R E S E A R C H
Open Access
Difference of composition operators on
weighted Bergman spaces over the half-plane
Maocai Wang
1*and Changbao Pang
2**Correspondence:
[email protected]; [email protected] 1School of Computer, China
University of Geosciences, Wuhan, 430074, China
2School of Mathematics and
Statistics, Wuhan University, Wuhan, 430072, China
Abstract
Recently, the bounded, compact and Hilbert-Schmidt difference of composition operators on the Bergman spaces over the half-plane are characterized in (Choeet al. in Trans. Am. Math. Soc., 2016, in press). Motivated by this, we give a sufficient condition when two composition operatorsCϕandCψare in the same path
component under the operator norm topology and show that there is no cancellation property for the compactness of double difference of composition operators. More precisely, we show that ifCϕ1,Cϕ2, andCϕ3are distinct and bounded, then (Cϕ1–Cϕ2) – (Cϕ3–Cϕ1) is compact if and only if bothCϕ1–Cϕ2andCϕ1–Cϕ3are compact on weighted Bergman spaces over the half-plane. Moreover, we prove the strong continuity of composition operators semigroup induced by a one-parameter semigroup of holomorphic self-maps of half-plane.
MSC: Primary 47B33; secondary 30H20; 32A35
Keywords: Bergman space; composition operator; Hilbert-Schmidt; semigroup
1 Introduction
Let+be the upper half of the complex plane, that is,+:={z∈C:Imz> }, and letS(+)
be the set of all holomorphic self-maps of+. Forϕ∈S(+), the composition operator
Cϕis defined by
Cϕf =f◦ϕ
for functionsf holomorphic on+. It is clear thatCϕ maps the space of holomorphic
functions on+into itself. Our purpose in this paper is to study composition operators
acting on the weighted Bergman spaces over+. Forα> –, let
dAα(z) :=cα(Imz)αdA(z),
wherecα=
α(α+)
π is a normalizing constant andAis the area measure on
+. The weighted
Bergman spaceA
α(+) consists of holomorphic functionsf on+such that the norm
f:=
+
f(z)dAα(z)
is finite. It is well known thatA
α(+) is a Hilbert space with the inner product
f,g=
+f(z)g(z)dAα
forf,g∈A
α(+).
An extensive study on the theory of composition operators has been established dur-ing the past four decades on various settdur-ings. We refer to [–] for various aspects on the theory of composition operators acting on holomorphic function spaces. With the ba-sic questions such as boundedness and compactness settled on some symmetric regions [], it is natural to look at the topological structure of the composition operators under the operator norm topology, and this topic is one of continuing interests in the theory of composition operators. Berkson [] focused attention on topological structure with his isolation results onHp(D), where D is the unit disk of the complex plane, and <p<∞. Especially, we mention that Choe-Hosokawa-Koo [] studied the topological structure of the space of all composition operators under the Hilbert-Schmidt norm topology and gave a characterization of components and some sufficient conditions for isolation or noniso-lation. In relation to the study of the topological structure, the difference or the linear sum of composition operators in various settings has been a very active topic [–].
Recently, composition operators on upper half-plane have received more attention; for instance, refer to [–]. Especially, Elliott and Wynn [] characterized bounded com-position operators and showed that there is no compact comcom-position operator onA
α(+).
Choe-Koo-Smith [] studied the bounded and compact difference of composition opera-tors onAα(+). They also obtained conditions under which the difference of composition
operators is Hilbert-Schmidt.
In this paper, we proceed along this line to give a sufficient condition when the compo-sition operators Cϕ andCψ are in the same path component under the operator norm
topology. Moreover, we show that the cancellation of double difference cannot occur onAα(+). More precisely, for distinct and boundedCϕ, Cϕ, and Cϕ, the difference
(Cϕ–Cϕ) – (Cϕ–Cϕ) is compact onA
α(+) if and only if bothCϕ–CϕandCϕ–Cϕ
are compact. We also study the linear sum of composition operators induced by some special classes of holomorphic self-maps. In addition, we prove the strong continuity of composition operators semigroups induced by one-parameter semigroups of holomor-phic self-maps of+. Due to the unboundedness of the domain, some special techniques are needed.
