1. No category

GENERALIZED COMPOSITION OPERATORS ON
ZYGMUND-ORLICZ TYPE SPACES AND BLOCH-ORLICZ TYPE
SPACES∗
CONGLI YANG1 , FANGWEI CHEN2 AND PENGCHENG WU3
Abstract. The boundedness and compactness of a generalized composition
operators on Zygmund-Orlicz type spaces and Bloch-Orlicz type spaces are
established in this paper.
1. introduction
Let D be a unit disk in complex plane C, H(D) be the space of all holomorphic
functions on D with the topology of uniform convergence on compact subsets of D.
The Bloch space, B, consists of all functions f ∈ H(D) for which
||f ||b := sup(1 − |z|2 )|f 0 (z)| < ∞.
(1.1)
z∈D
B becomes a Banach space when it is equiped with the norm ||f ||B := |f (0)| + ||f ||b
(see e.g., [3]).
Let α > 0, the α-Bloch space, denoted as B α , consists of all holomorphic functions f on D such that
(1.2)
||f ||α := sup(1 − |z|2 )α |f 0 (z)| < ∞.
z∈D
α-Bloch space is introduced and studied by numerous authors. The general theory
of α-Bloch fucntion spaces refer to [20]. Recently, many authors studied different
class of Bloch-type spaces, where the typical weight function, w(z) = 1 − |z|2 (z ∈
D) is replaced by a bounded continuous positive function µ defined on D. More
precisely, a function f ∈ H(D) is called a µ-Bloch function, denoted as f ∈ B µ , if
||f ||µ := sup µ(z)|f 0 (z)| < ∞.
(1.3)
z∈D
α
Clearly, if µ(z) = w(z) with α > 0, B µ is just the α-Bloch space B α . It is readily
to see that B µ is a Banach space with the norm
||f ||Bµ := |f (0)| + ||f ||µ .
µ
B spaces appears in the literature of a natural way when one studies the properties
of some operators of holomorphic functions in a certain spaces. For instance, Attele
in [1] proves that the Hankel operator induced by a function f in the Bergman space
2010 Mathematics Subject Classification. Primary 47B33; Secondary 30H99.
Key words and phrases. Zygmud-Orlicz type space, Bloch-Orlicz type space, Generalized composition operators.
∗ The work is supported in part by CNSF (Grant No. 111010997, Grant No.11161007 and
Grant No. 11171080), Guizhou Foundation for Science and Technology (Grant No. [2012] 2273
and No. [2014] 2044), Guizhou technology foundation for selected overseas scholars, and Ph.D
research foundation of Guizhou Normal University.
1
2
CONGLI YANG1 , FANGWEI CHEN2 AND PENGCHENG WU3
z
is bounded if and only if f ∈ B µ1 (where µ1 = w(z) log w(z)
, with z ∈ D). The space
µ1
B is known as the Log-Bloch space or the weighted Bloch space. The logarithmic
Bloch-type space is introduced by Krantz and Stevi´c [6, 16], where some properties
of this space are studied. In the last decade, there were a big interest in the
investigation of Bloch-type spaces and various concrete linear operators L : X → Y ,
where at least one of the spaces X and Y is Bloch space. For some other recent
results in the areas, see for example [6, 8, 11, 12, 14, 22] and a lot of reference
therein.
Recently, Ramos-Fern´
andz in [12] use Young’s functions to define the BlochOrlicz space as a generalization of Bloch space. More precisely, let ϕ : [0, ∞) →
[0, ∞) be an N -function, that is, ϕ is a strictly increasing convex function such that
t
= lim ϕ(t)
= 0. The Bloch-Orlicz space associated to the
ϕ(0) = 0 and lim ϕ(t)
t
t→∞
t→0
function ϕ, denoted as B ϕ , is the class of all analytic functions f ∈ D such that
(1.4)
sup(1 − |z|2 )ϕ(λ|f 0 (z)|) < ∞,
z∈D
for some λ > 0 depending on f . Without loss of generality, we can suppose that ϕ−1
is continuously and differentiable. In fact, if ϕ−1 is not differentiable everywhere,
we set the function
Z t
ϕ(x)
(1.5)
ψ(t) =
dx
(t ≥ 0),
x
0
then ϕ is differentiable, whence ψ −1 is differentiable everywhere on [0, ∞). Furthermore, ϕ is a strictly increasing and convex function satisfying ϕ(0) = 0, then
the function ϕ(t)
t , t > 0, is increasing and
Z t
t
ϕ(x)
ϕ(t) ≥ ψ(t) ≥
dx ≥ ϕ
(1.6)
,
x
2
t/2
for all t ≥ 0. Hence B ϕ = B ψ . By the convexity of ϕ, it is not difficult to see that
the Minkowski’s functional
0
f
ϕ
(1.7)
≤1 ,
||f ||b = inf k > 0 : Sϕ
k
defines a seminorm for B ϕ , which in this case is known as Luxemburg’s seminorm,
where
(1.8)
Sϕ (f ) := sup (1 − |z|2 )ϕ(|f (z)|).
