Ann. Funct. Anal. 4 (2013), no. 1, 163–174
A nnals of F unctional A nalysis
ISSN: 2008-8752 (electronic)
URL:www.emis.de/journals/AFA/
WEIGHTED COMPOSITION OPERATORS FROM CAUCHY
INTEGRAL TRANSFORMS TO LOGARITHMIC
WEIGHTED-TYPE SPACES
AJAY K. SHARMA
Communicated by K. Gurlebeck
Abstract. We characterize boundedness and compactness of weighted composition operators from the space of Cauchy integral transforms to logarithmic
weighted-type spaces. We also manage to compute norm of weighted composition operators acting between these spaces.
1. Introduction and Preliminaries
Let D be the open unit disk in the complex plane C, ∂D its boundary, dA(z) the
normalized area measure on D (i.e. A(D) = 1), H(D) the class of all holomorphic
functions on D, H ∞ the space of all bounded holomorphic functions on D with
the norm kf k∞ = supz∈D |f (z)| and M the space of all complex Borel measures
on ∂D. Let
a−z
ηa (z) =
, a, z ∈ D,
1 − āz
that is, the involutive automorphism of D interchanging points a and 0. Let ν be
a positive continuous function on D (weight). A weight ν is called typical if it is
radial, i.e. ν(z) = ν(|z|), z ∈ D and ν(|z|) decreasingly converges to 0 as |z| → 1.
A positive continuous function ν on the interval [0, 1) is called normal if there are
δ ∈ [0, 1) and τ and t, 0 < τ < t such that
ν(r)
ν(r)
is
decreasing
on
[δ,
1)
and
lim
= 0;
r→1 (1 − r)τ
(1 − r)τ
Date: Received: 18 August 2012; Accepted: 6 November 2012.
2010 Mathematics Subject Classification. Primary 47B33, 46E10; Secondary 30D55.
Key words and phrases. Weighted composition operator, Cauchy integral transforms, logarithmic weighted-type space, little logarithmic weighted-type space.
163
164
A.K. SHARMA
ν(r)
ν(r)
is increasing on [δ, 1) and lim
= ∞.
t
r→1
(1 − r)
(1 − r)t
If we say that a function ν : D → [0, ∞) is a normal weight function, then we also
assume that it is radial. We denote by LAln (ν) the logarithmic weighted-type
space of functions f ∈ H(D) for which
2
||f ||LAln (ν) = sup ν(|z|)|f (z)| ln
< ∞.
1 − |z|2
z∈D
Likewise we write LAln,0 (ν) for little logarithmic weighted-type space of holomorphic functions f on D for which
2
lim ν(|z|)|f (z)| ln
= 0.
|z|→1
1 − |z|2
With the norm k · kLAln (ν) , the space LAln (ν) is a Banach space and the little
logarithmic weighted space LAln,0 (ν) is a closed subspace of LAln (ν).
A function f ∈ H(D) is in the space of Cauchy integral transforms K, if it admits
a representation of the form
Z
dµ(ζ)
f (z) =
(1.1)
∂D 1 − ζz
where µ ∈ M. The space K becomes a Banach space under the norm
Z
dµ(ζ)
kf kK = inf kµk : f (z) =
,
∂D 1 − ζz
where kµk denotes the total variation of measure µ. It is clear that the Banach
space K is the quotient of Banach space M of Borel measures by the subspace of
measures whose Cauchy transforms vanish. By the E. and M. Riesz theorem it
follows that the Borel measure µ has a vanishing Cauchy transform if and only
if it has the form dµ = f dm, where f ∈ H01 , the subspace of L1 consisting of
functions with mean value 0 whose conjugate belongs to the Hardy space H 1 ,
and dm is the normalized Lebesgue measure on ∂D. Hence K is isometrically
isomorphic to M/H01 . Since M has a decomposition M = L1 ⊕Ms , where Ms is the
space of all Borel measures which are singular with respect to Lebesgue measure,
and H01 ⊂ L1 , it follows that K is isometrically isomorphic to L1 /H01 ⊕ Ms .
Consequently, K has an analogous decomposition K = Ka ⊕ Ks , where Ka is
isometrically isomorphic to L1 /H01 and Ks is isometrically isomorphic to Ms . It
is known that
H 1 ⊂ K ⊂ ∩0<p<1 H p ,
where H p is the Hardy space. For more about the space K, see [3], [4], [5] and
[9].
