if and only if

(B) sup
x∈BX

(1 − ‖x‖)|v′(x)|

v(x)
< ∞.

Finally, to study isometries we need some geometric tools. The generalized pseudohyperbolic

distance of two points z,p ∈ BX is given by

d(z,p) := sup {ρ(h(z),h(p)); h : BX → D holomorphic }

3 Boundedness

As we said before the following proof is very similar to the proof of Proposition 2.3 in [14]. Nev-

ertheless we give it here for the sake of completeness.

Proposition 3.1. Let v,w be two weights and φ : BX → BY be holomorphic. Moreover, let
ψ ∈ H(BX). Then the following are equivalent:

(a) Cφ,ψ : Hv(BY) → Hw(BX) is well-defined and bounded.

(b) supx∈BX
w(x)|ψ(x)|

ṽ(φ(x))
< ∞.

Proof. Let us first suppose that the operator is bounded. We assume to the contrary that (b) does

not hold. Then we can find a sequence (xn)n ⊂ BX such that

w(xn)|ψ(xn)|

ṽ(φ(xn))
≥ n for every n ∈ N.

Now, for each n ∈ N we can select fn ∈ Hv(BX) with ‖fn‖v ≤ 1 such that |fn(φ(xn))| =
1

ṽ(φ(xn))
.

Since Cφ,ψ : Hv(BY) → Hw(BX) is bounded, there is C > 0 such that

C ≥ ‖Cφ,ψfn‖w ≥
w(xn)|ψ(xn)|

ṽ(φ(xn))
≥ n

for every n ∈ N, which is a contradiction. Conversely, let f ∈ Hv(BY). Then we obtain for every

x ∈ BX

w(x)|ψ(x)||f(φ(x))| =
|ψ(x)|w(x)

ṽ(φ(x))
ṽ(φ(x)) ≤ M‖f‖ṽ = M‖f‖v.

Thus, the claim follows.



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Isometric weighted composition operators on weighted Banach ... 47

4 Isometries

We obtain the following lemma which was shown for the setting of the spaces H∞v in [7]. However,

in this setting there occur several different phenomena.

Lemma 4.1. Let v be a weight on BX such that v is norm-radial and satisfies condition (B).

Moreover, let f ∈ H∞v . Then there is a finite constant M > 0 independent of f ∈ H
∞

v such that

|v(a)f(a) − v(b)f(b)| ≤ M‖f‖vd(a,b)

for every a,b ∈ BX.

Proof. We fix a,b ∈ BX with a 6= b. Now, there are n1,n2 ∈ N such that

‖a‖X < 1 −
1

n1
and ‖b‖X < 1 −

1

n2
.

Then we can find ε > 0 such that

h : D → BX, h(t) = (t − ε)b + (1 − (t − ε))a.

Moroever h(ε) = a and h(1 − ε) = b. Now, by Cauchy’s formula we obtain

|(f ◦ h)′(ε)| =
1

2π

∣

∣

∣

∣

∣

∫

|ξ−ε|=(1−|ε|)r

(f ◦ h)(ξ)

|ξ − ε|
dξ

∣

∣

∣

∣

∣

≤
1

2πr

1

(1 − |ε|)2
‖f‖v

∫

|ξ−ε|=(1−|ε|)r

|dξ|

v(h(ξ))
.

Now, since v(a) < M and ‖h(ξ)‖X ≤ r0 < 1 for every ξ with |ξ − ε| = (1 − |ε|)r. Hence there is

C > 0 such that
v(a)

v(h(ξ))
=
v(h(ε))

v(h(ξ))
≤ C

for every ξ with |ξ − ε| = (1 − |ε|)r. Thus,

|(f ◦ h)′(ε)| ≤
C

2πr2
1

(1 − |ε|)2
‖f‖v
v(h(ε))

2π(1 − |ε|)r

=
C‖f‖v

r(1 − ε)v(h(ε))
.

Next, we consider

k(q) := v(q)f(q) for every q ∈ BX.

Then the total differential of k ◦ h is given by

d(k ◦ h) =
∂(k ◦ h)

∂t
dt +

∂(k ◦ h)

∂t
dt.



48 Elke Wolf CUBO
15, 2 (2013)

Now, for every t ∈ D we obtain

∂(k ◦ h)

∂t
= (v ◦ h)′(t)f(h(t)) + v(h(t))(v ◦ h)′(t)

and
∂(k ◦ h)

∂t
= 0

This yields

|d(k ◦ h)(t)| ≤ [|(v ◦ h)′(t)|f(h(t))| + |v(h(t))||(f ◦ h)′(t)|] |dt|

≤

[

|
(v ◦ h)′(t)

v(h(t))
‖f‖v +

C‖f‖v
r(1 − |t|)v(h(t))

]

|dt|

By condition (B) we can find C1 > 0 such that

|v′(h(t))|

v(h(t))
‖b − a‖ =

|(v ◦ h)′(t)|

(v ◦ h)(t)|
≤

C1

1 − |h(t)|
.

Therefore

|d(k ◦ h)(t)| ≤

(

C1 +
C

r

)

‖f‖v
1 − |t|

|dt|.

If d(h(p),h(q)) ≤ r, then ρ(p,q) ≤ r and by using

1 − ρ(p,q)2 =
(1 − |q|2)(1 − |p|2)

|1 − pq|2

we have that
|q − p|

1 − |p|
∼ ρ(p,q).