z∈D
In fact, it can be shown that B ϕ is a Banach space with the norm
(1.9)
||f ||Bϕ = |f (0)| + ||f ||bϕ .
We observe that for any f ∈ B ϕ \ {0}, the following relation
f0
(1.10)
Sϕ
≤ 1,
||f ||Bϕ
holds.
The inequality above allow us to obtain
1
0
−1
(1.11)
|f (z)| ≤ ϕ
||f ||Bϕ ,
1 − |z|2
GENERALIZED COMPOSITION OPERATORS
3
for all f ∈ B ϕ and for all z ∈ D. This last inequality implies that the evaluation
function defined as Tz (f ) := f (z), where z ∈ D is fixed and f ∈ B ϕ is continuous
on B ϕ . In fact, let z ∈ D be fixed and any f ∈ B ϕ , we have
Z
|Tz (f )| = |f (z)| ≤ |f (0)| +
|f 0 (s)||ds|
[0,z]
≤
(1.12)
1
Z
−1
1+
ϕ
0
1
1 − |z|2 t2
dz ||f ||Bϕ .
Form the definition of Luxemburg seminorm and the expression (1.10), we have
Sϕ (f 0 ) ≤ 1 ⇐⇒ ||f ||Bϕ ≤ 1,
(1.13)
for any f ∈ B ϕ .
As an easy consequence of (1.10), we have that the Bloch-Orlicz space is isometrically equal to µ-Bloch space when
1
(1.14)
,
µ(z) = −1
1
ϕ ( 1−|z|
2)
with z ∈ D. Thus, for any f ∈ B ϕ , we have
||f ||Bϕ = sup µ(z)|f 0 (z)|.
z∈D
Denote by Z the class of all f ∈ H(D) ∩ C(D) such that
|f (ei(θ+h) ) + f (ei(θ−h) ) − 2f (eiθ )|
< ∞,
h
where the supremum is taken over all eiθ ∈ ∂D and h > 0. From the Theorem of
Zygmund (see [5], Theroem 5.3) and the closed graph Theorem, we see that f ∈ Z
if and only if supz∈D (1 − |z|2 )|f 00 (z)| < ∞. It is easy to see that Z is a Banach
space under the norm || · ||Z , where
(1.15)
(1.16)
||f || = sup
||f ||Z = |f (0)| + |f 0 (0)| + sup(1 − |z|2 )|f 00 (z)|.
z∈D
From (1.16) it is easy to obtain
(1.17)
|f 0 (z) − f 0 (0)| ≤
1
1 + |z|
||f ||Z log
2
1 − |z|
f or f ∈ Z.
For some other information and operators on this space, see, for example, [8, 9, 10].
Inspired by the way as Bloch-Orlicz spaces were defined (see [12, 13]), we define
the Zygmund-Orlicz space, which is denoted by Z ϕ , is the class of all analytic
functions f in D such that
(1.18)
sup(1 − |z|2 )ϕ(λ|f 00 (z)|) ≤ ∞,
z∈D
for some λ > 0 depending on f . The same as the Bloch-Orlicz space, since ϕ is
convex, it is not difficult to see that the Minkowski functional
00 f
(1.19)
||f ||zϕ = inf{k > 0 : Sϕ
≤ 1},
k
defines a seminorm for Z ϕ . Furthermore, it can be shown that Z ϕ is a Banach
space with the norm
(1.20)
||f ||Z ϕ = |f (0)| + |f 0 (0)| + ||f ||zϕ .
CONGLI YANG1 , FANGWEI CHEN2 AND PENGCHENG WU3
4
It is easily obtained that the following useful lemmas.
Lemma 1.1. [18] If f ∈ Z, then
(1.21)
(1.22)
(i),
(ii),
|f (z)| ≤ ||f ||Z
e
|f (z)| ≤ log(
)||f ||Z
1 − |z|2
0
f or every z ∈ D,
f or every z ∈ D.