In Section , we recall some basic facts to be used in later sections. In Section ., we prove our sufficient condition for the path component of composition operators. In Sec-tion ., we prove that there is no cancellaSec-tion property for the compactness of double difference of composition operators. The continuity of composition operators semigroup is proved in Section ..
In the rest of the paper,Cwill denote a positive constant, the exact value of which will vary from one appearance to the next. We use the notationXYorYXfor nonneg-ative quantitiesXandYto meanX≤CY for some inessential constantC> . Similarly, we say thatX≈Yif bothXYandYXhold.
2 Preliminaries
In this section, we give some notation and well-known results onA
α(+). Recall that for a
bounded set into a relatively compact set. Due to the metric topology ofX,Tis compact if and only if the image of every bounded sequence has a convergent subsequence.
The following lemma gives a convenient compact criterion for a linear combination of composition operators acting onA
α(+).
Lemma . Let T be a linear combination of composition operators and assume that T is bounded on A
α(+).Then T is compact on Aα(+)if and only if Tfn→in Aα(+)for any bounded sequence{fn}in Aα(+)satisfying fn→uniformly on compact subsets of+.
A proof can be found in [], Proposition ., for composition operators on a Hardy space over the unit disk, and it can be easily modified for composition operators onA
α(+).
The pseudo-hyperbolic distance is defined as follows:
σ(z,w) :=z–w
z–w
, z,w∈+.
Note thatσis invariant under dilation and horizontal translation. We know from [] that
σ(z,w) < , z,w∈+.
Givenϕ,ψ∈S(+), we put
σ(z) :=σϕ(z),ψ(z).
Forz∈+and <δ< , letEδ(z) denote the pseudo-hyperbolic disk centered atzwith
radiusδ. We may check by an elementary calculation thatEδ(z) is actually a Euclidean disk
centered atx+i+–δδy, of radius
δ
–δy, wherex=Rezandy=Imz.
We will often use the following submean value type inequality:
f(z)≤ C (Imz)α+
Eδ(z)
f(z)dAα(z), z∈+ (.)
for allf ∈A
α(+) and some constantC=C(α,δ); see [] for more details. In particular, we
have
f(z)≤ C (Imz)α+f
, z∈+ (.)
forf∈A
α(+).
Given α> –, it follows from (.) that each point evaluation is a continuous linear functional onAα(+). Thus, for eachz∈+, there exists a unique reproducing kernel Kz(α)∈Aα(+) that has the reproducing property
f(z) =
+f(w)K
(α)
z (w)dAα(w)
forf∈A
α(+). The explicit formula ofK
(α)
z is given as
K(α)
z (w) =
i w–z
α+
wherei= –. Notice that
Kz(α)=
Kz(α)(z) =
Imz
α+
.
Thus, the normalized reproducing kernel K
(α)
z
K(α)
z
converges to uniformly on compact
sub-sets of+ asz→∂+. Here,+:=+∪ {∞}, and∂+ is the boundary of+. In the sequel, we usually writeKz=Kz(α)andkz= K
(α)
z
K(α)
z
for simplicity.
Before introducing angular derivatives in the half-plane setting, we first clarify the no-tion of nontangential limits at boundary points of+. Of course, those at a finite
bound-ary point refer to the standard notion. Meanwhile, those at∞ ∈∂+refer to those as-sociated with nontangential approach regions,> , consisting of allz∈Csuch that
Imz>|Rez|. For a functionϕ:+→+andx∈∂+, we writeϕ(x) =η(possibly∞) if ϕhas a nontangential limitη, that is,∠limz→xϕ(z) =η. For a holomorphic self-mapψ of D, the angular derivative ofψexists atζ∈∂Dif there isη∈∂Dsuch that a nontangential limit of η–ζψ–z(z) exists as a finite complex value asz→ζ. Now we introduce the notion of angular derivatives on+via the Caley transformation
γ(z) =i +z
–z, z∈D,
which conformally maps D onto+. Note that a regionis contained in a nontangential
approach region in D if and only if γ() is contained in some nontangential approach region in+.