Here the constants only depend on r. By integration on both sides we can find constants C2,C3 > 0

with

|k(h(q)) − k(h(p))| ≤ C2‖f‖v
1

1 − |p|
|q − p| ≤ C3‖f‖vρ(p,q) ≤ C3‖f‖vd(h(p),h(q))

for all p,q with d(h(p),h(q)) ≤ r. If d(h(p),h(q)) > r then

|v(p)f(p) − v(q)f(q)| ≤ 2‖f‖v ≤
2

r
‖f‖vd(p,q)

and the claim follows.

The main ideas of the proof of the following theorem are taken from [8] but there also occur

new phenomena.

Theorem 4.2. Let φ be an analytic self-map of BX and ψ ∈ H(BX). Moreover, assume that v is

a norm-radial weight satisfying condition (B) such that v is continuously differentiable.



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15, 2 (2013)

Isometric weighted composition operators on weighted Banach ... 49

(a) If supx∈BX
|ψ(x)|v(x)

ṽ(φ(x))
≤ 1 and

(M) for every a ∈ BX there is (xn)n ⊂ BX such that

d(φ(xn),a) → 0 and
|ψ(xn)|v(xn)

ṽ(φ(xn))
→ 1

then Cφ,ψ : Hv(BX) → Hv(BX) is an isometry.

(b) Let v be a norm-radial weight with v = ṽ such that for each h : BX → D holomorphic
w(x) :=

v(x)

1−|h(x)|2)p
for every x ∈ BX is a weight for some 0 < p < ∞ and w = w̃. If Cφ,ψ :

Hv(BX) → Hv(BX) is an isometry, then condition (M) holds and supx∈BX
|ψ(x)|v(x)

ṽ(φ(x))
≤ 1.

Proof. We first show (a). For every f ∈ Hv(BX) we have that

‖Cφ,ψ‖v = sup
z∈BX

|ψ(x)|v(x)

v(φ(x))
v(φ(x))|f(φ(x))| ≤ ‖f‖v.

Now, let f ∈ Hv(BX). Then ‖f‖v = limm→∞ v(am)|f(am)| for some sequence (am)m. Let m ∈ N

be fixed. Hence, by condition (M), there is (xmn )n ⊂ BX such that d(φ(x
m
n ),am) → 0 and

|ψ(x
m
n )|v(x

m
n )

v(φ(xmn ))
→ 1 when n → ∞. By the previous lemma, for all m and n

|v(am)f(am) − v(φ(x
m
n ))f(φ(x

m
n ))| ≤ M‖f‖vd(am,φ(x

m
n )).

Hence

‖Cφ,ψ‖v = sup
x∈BX

|ψ(x)|v(x)

v(φ(x))
(|f(am)|v(am) − M‖f‖vd(φ(x

m
n ),am)) = v(am)|f(am)|.

Since this is true for all m, we have ‖Cφ,ψf‖v ≥ ‖f‖v.

Next, we show (b). We choose p > 0 and fix h : BX → D holomorphic such that w(x) =
v(x)

(1−|h(x)|2)p

is a weight on BX with w = w̃. By assumption ‖Cφ,ψf‖v = ‖f‖v for all f ∈ Hv(BX). Thus,

‖Cφ,ψ‖ = sup
x∈BX

|ψ(x)|v(x)

ṽ(φ(x))
≤ 1.

Next, fix a ∈ BX and h : BX → D. Then there exists ga ∈ Hw(BX) with ‖ga‖w ≤ 1 such that
ga(a) = w̃(a). Put

fa(z) = ga(z)

(

(1 − |h(a)|2)

(1 − h(z)h(a))2

)p

.

Now, ‖fa‖v = 1 since |fa(a)|v(a) = 1. This means, that we can pick a sequence (xn)n ⊂ BX so

that |ψ(xn)|fa(φ(xn))|v(xn) → 1 when n → ∞. Hence

1 ≥
|ψ(xn)|v(xn)

ṽ(φ(xn))
≥

|ψ(xn)|v(xn)

ṽ(φ(xn))
|fa(φ(xn))|ṽ(φ(xn)) = |ψ(xn)|v(xn)|fa(φ(xn))|.

Finally,

lim
n→∞

|ψ(xn)|v(xn)

ṽ(φ(xn))
= 1.



50 Elke Wolf CUBO
15, 2 (2013)

Further,

1 ≥ (1 − |σh(a)(h(φ(zn))|
2)p =

(1 − |h(a)|2)p(1 − |h(φ(xn))|
2)p

|1 − h(φ(xn))h(a)|
2p

=
|fa(φ(xn))|v(φ(xn))(1 − |h(φ(xn))|

2)p

ga(h(φ(xn)))v(φ(xn))
≥ |fa(φ(xn))|v(φ(xn)).

Since, |fa(φ(xn))|ṽ(φ(xn)) → 1 when n → ∞, we conclude, as v = ṽ, that limn→∞(1 −
σh(a)(h(φ(xn))|

2)p = 1 and ρ(h(φ(xn)),h(a)) → 0 when n → ∞. Since h : BX → D holomorphic
was arbitrary the claim follows.

Example 4.3. Let X be an arbitrary complex Banach space, h : BX → D be holomorphic and select
v(x) = (1− |h(x)|2)p. For fixed b ∈ BX we put φ(x) := σh(b)(h(x)) and ψ(x) := (σh(b))

′(h(x)) for

every x ∈ BX. Then easy calculations show that the corresponding weighted composition operator

is an isometry.

Received: March 2012. Accepted: September 2012.

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