Also we can observe the following property:
Lemma 1.2. For any f ∈ Z ϕ \ {0}, the following relation
f 00
Sϕ
(1.23)
≤1
||f ||Z ϕ
holds.
Proof. The same way as the case of Bloch-Orlicz space, so the details are omitted
here.
Lemma 1.2 allows us to obtain that
(1.24)
00
−1
|f (z)| ≤ ϕ
1
1 − |z|2
||f ||Z ϕ ,
for all f ∈ Z ϕ and all z ∈ D.
From the definition of Luxemburg Seminorm and the expression (1.23), we have
(1.25)
Sϕ (f 00 ) ≤ 1 ⇐⇒ ||f ||Z ϕ ≤ 1.
Also, as an easy consequence of (1.23), we have that Zygmund-Orlicz space is
isometrically equal to µ-Zygmund space, where µ(z) = −1 1 1 with z ∈ D.
ϕ
1−|z|2
µ
More information about Z , see [11, 21, 22].
Specially, if ϕ is an N -function such that
Z 1
1
Iϕ (z) =
(1.26)
ϕ−1
dt,
1 − |z|2 t2
0
is bounded for all z ∈ D. Then we get Z ϕ ⊂ H ∞ , the space of all bounded analytic
functions on D. However, there exists N -functions for which Iϕ (z) is not a bounded
function, for instance, consider ϕ(t) = t log(1 + t) with t ≥ 0.
Let φ be an analytic self-map of D, then the composition operator on H(D) is
given by
(1.27)
Cφ f = f ◦ φ.
Composition operators acting on various spaces of analytic functions have been
the object for recent years. Especially, the problems of relating operator-theoretic
properties of Cφ to function-theoretic properties of φ are interesting and been widely
discussed. See the book of Cowen and Maccuer [4] and Shapiro [14] for discussions
of composition operators classical spaces of analytic functions.
Assume that g : D → C be a holomorphic map of the disk D, for f ∈ H(D), we
define a linear operator as follows
Z z
(Cφg f )(z) =
f 0 (φ(ξ))g(ξ)dξ,
z ∈ D.
0
The operator Cφg is called the generalized composition operator, when g = φ0 .
We see that this operator is essentially composition operator, since the difference
GENERALIZED COMPOSITION OPERATORS
5
Cφg − Cφ is constant. Therefore, Cφg is a generalization of the composition operator,
which was introduced in [8].
Recall that if X and Y are Banach spaces, and L : X → Y be a linear operator, then L is said to be compact if for every bounded sequence {xn } in X, the
sequence(L{xn }) has a convergent subsequence. The book [4] contains plenty of information on this topic. By the standard arguments (see, for example, Proposition
3.11 in [4] ), the following lemma follows.
Lemma 1.3. Let ϕ : [0, ∞) → [0, ∞) be an N -function, g ∈ H(D) and φ be an
analytic self-map of D. Let X = Z, Y = B ϕ or Z ϕ . Then Cφg : X → Y is compact
if and only if Cφg : X → Y is bounded and for any bounded sequence (fk )k∈N in
X which converges to Zero uniformly on D as k → ∞, we have ||Cφg fk ||Y → 0 as
k → ∞.
Some characterization of the boundedness and compactness of the composition
operator, as well as Volterra type operator, on Bloch-Orlicz-type space and Zygmund space can be found in [2, 7, 15, 17, 19, 20]. In [8], the boundedness and
compactness of the generalized composition operator on Zygmund space and Bloch
type spaces are characterized by Songxiao Li and Stevo Stevi´c.
In this paper, we devote to investigating the boundedness and compactness
of generalized composition operators between Zygmound-Orlicz type spaces and
Bloch-Orlicz type spaces. The paper is organized as follows. In section 2 we give
the necessary and sufficient conditions for the boundedness and compactness of the
operator Cφg : Z → B ϕ . Section 3 we obtain the necessary and sufficient conditions
for the boundedness and compactness of the operator Cφg on Zygmund type spaces.
Throughout this paper, we use the letter C to denote a generic positive constant
that can change its value at each occurrence. The notation a ≤ b means that there
is a positive constant C such that a ≤ Cb. If a ≤ b and b ≤ a hold, then one says
that a b.
2. the boundedness and compactness of Cφg : Z → B ϕ
Now, we are ready to state and prove the main results in this section.
Theorem 2.1. Let ϕ : [0, ∞) → [0, ∞) be an N -function, g ∈ H(D) and φ be an
analytic self-map of D. Then Cφg : Z → B ϕ is bounded if and only if
(2.1)
L := sup ϕ−1
z∈D
1
1 − |z|2
−1
|g(z)| log
e
1 − |φ(z)|2
< ∞.