Let ψ ∈ H(D) and ϕ be a holomorphic self-map of D. Define a linear operator
Wψ,ϕ f (z) = ψ(z)f (ϕ(z))
for f ∈ H(D) and z ∈ D. The operator Wψ,ϕ is called a weighted composition
operator. We can regard this operator as a generalization of a multiplication
operator Mψ induced by ψ and a composition operator Cϕ induced by ϕ, where
Mψ f (z) = ψ(z)f (z) and Cϕ f (z) = f (ϕ(z)). In fact, Wψ,ϕ = Mψ Cϕ . For more
WEIGHTED COMPOSITION OPERATORS
165
about these operators, see [6] and [15].
It is well known that every holomorphic self-map ϕ of D induces a bounded
composition operator on K. In fact, Bourdon and Cima [3] proved that
√
2+2 2
kCϕ kK→K ≤
1 − |ϕ(0)|
which was improved to
kCϕ kK→K ≤
1 + 2|ϕ(0)|
1 − |ϕ(0)|
(1.2)
by Cima and Matheson [4]. Moreover, equality is attained for certain linear
fractional maps.
Isometries in many Banach spaces of analytic functions are weighted composition
operators for example see [7] and [8]. It is of interest to provide function-theoretic
characterizations indicating when ψ and ϕ induce bounded or compact weighted
composition operators on spaces of holomorphic functions. For some recent results
in this area, see [1],[2], [10]-[14], [16]-[26] and the references therein. In this paper,
we provide, in a concise way, a function theoretic characterizations indicating
when ψ and ϕ induce bounded or compact weighted composition operators from
the space of Cauchy integral transforms to logarithmic weighted-type spaces. We
also manage to calculate the norm of the operator Wψ,ϕ : K → LAln (ν) which
is one of the problems that recently attracted some attention. Throughout this
paper constants are denoted by C, they are positive and not necessarily the same
at each occurrence. We write A B if there is a positive constant C such that
CA ≤ B ≤ A/C.
2. Boundedness and Compactness of Wψ,ϕ : K → LAln (ν) and
Wψ,ϕ : K → LAln,0 (ν)
In this section, we characterize the boundedness and compactness of Wψ,ϕ :
K → LAln (ν) and Wψ,ϕ : K → LAln,0 (ν).
Theorem 1. Let ν : D → [0, ∞) be a normal weight function, ψ ∈ H(D) and ϕ
a holomorphic self-map of D. Then Wψ,ϕ : K → LAln (ν) is bounded if and only
if
M := sup sup
ζ∈∂D z∈D
ν(|z|)
2
< ∞.
|ψ(z)| ln
1 − |z|2
|1 − ζϕ(z)|
(2.1)
Moreover, if Wψ,ϕ : K → LAln (ν) is bounded then
kWψ,ϕ kK→LAln (ν) = M.
(2.2)
Proof. First suppose that Wψ,ϕ : K → LAln (ν) is bounded. Consider the family
of functions
1
fζ (z) =
, ζ ∈ ∂D.
(2.3)
1 − ζz
166
A.K. SHARMA
Then kfζ kK = 1, for each ζ ∈ ∂D (see, e.g., [3, p. 468]). Thus by the boundedness
of Wψ,ϕ : K → LAln (ν) we have that
kWψ,ϕ fζ kLAln (ν) ≤ kWψ,ϕ kK→LAln (ν) ,
for every ζ ∈ ∂D and so
M := sup sup
ζ∈∂D z∈D
ν(|z|)
2
|ψ(z)| ln
≤ kWψ,ϕ kK→Wψ,ϕ LAln (ν) .
1 − |z|2
|1 − ζϕ(z)|
(2.4)
Conversely, suppose that (2.1) holds. Let f ∈ K. Then there is a µ ∈ M such
that kµk = kf kK and
Z
dµ(ζ)
.