Forϕ∈S(+), let
ϕγ =γ–◦ϕ◦γ. (.)
We say thatϕhas finite angular derivative atx∈∂+ifϕ
γ has a finite angular derivative
atx:=γ–(x), whereγ–(w) =w–i
w+i,w∈+.
The following Julia-Carathéodory theorem for the upper half-plan is proved in [].
Proposition . Forϕ∈S(+),the following statements are equivalent:
(a) ϕ(∞) =∞,andϕ(∞)exists; (b) supz∈+ImImϕ(zz)<∞;
(c) lim supz→∞ImImϕz(z)<∞.
Moreover,the quantities in(b)and(c)are equal toϕ(∞).
Elliott and Wynn [] gave the following characterization of bounded composition op-erators by means of angular derivatives.
Theorem . Letα> –andϕ∈S(+).Then Cϕis bounded on Aα(+)if and only ifϕ has a finite angular derivativeϕ(∞) =λ∈(,∞).Moreover,Cϕ=λ
α+ .
Forz∈+, letτzbe the function on+defined by
τz(w) :=
In [], the authors showed thatτs
z∈Aα(+) if and only if s>α+ . In this case,
τzs= C
(Imz)s–α–, (.)
whereCis a constant. Thus, τzs
τzs→ uniformly on compact subsets of
+asz→∂+.
3 Main results
3.1 Path component of composition operators
In this subsection, we give a sufficient condition for composition operatorsCϕandCψto
be in the same path component under the operator norm topology. To this end, we recall the definition of Hilbert-Schmidt operators onA
α(+). A bounded linear operatorT on
a separable HilbertHis Hilbert-Schmidt if
THS:= ∞
j=
TejH=
∞
j,n=
Tej,enH
<∞
for any (or some) orthonormal basis{en} ofH. As is well known, the value of the sum above does not depend on the choice of orthonormal basis{en}ofH, andT ≤ THS. We know that every Hilbert-Schmidt operator is compact; see [] for more details. Let
C(A
α(+)) be the space of all bounded composition operators onAα(+) endowed with
norm topology. The following theorem is due to Choe-Koo-Smith [], Theorem ..
Theorem . Letα> –andϕ,ψ∈S(+).Then Cϕ–Cψ is Hilbert-Schmidt on Aα(+)if and only if
+
Imϕ(z)+
Imψ(z) α+
σ(z)dAα(z) <∞,
whereσ(z) :=|ϕ(z)–ψ(z)
ϕ(z)–ψ(z)|.
WriteCϕ∼CψifCϕandCψare in the same path component ofC(Aα(+)). Fort∈[, ],
we putϕt = ( –t)ϕ+tψ. It is easy to see thatϕt∈S(+). In order to give a sufficient condition of path connected of two compositions, we need the following lemma.
Lemma . Letα> – andϕ,ψ ∈S(+). Assume that Cϕ–Cψ is Hilbert-Schmidt on
Aα(+).Letϕt= ( –t)ϕ+tψ for t∈[, ].Then Cϕs–Cϕt is Hilbert-Schmidt on Aα(+) for any s,t∈[, ].
Proof SinceCϕs–CϕtHS≤ Cϕ–CϕsHS+Cϕ–CϕtHS, it is enough to prove thatCϕ– Cϕtis Hilbert-Schmidt onAα(+) for everyt∈[, ].