Proof. Suppose that (2.1) holds. For arbitrary z ∈ D and f ∈ Z, by Lemma 1.1,
we have the following estimate.
CONGLI YANG1 , FANGWEI CHEN2 AND PENGCHENG WU3
6
Sϕ
(Cφg f )0 (z)
!
2
= sup(1 − |z| )ϕ
L||f ||Z
z∈D
|f 0 (φ(z))||g(z)|
L||f ||Z
|f 0 (φ(z))|
1
)
≤ sup(1 − |z| )ϕ ϕ (
e
2
1 − |z| log( 1−|φ(z)|
2 )||f ||Z
z∈D
1
)
≤ sup(1 − |z|2 )ϕ ϕ−1 (
1 − |z|2
z∈D
≤ 1.
!
−1
2
Where we use the Lemma 1.1 (the relation (ii)) in the last inequality. From here,
we can conclude that ||Cφg ||ϕ ≤ L||f ||Z . Since Cφg f (0) = 0, it follows that the
generalized composition operators Cφg : Z → B ϕ is bounded.
Now, suppose that there exists a constant C > 0 such that
||Cφg ||ϕ ≤ C||f ||Z
f or all f ∈ Z.
Let
"
h(z) = (z − 1)
(2.2)
#
e
1 + log
1−z
2
log
−1
+1 ,
and put
(2.3)
fa (z) =
h(az)
a
e
1 − |a|2
,
√
for any a ∈ D such that 1/ 2 < |a| < 1. Then we have
2 −1
e
e
0
log
(2.4)
fa (z) = log
,
1 − az
1 − |a|2
and
fa00 (z)
2a
=
1 − az
e
log
1 − az
log
e
1 − |a|2
−1
.
Which implies that
|fa00 (z)|
2
≤
1 − |z|
e
C + log
1 − |a|
log
e
1 − |a|2
−1
≤
2
,
1 − |z|
√
for 1/ 2 < |a| < 1 and sup1/√2<|a|<1 ||fa ||Z < ∞. Therefore we have
(2.5)
||Cφg fa ||ϕ ≤ C||fa ||Z ≤ L < ∞.
It follows that
Sϕ
(Cϕg fa )0 (z)
L
That is
2
(1 − |z| )ϕ
≤ 1.
(Cϕg fa )0 (z)
L
≤ 1,
for any a ∈ D. But, on the other hand,
|g(z)| 0
1
ϕ
|fa (φ(z))| ≤
,
L
1 − |z|2
GENERALIZED COMPOSITION OPERATORS
then
|g(z)| 0
|fa (φ(z))| ≤ ϕ−1
L
1
1 − |z|2
7
.
So we have
−1
1
|g(z)||fa0 (φ(z))| ≤ L,
1 − |z|2
for all a ∈ D. In particular for a = φ(z), we have
−1
e
1
|g(z)| log
≤ L.
ϕ−1
1 − |z|2
1 − |φ(z)|2
ϕ−1
This conclude the proof of the theorem.
Theorem 2.2. Let ϕ : [0, ∞) → [0, ∞) be an N -function, g ∈ H(D) and φ be an
analytic self-map of D. Then Cφg : Z → B ϕ is compact if and only if Cφg : Z → B ϕ
is bounded and
e
(2.6)
lim µ(z)|g(z)| log
= 0.
1 − |φ(z)|2
|φ(z)|→1
Proof. First assume that Cφg : Z → B ϕ be bounded and (2.6) holds. By the
boundedness of Cϕg with f (z) = z, we see that
L := sup µ(z)|g(z)| < ∞.
(2.7)
z∈D
Let {fn } be a sequence in Z such that supn∈N ||fn ||Z ≤ K and fn → 0 uniformly
on compact subsets of D as n → ∞. Then by Lemma 1.3, it is suffices to show
that ||Cφg fn ||µ → 0 as n → ∞. By (2.6), we have that for every ε > 0, there is a
constant r ∈ (0, 1), such that r < |φ(z| ≤ 1, which implies
e
ε
µ(z)|g(z)| log
< ,
1 − |φ(z)|2
k
for any z ∈ D. From here, we have that
µ(z)|(Cφg fn )0 (z)| = µ|g(z)||fn0 (φ(z))|
e
||fn ||Z ≤ ε,
≤ µ(z)|g(z)| log
1 − |φ(z)|2
wherer < |φ(z)| < 1.