(2.5)
f (z) =
∂D 1 − ζz
Replacing z in (2.5) by ϕ(z), using a well-known inequality and multiplying
2
such obtained inequality with ν(|z|)|ψ(z)| ln 1−|z|
2 , we obtain
Z
ν(|z|)
2
2
|f (ϕ(z))| ≤
d|µ|(ζ). (2.6)
ν(|z|)|ψ(z)| ln
|ψ(z)| ln
2
1 − |z|
1 − |z|2
∂D |1 − ζϕ(z)|
Thus
Z
2
2
ν(|z|)
ν(|z|) ln
|ψ(z)| ln
|Wψ,ϕ f (z)| ≤ sup sup
d|µ|(ζ)
1 − |z|2
1 − |z|2 ∂D
ζ∈∂D z∈D |1 − ζϕ(z)|
2
ν(|z|)
kf kK .
|ψ(z)| ln
= sup sup
1 − |z|2
ζ∈∂D z∈D |1 − ζϕ(z)|
Taking the supremum in the last inequality over all z ∈ D it follows that
||Wψ,ϕ f ||LAln (ν) ≤ sup sup
ζ∈∂D z∈D
ν(|z|)
2
kf kK .
|ψ(z)| ln
1 − |z|2
|1 − ζϕ(z)|
(2.7)
This shows that kWψ,ϕ f kLAln (ν) ≤ M kf kK , hence Wψ,ϕ : K → LAln (ν) is bounded.
From (2.4) and (2.7), equality (2.2) follows.
In the following corollary we give another necessary and sufficient condition for
the boundedness of the operator Wψ,ϕ : K → LAln (ν).
Corollary 1. Let ν : D → [0, ∞) be a normal weight function, ψ ∈ H(D), ϕ
a holomorphic self-map of D and dλ(z) = dA(z)/(1 − |z|2 )2 . Then Wψ,ϕ : K →
LAln (ν) is bounded if and only if
Z
ν(|z|)
2
|ψ(z)| ln
N := sup sup
(1 − |ηa (z)|2 )2 dλ(z) < ∞. (2.8)
1 − |z|2
ζ∈∂D z∈D D |1 − ζϕ(z)|
Moreover, N M.
Proof. First assume that the operator Wψ,ϕ : K → LAln (ν) is bounded. Using
Theorem 1 and the identity
(1 − |z|2 )|ηa0 (z)| = 1 − |ηa (z)|2 =
(1 − |a|2 )(1 − |z|2 )
,
|1 − āz|2
WEIGHTED COMPOSITION OPERATORS
167
we have
Z
N ≤ M sup
z∈D
D
(1 − |a|2 )2
dA(z) = M C < ∞.
|1 − az|4
(2.9)
Conversely, assume that (2.8) holds. Let D(a, (1 − |a|)/2) = z ∈ D : |z − a| <
(1 − |a|)/2 . Since ν : D → [0, ∞) is a normal weight function, so
ν(|a|) ln
2
2
ν(|z|)
ln
,
1 − |a|2
1 − |z|2
(2.10)
for z ∈ D(a, (1 − |a|)/2). Using (2.9), (2.10) and the subharmonicity of the function |ψ|/|1 − ζϕ|, we have
Z
ν(|z|)
2
|ψ(z)| ln
N ≥ sup sup
(1 − |ηa (z)|2 )2 dλ(z)
2
1
−
|z|
ζ∈∂D z∈D D(a,(1−|a|)/2) |1 − ζϕ(z)|
2
ν(|a|)
= CM,
|ψ(a)| ln
≥ C sup sup
1 − |a|2
ζ∈∂D a∈D |1 − ζϕ(a)|
so that (2.1) holds. Thus by Theorem 1, we have that the operator Wψ,ϕ : K →
LAln (ν) is bounded and M N , as desired.
Lemma 1. Let ν : D → [0, ∞) be a normal weight function and dλ(z) =
dA(z)/(1 − |z|2 )2 . Then f ∈ LAln (ν) if and only if
Z
2
I := sup |f (z)|ν(|z|) ln
(1 − |ηa (z)|2 )2 dλ(z) < ∞.
(2.11)
2
1 − |z|
a∈D D
Moreover, the following asymptotic relationship holds
kf kLAln (ν) I.