On the other hand, let Ω = {ω ∈ D : |ω| ≤ r}, by the cauchy estimate, if
{fn }n∈N is a sequence converging to zero on compact subsets of D, then the sequence {fn0 }n∈N also converges to zero on compact subsets of D as n → ∞. In
particular, since Ω is compact, it follows that lim sup |fn0 (ω)| = 0. Using these
n→∞ ω∈Ω
facts and letting n → ∞, we obtain
sup µ(z)|(Cφg fn )0 (z)| =
|φ(z)|≤r
sup µ(z)|g(z)|fn0 (z)|
|φ(z)|≤r
≤ L sup |fn0 (ω)| → 0.
as n → ∞.
ω∈Ω
Since (Cφg fn )(0) → 0 as n → ∞, for given ε > 0, there exists an n ∈ N , such that
||Cφg fn ||µ = |Cφg fn (0)| + sup µ(z)|(Cφg fn )0 (z)| ≤ 3ε,
z∈D
whenever n ≥ N , which means that
Cφg
: Z → B ϕ is a compact operator.
CONGLI YANG1 , FANGWEI CHEN2 AND PENGCHENG WU3
8
To prove the convese, suppose that there exists an ε0 > 0 such that
e
sup µ(z)|g(z)| log
≥ ε0 ,
1 − |φ(z)|2
|φ(z)|≥r
for any r ∈ (0, 1). Given a sequence of real number {rn } ⊂ (0, 1) such that rn → 1
as n → ∞. We can find a sequence {zn } ⊂ D such that |φ(zn )| ≥ rn and
1
e
≥ ε0 ,
µ(zn )|g(zn )| log
2
1 − |ω|
2
where ωn = φ(zn ), if necessary, we may suppose that |ωn | → 1 as n → ∞.
Note that fa , defined by (2.3), converges to zero uniformly on compact subset
of D as |a| → 1 and
e
,
f or a ∈ D \ {0}.
fa0 (a) = log
1 − |a|2
Now, we choose functions {fn }n∈N defined by
"
#
2
−1
wn z − 1
e
e
fn (z) =
+ 1 log
.
1 + log
w
1 − wn z
1 − |wn |2
From the proof of Theorem 2.1, we see that
sup ||f ||Z ≤ C.
(2.8)
n∈N
Moreover, we can see that {fn } converges to zero uniformly on compact subsets of
D, and satisfying
(2.9)
||Cφg fn ||µ
≥
µ(zn )|g(zn )||fn0 (wn )|
= µ(zn )|g(zn )| log
e
1 − |wn |2
≥
1
ε0 > 0.
2
Therefore, Cφg : Z → B ϕ is not a compact operator. This completes the proof of
the theorem.
3. The boundedness and compactness of Cφg : Z → Z ϕ
In this section, we characterize the boundedness and compactness of the operator
Cφg on Zygmund-Orlicz type spaces.
Theorem 3.1. Let ϕ : [0, ∞) → [0, ∞) be an N -function, g ∈ H(D) and φ be an
analytic self-map of D. Then Cφg : Z → Z ϕ is bounded if and only if
|φ0 (z)||g(z)|
< ∞,
1
2)
z∈D ϕ−1
(1
−
|φ(z)|
2
1−|z|
e
0
|g (z)| log 1−|φ(z)|
2
k2 = sup
< ∞.
1
2
z∈D ϕ−1
1−|z|2 (1 − |φ(z)| )
k1 = sup
(3.1)
Proof. Suppose first that (3.1) holds. For each f ∈ Z, we have the following
estimate.
GENERALIZED COMPOSITION OPERATORS
(Cφg f )00 (z)
!
9
|f 00 (φ(z))||g(z)| + |f 0 (φ(z))||g 0 (z)|
Sϕ
≤ sup(1 − |z| )ϕ
||f ||Z
C||f ||Z
z∈D
 


1
1
ϕ−1 1−|φ(z)|
ϕ−1 1−|φ(z)|
|f 00 (φ(z))||φ0 (z)||g(z)|
|f 0 (φ(z))||g 0 (z)|
2
2
+

= sup(1 − |z|2 )ϕ 
1
1
z∈D
ϕ−1 1−|z|
C||f ||Z
ϕ−1 1−|z|
C||f ||Z
2
2
!