Proof. Assume that (2.11) holds. Let E(a, 1/2) = z ∈ D : |ηa (z)| < 1/2 . Then
Z
|f (η(z))|dA(z)
|f (a)| = |fa (η(0))| ≤ 4
|z|<1/2
Z
|f (z)||ηa0 (z)|2 dA(z)
=4
E(a,1/2)
Since ν : D → [0, ∞) is a normal weight function, so
2
2
ν(|a|) ln
ν(|z|) ln
2
1 − |a|
1 − |z|2
for z ∈ E(a, 1/2). Thus
Z
C
2
|f (a)| ≤
|f (z)|ν(|z|) ln
(1 − |ηa (z)|2 )2 dλ(z).
2
1 − |z|2
ν(|a|) ln 1−|a|2 E(a,1/2)
Hence
2
|f (a)|
1 − |a|2
Z
≤ C sup |f (z)|ν(|z|) ln
ν(|a|) ln
a∈D
D
2
(1 − |ηa (z)|2 )2 dλ(z),
1 − |z|2
168
A.K. SHARMA
which implies that if I is finite, then f ∈ LAln (ν) and kf kLAln (ν) ≤ CI. Conversely,
assume that f ∈ LAln (ν), then, we get
Z
(1 − |a|2 )2
I ≤ kf kLAln (ν) sup
dA(z) ≤ Ckf kLAln (ν) < ∞.
4
a∈D D |1 − āz|
Hence kf kLAln (ν) I, as desired.
Using the fact that the family of functions
o
n
1
: ζ ∈ ∂D
fζ =
1 − ζz
satisfies kfζ kK = 1, ζ ∈ D, by Corollary 1 and Lemma 1, we easily obtain the
following result.
Corollary 2. Let ν : D → [0, ∞) be a normal weight function, ψ ∈ H(D) and ϕ
a holomorphic-self map of D. Then Wψ,ϕ : K → LAln (ν) is bounded if and only if
the family of functions
n ψ
o
: ζ ∈ ∂D
1 − ζϕ
is norm-bounded in LAln (ν).
By (1.1), it is easy to see that the unit ball of K is a normal family of holomorphic functions. A standard normal family argument then yields the proof of
the following lemma (see, e.g. Proposition 3.11 of [6] or Lemma 3 in [13]).
Lemma 2. Let ν : D → [0, ∞) be a normal weight function, ψ ∈ H(D) and ϕ
a holomorphic self-map of D. Then Wψ,ϕ : K → LAln (ν) is compact if and only
if for any sequence {fn }n∈N in K with supn∈N kfn kK ≤ 1 converging to zero on
compacts of D, we have limn→∞ kWψ,ϕ fn kLAln (ν) = 0.
Lemma 3. Let f ∈ BK , the unit ball in K and ft (z) = f (tz), 0 < t < 1. Then
ft ∈ K and sup0<t<1 kft kK ≤ kf kK .
Proof. Let f ∈ BK and ft (z) = f (tz), 0 < t < 1. For 0 < t < 1, let ϕt be defined
on D as ϕt (z) = tz. Then ϕt is a holomorphic self-map of D and ϕt (0) = 0. Also
ft (z) = f (tz) = (f ◦ ϕt )(z) = Cϕt f (z) for all z ∈ D. Therefore, ft = Cϕt f. Since
every self-map of D induces bounded composition operator on K, we have that
Cϕt is bounded on K. Moreover, by (1.2), we have
kft kK = kCϕt f kK ≤
1 + 2|ϕt (0)|
kf kK = kf kK .
1 − |ϕt (0)|
Taking supermum over t, 0 < t < 1, we get the desired result.
Theorem 2. Let ν : D → [0, ∞) be a normal weight function, ψ ∈ H(D) and ϕ
a holomorphic-self map of D. Then Wψ,ϕ : K → LAln (ν) is compact if and only if
Z
2
M1 := sup ν(|z|)|ψ(z)| ln
(1 − |ηa (z)|2 )2 dλ(z) < ∞
(2.12)
1 − |z|2
a∈D D
WEIGHTED COMPOSITION OPERATORS
169
and
Z
lim sup sup
r→1 ζ∈∂D a∈D
|ϕ(z)|>r
ν(|z|)
2
|ψ(z)| ln
(1 − |ηa (z)|2 )2 dλ(z) = 0 (2.13)
1 − |z|2
|1 − ζϕ(z)|
for every ζ ∈ ∂D.
Proof. First suppose that (2.12) and (2.13) hold. Let {fm }m∈N be a bounded
sequence in K, say by L and converging to 0 uniformly on compacts of D as
m → ∞. By Lemma 2, we have to show that kWψ,ϕ fm kLAln (ν) → 0 as m → ∞.
For each m ∈ N, we can find µm ∈ M with kµm k = kfm kK such that
Z
dµm (ζ)
.
fm (z) =
∂D 1 − ζz
By (2.13), we have for every > 0, there is an r1 ∈ (0, 1) such that for r ∈ (r1 , 1),
we have
Z
ν(|z|)
2
sup sup
|ψ(z)| ln
(1 − |ηa (z)|2 )2 dλ(z) < .
2
1
−
|z|
ζ∈∂D a∈D |ϕ(z)|>r |1 − ζϕ(z)|
By Lemma 1, applied to the function ψ(fm ◦ ϕ) we have
Z
Z
kWψ,ϕ fm kLAln (ν) sup
+
|fm (ϕ(z))|
a∈D
|ϕ(z)|≤r
× ν(z)|ψ(z)| ln
|ϕ(z)|>r
2
(1 − |ηa (z)|2 )2 dλ(z).
1 − |z|2
Since the set |w| ≤ r is compact we have sup|ϕ(z)|≤r |fm (ϕ(z))| < for sufficiently
large m, say m ≥ m0 . Thus by Fubini’s theorem, we have
Z
kWψ,ϕ fm kLAln (ν) ≤ C sup |fm (ϕ(z))| sup
|ψ(z)|
a∈D
|ϕ(z)|≤r
|ϕ(z)|≤r
2
× ν(z) ln
(1 − |ηa (z)|2 )2 dλ(z)
1 − |z|2
Z Z
ν(|z|)
+ sup sup
ζ∈∂D a∈D ∂D |ϕ(z)|>r |1 − ζϕ(z)|
2
× |ψ(z)| ln
(1 − |ηa (z)|2 )2 dλ(z)d|µm |(ζ)
1 − |z|2
Z
≤ C(M1 +
d|µm |(ζ)) ≤ C(M1 + kfm kK )
∂D
≤ C(M1 + L)
for m ≥ m0 . Since > 0 is arbitrary, Wψ,ϕ : K → LAln (ν) is compact.
Conversely, suppose that Wψ,ϕ : K → LAln (ν) is compact. By choosing f (z) = 1
in K, we have
Z
2
M := sup ν(|z|)|ψ(z)| ln
(1 − |ηa (z)|2 )2 dλ(z) < ∞.
2
1 − |z|
a∈D D
Let fm (z) = z m , m ∈ N. It is easy to see that {fm }m∈N is a norm bounded
sequence in K converging to zero uniformly on compact subsets of D. Hence by
170
A.K. SHARMA
Lemma 2, it follows that kWψ,ϕ fm kLAln (ν) → 0 as m → ∞. Thus for every > 0,
there is an m0 ∈ N such that for m ≥ m0 , we have
Z
2
sup |ϕ(z)|2m ν(|z|)|ψ(z)| ln
(1 − |ηa (z)|2 )2 dλ(z) < .
(2.14)
2
1 − |z|
a∈D D
From (2.14), we have for each r ∈ (0, 1)
Z
2
2m
(1 − |ηa (z)|2 )2 dλ(z) < .
r sup ν(|z|)|ψ(z)| ln
1 − |z|2
a∈D D
Hence for r ∈ (1/21/(2m0 ) , 1), we have
Z
2
sup ν(|z|)|ψ(z)| ln
(1 − |ηa (z)|2 )2 dλ(z) < 2.
2
1
−
|z|
a∈D D
(2.15)
Let f ∈ BK and ft (z) = f (tz), 0 < t < 1, then by Lemma 3, we have that
sup0<t<1 kft kK ≤ kf kK , ft ∈ K, t ∈ (0, 1) and ft → f uniformly on compact
subsets of D as t → 1. The compactness of Wψ,ϕ : K → LAln (ν) implies that
lim kWψ,ϕ ft − Wψ,ϕ f kLAln (ν) = 0.