1
1
2
00
0
k1 ϕ−1 ( 1−|z|
k2 ϕ−1 ( 1−|z|
2 )(1 − |φ(z)| )|f (φ(z))|
2 )|f (φ(z))|
2
≤ sup(1 − |z| )ϕ
+
e
C||f ||Z
log( 1−|φ(z)|
2 )C||f ||Z
z∈D
k1 C1 + k2 C2 −1
1
= sup(1 − |z|2 )ϕ
ϕ
C
1
−
|z|2
z∈D
1
≤ sup(1 − |z|2 )ϕ ϕ−1
1 − |z|2
z∈D
= 1.
2
Where C1 and C2 are constants, such that k1 C1 + k2 C2 ≤ C. Here we use the
relation (1.24) and (1.22) in the last seconded inequality. Now, we can conclude
that
||Cφg f ||Z φ ≤ C||f ||Z ,
(3.2)
and the generalized composition operators Cφg : Z → Z is bounded.
Now assume that Cφg : Z → Z is bounded, i.e., there exists a constant C such that
||Cφg f ||Z φ ≤ C||f ||Z for all f ∈ Z. Taking the functions f (z) = z and f (z) = z 2 ,
respectively, we obtain
0
|g (z)|
2
(1 − |z| )ϕ
≤ 1.
C
Which implies
(3.3)
sup
z∈D
|g 0 (z)|
< ∞,
ϕ−1
1
1−|z|2
and
2
(1 − (|z| ))ϕ
Then
|φ0 (z)||g(z)| + |φ(z)||g 0 (z)|
C(1 − |z|2 )
≤ 1.
|φ0 (z)||g(z)| + |φ(z)||g 0 (z)|
≤ C.
1
ϕ−1 1−|z|
2
So we obtain
sup
z∈D
|φ0 (z)||g(z)| + |φ(z)||g 0 (z)|
≤ ∞.
1
ϕ−1 1−|z|
2
Using these facts and the boundedness of the function φ(z) we get
(3.4)
sup
z∈D
|φ0 (z)||g(z)|
< ∞.
1
ϕ−1 1−|z|
2
CONGLI YANG1 , FANGWEI CHEN2 AND PENGCHENG WU3
10
Let h(z) = (z − 1)
1 + log
e
1−z
2
+ 1 and put fa (z) =
h(az)
a
log
e
1−|a|2
−1
for
a ∈ D such that |a| > 1/2. We have

(1 − |λ|2 )ϕ 
|g 0 (λ)| log
e
1−|φ(z)|2
|φ(λ)|
− 2|φ0 (λ)||g(λ)| 1−|φ(z)|
2
C||Cφg ||Z→Z ϕ

 ≤ 1.
Then

|g 0 (λ)| log
e
1−|φ(z)|2
|φ(λ)|
− 2|φ0 (λ)||g(λ)| 1−|φ(z)|
2
C||Cφg ||Z→Z ϕ


 ≤ ϕ−1
1
1 − |λ|2
,
e
|g 0 (λ)| log 1−|φ(λ)|
2
|φ0 (λ)||g(λ)||φ(λ)|
−2
≤ C||Cφg ||Z→Z ϕ .
1
1
−1
2)
ϕ−1 1−|λ|
ϕ
(1
−
|φ(λ)|
2
1−|λ|2
So we obtain
(3.5)
e
|g 0 (λ)| log 1−|φ(λ)|
2
|φ0 (λ)||g(λ)||φ(λ)|
≤ C||Cφg ||Z→Z ϕ + 2
.
1
1
ϕ−1 1−|λ|
ϕ−1 1−|λ|
(1 − |φ(λ)|2 )
2
2
Set
(3.6)
ha (z) =
h(az)
a
log
1
1 − |a|2
−1
Z
−
z
log
0
1
dw,
1 − aw
for a ∈ D, such that |a| > 1/2. Then,
2 −1
1
1
e
0
ha (z) = log
(3.7)
log
,
− log
2
1 − az
1 − |a|
1 − az
and
(3.8)
h00a (z) =
2a
1 − az
log
1
1 − az
log
1
1 − |a|2
−1
−
a
.
1 − az
Similar to the case of fa , we have ha ∈ Z and M1 = sup1/2<|a|<1 ||ha ||Z < ∞.
a
From this and by the facts that h0a (a) = 0, h00a (a) = (1−|a|
2 ) , it follows that
C||Cφg ||Z→Z ϕ ≥
(3.9)
|φ0 (λ)||g(λ)||φ(λ)|
.