t→1
Hence for every > 0, there is a t ∈ (0, 1) such that
Z
2
(1 − |ηa (z)|2 )2 dλ(z) < . (2.16)
sup |ft (ϕ(z)) − f (ϕ(z))|ν(|z|)|ψ(z)| ln
1 − |z|2
a∈D D
From (2.15) and (2.16), we have
Z
2
(1 − |ηa (z)|2 )2 dλ(z)
sup
|f (ϕ(z))|ν(|z|)|ψ(z)| ln
2
1
−
|z|
a∈D |ϕ(z)|>r
Z
2
≤ sup |ft (ϕ(z)) − f (ϕ(z))|ν(|z|)|ψ(z)| ln
(1 − |ηa (z)|2 )2 dλ(z)
2
1 − |z|
a∈D D
Z
2
+ sup |ft (ϕ(z))|ν(|z|)|ψ(z)| ln
(1 − |ηa (z)|2 )2 dλ(z)
2
1 − |z|
a∈D D
≤ (2 + ||ft ||∞ ).
Thus we conclude that for every f ∈ BK , there is an r0 ∈ (0, 1) such that for
r ∈ (r0 , 1), we have
Z
2
(1 − |ηa (z)|2 )2 dλ(z) < C. (2.17)
sup
|f (ϕ(z))|ν(|z|)|ψ(z)| ln
2
1 − |z|
a∈D |ϕ(z)|>r
Since Wψ,ϕ : K → LAln (ν) is compact, we have for every > 0, there is a finite
collection of functions f1 , f2 , · · · , fk ∈ BK such that for each f ∈ BK , there is a
j ∈ {1, 2, · · · , k} such that
Z
2
sup |f (ϕ(z)) − fj (ϕ(z))|ν(|z|)|ψ(z)| ln
(1 − |ηa (z)|2 )2 dλ(z) < . (2.18)
2
1
−
|z|
a∈D D
On the other hand from (2.17) it follows that if δ := max1≤j≤k δj (, fj ), then for
r ∈ (δ, 1) and all j ∈ {1, 2, · · · , k} we have
Z
2
sup
|fj (ϕ(z))|ν(|z|)|ψ(z)| ln
(1 − |ηa (z)|2 )2 dλ(z) < . (2.19)
2
1 − |z|
a∈D |ϕ(z)|>r
WEIGHTED COMPOSITION OPERATORS
From (2.18) and (2.19) we have for r ∈ (δ, 1) and every f ∈ BK
Z
2
|f (ϕ(z))|ν(|z|)|ψ(z)| ln
sup
(1 − |ηa (z)|2 )2 dλ(z) < C.
2
1 − |z|
a∈D |ϕ(z)|>r
171
(2.20)
If we apply (2.20) to the function fζ (z) = 1/(1 − ζz), we have
Z
ν(|z|)
2
sup
|ψ(z)| ln
(1 − |ηa (z)|2 )2 dλ(z) < C,
2
1
−
|z|
a∈D |ϕ(z)|>r |1 − ζϕ(z)|
from which (2.13) follows as desired.
Theorem 3. Let ν : D → [0, ∞) be a normal weight function, ψ ∈ H(D) and ϕ
a holomorphic-self map of D. Then Wψ,ϕ : K → LAln,0 (ν) is bounded if and only
if
M := sup sup
ζ∈∂D z∈D
2
ν(|z|)
<∞
|ψ(z)| ln
1 − |z|2
|1 − ζϕ(z)|
(2.21)
and
ν(|z|)
2
|ψ(z)| ln
=0
|z|→1 |1 − ζϕ(z)|
1 − |z|2
lim
(2.22)
for every ζ ∈ ∂D.
Proof. First suppose that Wψ,ϕ : K → LAln,0 (ν) is bounded. Once again consider
the family of test functions in (2.3). Then kfζ kK = 1. Thus by the boundedness
of Wψ,ϕ : K → LAln,0 (ν), we have Wψ,ϕ fζ ∈ LAln,0 (ν) for every ζ ∈ ∂D and so
2
ν(|z|)
|ψ(z)| ln
=0
|z|→1 |1 − ζϕ(z)|
1 − |z|2
lim
for every ζ ∈ ∂D. Again, if Wψ,ϕ : K → LAln,0 (ν) is bounded, then for every
f ∈ K, we have that Wψ,ϕ f ∈ LAln,0 (ν) ⊂ LAln (ν). So by the closed graph
theorem Wψ,ϕ : K → LAln (ν) is bounded. Thus by Theorem 1, we have (2.21).