1
ϕ−1 1−|λ|
(1 − |φ(λ)|2 )
2
From (3.9) we have
|φ0 (λ)||g(λ)||φ(λ)|
|φ0 (λ)||g(λ)||φ(λ)|
<2
1
1
2
|φ(λ)|>1/2 ϕ−1
ϕ−1 1−|λ|
(1 − |φ(λ)|2 )
2
1−|λ|2 (1 − |φ(λ)| )
sup
≤
(3.10)
sup
|φ(λ)|>1/2
C||Cφg ||Z→Z ϕ < ∞.
By (3.4) we see that
(3.11)
sup
|φ(λ)|≤1/2
ϕ−1
4 |φ0 (λ)||g(λ)|
|φ0 (λ)||g(λ)|
< ∞.
≤ sup
1
1
2
|φ(λ)|≤1/2 3 ϕ−1
1−|λ|2 (1 − |φ(λ)| )
1−|λ|2
GENERALIZED COMPOSITION OPERATORS
11
From (3.10) and (3.11) we obtain the first inequality of (3.1). Similarly, from (3.3)
and (3.5) the second inequality in (3.1) follows as desired.
Theorem 3.2. Let ϕ : [0, ∞) → [0, ∞) be an N -function, g ∈ H(D) and φ be an
analytic self-map of D. Then Cφg : Z → Z ϕ is compact if and only if Cφg : Z → Z ϕ
is bounded and
|φ0 (z)||g(z)|
lim
= 0,
1
|φ(z)|→1 ϕ−1
2
1−|z|2 (1 − |φ(z)| )
e
|g 0 (z)| log 1−|φ(z)|
2
= 0.
1
|φ(z)|→1
ϕ−1 1−|z|
2
(3.12)
lim
Proof. Suppose that Cφg : Z → Z ϕ be compact. It is clear that Cφg : Z → Z is
bounded. Let {Zn }n∈N be a sequence in D such that |φ(zn )| → 1 as n → ∞ and
(fn )n∈N be defined by
"
#
−1
φ(zn )z − 1
e
e
(3.13)
.
1 + log
log
fn (z) =
1 − |φ(zn )|2
φ(zn )
1 − φ(zn )z
Then supn∈N ||fn ||Z < ∞, and fn → 0 uniformly on compact subsets of D as
n → ∞. Since Cφg : Z → Z ϕ is compact, it gives limn→∞ ||Cφg fn ||Z ϕ = 0. Note
that
e
fn0 (φ(zn )) = log
,
1 − |φ(zn )|2
fn00 (φ(zn )) =
2φ(zn )
.
1 − |φ(zn )|2
We have
(3.14)
||Cφg fn ||Z ϕ ≥
e
|g 0 (zn )| log 1−|φ(z
2
|φ0 (zn )||g(zn )||φ(zn )|
n )| ,
−
1
1
ϕ−1 1−|zn |2 (1 − |φ(zn )|2 )
ϕ−1 1−|zn |2
and consequently,
(3.15)
e
|g 0 (zn )| log 1−|φ(z
2
|φ0 (zn )||g(zn )||φ(zn )|
n )| .
= lim
lim
1
1
|φ(zn )|→1 ϕ−1
|φ(z
)|→1
2
−1
n
ϕ
1−|zn |2 (1 − |φ(zn )| )
1−|zn |2
If one of these two limits exists, set
(3.16)
hn (z) =
h(φ(zn )z)
φ(zn )
log
e
1 − |φ(zn )|2
−1
− log
e
1 − |φ(zn )|2
−2 Z
0
z
log3
e
1 − φ(zn )w
then, h0n (φ(zn )) = 0, supn∈N ||hn ||Z ≤ C and hn converges to 0 uniformly on
compact subsets of D as n → ∞. Since Cφg : Z → Z ϕ is compact, we have
lim ||Cφg hn ||Z ϕ = 0.
n→∞
On the other hand,
|φ0 (zn )||g(zn )||φ(zn )|
≤ ||Cφg hn ||Z ϕ .
ϕ−1 1−|z1 n |2 (1 − |φ(zn )|2 )
dw,
CONGLI YANG1 , FANGWEI CHEN2 AND PENGCHENG WU3
12
Hence
|φ0 (zn )||g(zn )||φ(zn )|
= 0.
n→∞ −1
1
2
ϕ
1−|zn |2 (1 − |φ(zn )| )
lim
Therefore
|φ0 (zn )||g(zn )||φ(zn )|
|φ0 (zn )||g(zn )||φ(zn )|
= lim
= 0.
1
1
k→∞ −1
|φ(zn )|→1 ϕ−1
2
2
ϕ
1−|zn |2 (1 − |φ(zn )| )
1−|zn |2 (1 − |φ(zn )| )
lim
This together with (3.15) imply
e
|g 0 (zn )| log 1−|φ(z
2
n )| = 0.
lim
1
|φ(zn )|→1
−1
ϕ
1−|zn |2
(3.17)
The implication follows from the last two equalities.