Conversely, suppose that (2.21) and (2.22) hold. By (2.22), the inner expression
in the second term of (2.6) tends to zero for every ζ ∈ ∂D, as |z| → 1. Also the
inner expression in the second term of (2.6) is dominated by
M := sup sup
ζ∈∂D z∈D
ν(|z|)
2
|ψ(z)| ln
.
1 − |z|2
|1 − ζϕ(z)|
Thus by the bounded convergence theorem, the second term of (2.6) tend to zero
as |z| → 1, so
2
lim ν(|z|) ln
|Wψ,ϕ f (z)| = 0.
|z|→1
1 − |z|2
Thus we conclude that if f ∈ K, then Wψ,ϕ f ∈ LAln,0 (ν). Therefore, the boundedness of Wψ,ϕ : K → LAln,0 (ν) follows by the closed graph theorem.
In order to prove the compactness of Wψ,ϕ : K → LAln,0 (ν), we require the
following lemma.
172
A.K. SHARMA
Lemma 4. A subset F of LAln,0 (ν) is compact if and only if it is closed, bounded
and satisfies
2
lim sup ν(z)|f (z)| ln
= 0.
|z|→1 f ∈F
1 − |z|2
The proof is similar to that of Lemma 1 in [13], we omit the details.
Theorem 4. Let ν : D → [0, ∞) be a normal weight function, ψ ∈ H(D) and ϕ
a holomorphic-self map of D. Then Wψ,ϕ : K → LAln,0 (ν) is compact if and only
if
ν(|z|)
2
|ψ(z)| ln
= 0.
|z|→1 ζ∈∂D |1 − ζϕ(z)|
1 − |z|2
lim sup
(2.23)
Proof. By Lemma 4, a closed set F in LAln,0 (ν) is compact if and only if it is
bounded and satisfies
2
lim sup ν(|z|)|f (z)| ln
= 0.
|z|→1 f ∈F
1 − |z|2
Thus the set {Wψ,ϕ f : f ∈ K, kf kK ≤ 1} has compact closure in LAln,0 (ν) if and
only if
2
|Wψ,ϕ f (z)| : f ∈ K, kf kK ≤ 1} = 0.
(2.24)
lim sup{ν(|z|) ln
|z|→1
1 − |z|2
Let f ∈ LAln,0 (ν) with kf kLAln (ν) ≤ 1. Then there is a µ ∈ M such that kµk =
kf kK and
Z
dµ(ζ)
f (z) =
.
∂D 1 − ζz
Then by (2.6), we have
Z
2
ν(|z|)
2
ν(|z|) ln
|ψ(z)| ln
|Wψ,ϕ f (z)| ≤
d|µ|(ζ)
2
1 − |z|
1 − |z|2
∂D |1 − ζϕ(z)|
2
ν(|z|)
≤ kµk sup
|ψ(z)| ln
1 − |z|2
ζ∈∂D |1 − ζϕ(z)|
ν(|z|)
2
= kf kK sup
|ψ(z)| ln
.
1 − |z|2
ζ∈∂D |1 − ζϕ(z)|
Hence by (2.23), we have
lim sup{ν(z)|Wψ,ϕ f (z)| ln
|z|→1
2
: f ∈ K, kf kK ≤ 1} = 0.
1 − |z|2
Conversely, suppose that Wψ,ϕ : K → LAln,0 (ν) is compact. Taking the test
functions in (2.3) and using the fact that kfζ kK = 1 we obtain that (2.23) follows
from (2.24).
Acknowledgements. The author would like to thank Prof. S. Stević and the
referee for their helpful comments and suggestions.
WEIGHTED COMPOSITION OPERATORS
173
References
1. R.F. Allen and F. Collona, Weighted composition operators from H ∞ to the Bloch space
of a bounded homogeneous domain, Integral Equations Operator Theory 66 (2010), no. 1,
21–40.
2. R.F. Allen and F. Collona, Weighted composition operators on the Bloch space of a bounded
homogeneous domain, Oper. Theory Adv. Appl. 202 (2010), 11–37.
3. P. Bourdon and J.A. Cima, On integrals of Cauchy-Stieltjes type, Houston J. Math. 14
(1988), no. 4, 465–474.
4. J.A. Cima and A.L. Matheson, Cauchy transforms and composition operators, Illinois J.
Math. 4 (1998), no. 1, 58–69.
5. J.A. Cima and T.H. MacGregor, Cauchy transforms of measures and univalent functions,
Lecture Notes in Math. 1275, Spinger-Verlag 1987, 78–88.
6. C.C. Cowen and B.D. MacCluer, Composition operators on spaces of analytic functions,
CRC Press Boca Raton, New York, 1995.
7. F. Forelli, The isometries of H p , Canad. J. Math. 16 (1964), 721–728.
8. F. Forelli, A theorem on isometries and the application of it to the isometries of H p (S) for
2 < p < ∞, Canad. J. Math. 25 (1973), 284–289.
9. R. Hibschweiler and E. Nordgren, Cauchy transforms of measures and weighted shift operators on the disc algebra, Rocky Mountain J. Math. 26 (1996), no. 2, 627–654.
10. S. Li and S. Stević, Generalized composition operators on Zygmund spaces and Bloch type
spaces, J. Math. Anal. Appl. 338 (2008), no. 2, 1282–1295.
11. S. Li and S. Stević, Products of composition and integral type operators from H ∞ and the
Bloch space, Complex Var. Elliptic Equ. 53 (2008), no. 5, 463–474.
12. S. Li and S. Stević, Products of Volterra type operator and composition operator from H ∞
and Bloch spaces to Zygmund spaces, J. Math. Anal. Appl. 345 (2008), no. 1, 40–50.
13. K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans.
Amer. Math. Soc. 347 (1995), no. 7, 2679–2687.
14. S. Ohno, K. Stroethoff and R. Zhao, Weighted composition operators between Bloch-type
spaces, Rocky Mountain J. Math. 33 (2003), no. 1, 191–215.
15. J.H. Shapiro, Composition operators and classical function theory, Springer-Verlag, New
York. 1993.
16. A.K. Sharma, Volterra composition operators between Bergman-Nevanlinna and Bloch-type
spaces, Demonstratio Math. 42 (2009), no. 3, 607–618.
17. A.K. Sharma, Products of multiplication, composition and differentiation between weighted
Bergman-Nevanlinna and Bloch-type spaces, Turkish. J. Math. 35 (2011), no. 2, 275–291.
18. A.K. Sharma and S. Ueki, Compactness of composition operators acting on weighted
Bergman Orlicz spaces, Ann. Polon. Math. 103 (2011), no. 1, 1–13.
19. A.K. Sharma and S. Ueki, Composition operators from Nevanlinna type spaces tp Bloch
type spaces, Banach J. Math. Anal. 6 (2012), no. 1, 112–123.
20. A. Sharma and A.K. Sharma, Carleson measures and a class of generalized integration
operators on the Bergman space, Rocky Mountain J. Math. 41 (2011), no. 5, 711–724.
21. S. Stević, Weighted composition operators between mixed norm spaces and Hα∞ spaces in
the unit ball, J. Inequal. Appl. 2007, Article ID 28629, 9 pp.
22. S. Stević, Generalized composition operators from logarithmic Bloch spaces to mixed-norm
spaces, Util. Math. 77 (2008) 167–172.
23. S. Stević, Weighted iterated radial composition operators between some spaces of holomorphic functions on the unit ball, Abstr. Appl. Anal. 2010 Article ID 801264, 14 pp.
24. S. Stević and A.K. Sharma, Essential norm of composition operators between weighted Hardy
spaces, Appl. Math. Comput. 217 (2011), no. 13, 6192–6197.
25. S. Stević and A.K. Sharma, Composition operators from the space of Cauchy transforms to
Bloch and the little Bloch-type spaces on the unit disk, Appl. Math. Comput. 217 (2011),
no. 24, 10187–10194.
174
A.K. SHARMA
26. S. Stević, A.K. Sharma and A. Bhat, Products of multiplication composition and differentiation operators on weighted Bergman spaces, Appl. Math. Comput. 217 (2011), no. 20,
8115–8125.
1
School of Mathematics, Shri Mata Vaishno Devi University, Kakryal, Katra182320, J& K, India.
E-mail address: aksju 76@yahoo.com; aksju76@gmail.com