Conversely, assume that Cφg : Z → Z ϕ be bounded and (3.12) holds. From the
proof of Theorem 3.1 we have
C1 = sup
z∈D
C2 = sup
(3.18)
z∈D
|g 0 (z)|
< ∞,
ϕ−1
1
1−|z|2
|φ0 (z)||g 0 (z)|
< ∞.
1
ϕ−1 1−|z|
2
On the other hand, from (3.12) we have that for every ε > 0, there is a δ ∈ (0, 1),
such that
e
|g 0 (z)| log 1−|φ(z)|
2
|φ0 (z)||g(z)|
< ε,
(3.19)
< ε and
ϕ−1 1−|z1 n |2 (1 − |φ(zn )|2 )
ϕ−1 1−|z1 n |2
when δ < |φ(z)| < 1.
Assume that {fn }n∈N be a sequence in Z such that supn∈N ||fn ||Z ≤ 1 and fn
converges to 0 uniformly on compact subsets of D as n → ∞. Let k = {z ∈ D :
|φ(z) ≤ δ|}. Then by (3.18) and (3.19) it follows that


g
00
0
0
0
00
|(C
f
)
(z)|
n
(φ(z))||g
(z)|
(φ(z))||φ
(z)||g(z)|
+
|f
|f
φ

= sup  n
n
||Cφg fn ||Z ϕ = sup
1
1
−1
z∈D ϕ−1
z∈D
ϕ
1−|z|2
1−|z|2
!
|fn00 (φ(z))||φ0 (z)||g(z)| + |fn0 (φ(z))||g 0 (z)|
≤ sup
1
ϕ−1 ( 1−|z|
z∈K
2)


0
00
0
0
(φ(z))||g
(z)|
|f
(φ(z))||φ
(z)||g(z)|
+
|f

n
+ sup  n
1
z∈D\K
ϕ−1 1−|z|
2
≤ C2 sup |fn00 (φ(z))| + C1 sup |fn0 (φ(z))| + C sup
z∈K
+ C sup
z∈D\K
z∈K
z∈D\K
e
|g 0 (z)| log 1−|φ(z)|
2
||fn ||Z
1
ϕ−1 1−|zn |2
≤ C2 sup |f 00 (w)| + C1 sup |fn0 (w)| + 2CεL.
|w|≤δ
|w|≤δ
ϕ−1
|φ0 (z)||g(z)|
||fn ||Z
1
(1 − |φ(zn )|2 )
2
1−|zn |
GENERALIZED COMPOSITION OPERATORS
13
So we obtain
(3.20)
||Cφg fn ||Z ϕ
(3.21)
=
|fn0 (φ(0))||f (0)|
|(Cφg fn )00 (z)|
+ sup
1
z∈D ϕ−1
1−|z|2
≤ C2 sup |fn00 (w)| + C1 sup |fn0 (w)| + 2Cε||fn ||Z + |fn0 (φ(0))||g(0)|.
|w|≤δ
|w|≤δ
By the Cauchy estimate, if {fn }n∈N is a sequence converging to zero on compact
subsets of D, then the sequence {fn }n∈N and {fn00 }n∈N also converges to zero on
compact subsets of D as n → ∞. In particular, since K is compact it follows that
limn→∞ supw∈K |fn0 (w)| = 0 and lim supw |f 00 (w)| = 0. Using these facts and
n→∞
letting n → ∞ in the last inequality, we obtain that
(3.22)
lim ||Cφg ||Z ϕ ≤ 2CεL.
n→∞
Since ε is an arbitrary positive number it follows that the last limit is equal to zero.
Employing Lemma 1.3 the implication follows.
Conflict of Interests: The authors declare that there is no conflict of interests
regarding the publication of this paper.
Acknowledgements: We would like to thank Professor Shengjian Wu for helpful conversations. Part of the work was done during the author stay at Peking
University, in the spring of 2014. We also thank the referee for some very valuable suggestions and comments that significantly improved the presentation of the
paper.
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1, 3. School of Mathematics and Computer Science, Guizhou Normal University,
Guiyang, Guizhou 550001, People’s Republic of China
E-mail address: [email protected],[email protected]
2. Department of Mathematics and Statistics, Guizhou University of Finance and
Economics, Guiyang, Guizhou 550004, People’s Republic of China.
E-mail address: [email protected]