Fixt∈[, ] and putσt(z) :=σ(ϕ(z),ϕt(z)). Then
σt(z) =
ϕϕ((zz) –) –ϕϕt(z) t(z)
= t(ϕ(z) –ψ(z))
ϕ(z) –ψ(z) – [( –t)ϕ(z) – ( –t)ψ(z)]
≤ tσ(z)
– ( –t)σ(z)<σ(z) (.)
for allz∈+.
Now, note that Imξ ≤Imz +Imw for anyξon the line segment connectingzandw. Thus, we have
Imϕt(z)
≤
Imϕ(z)+
Imψ(z) (.)
for all ≤t≤. By (.) and (.) we have
+
σt(z) (Imϕt(z))α+
dAα(z)
+
Imϕ(z)+
Imψ(z)
α+
σ(z)dAα(z).
Similarly,
+
σt(z)
(Imϕ(z))α+dAα(z)
+
Imϕ(z)+
Imψ(z)
α+
σ(z)dAα(z).
So, we conclude by Theorem . thatCϕ–Cϕtis Hilbert-Schmidt onAα(+).
Now, we give our first main result on sufficient conditions of path connected.
Theorem . Letα> –andϕ,ψ∈S(+).Assume that both C
ϕand Cψ are bounded on A
α(+)and Cϕ–Cψis Hilbert-Schmidt on Aα(+).Then Cϕ∼Cψ.
Proof Suppose thatCϕ–Cψis Hilbert-Schmidt onAα(+). Since
Cϕt ≤ Cϕt–Cϕ+Cϕ ≤ Cϕt–CϕHS+Cϕ, t∈[, ],
by Lemma . we obtain thatCϕt∈C(Aα(+)). We will show thatt∈[, ]→Cϕt is a
continuous path inC(A
α(+)). SinceCϕt–Cϕs ≤ Cϕt–CϕsHS, it is sufficient to prove
that
lim
t→sCϕt–CϕsHS= .
Givent,s∈[, ],t=s, putσt,s(z) =σ(ϕt(z),ϕs(z)),t,s∈[, ] andz∈+. From (.) we
have
σt,s(z)≤σt,(z) +σs,(z)≤σ(z). (.)
From [], Theorem ., we know that
Cϕt–CϕsHS≈
+
Imϕt(z)+
Imϕs(z) α+
Put
t,s(z) =
Imϕt(z)
+
Imϕs(z)
α+
σt,s(z), z∈+
for short. By (.) and (.) we have
t,s(z)
Imϕ(z)+
Imψ(z)
α+
σ(z) =,(z), z∈+.
SinceCϕ–Cψ is a Hilbert-Schmidt operator,, is integrable by Theorem .. Again
σt,s(z)→ in+ ast→s, and we conclude by the dominated convergence theorem
that
lim
t→sCϕt–CϕsHS= ,
which completes the proof.
Remark It follows immediately from the theorem above that
• GivenCϕ∈C(Aα(+)), the setN(ϕ) :={Cψ:Cϕ–CψHS<∞}is the a path-connected set inC(A
α(+))containingCϕ.
• The setN(ϕ)is ‘convex’ in the sense that ifCψ∈N(ϕ), then{Cϕt}t∈[,]∈N(ϕ).
3.2 Cancellation properties of composition operators
In this subsection, we study cancellation properties of composition operators onA
α(+).
The following lemma is cited from [], Corollary ..
Lemma . Letα> –andϕ,ψ∈S(+).Assume that both Cϕand Cψ are bounded on
Aα(+).Then Cϕ–Cψis compact on Aα(+)if and only if
lim
z→∂+
Imz
Imϕ(z)+
Imz
Imψ(z)
σ(z) = . (.)
Herelimz→∂+g(z) = means thatsup+\K|g| → as the compact setK⊂+expands
to the whole of+or, equivalently, thatg(z)→ asImz→+andg(z)→ as|z| → ∞.
The following theorem shows that there is no cancellation property for the compactness of double difference of composition operators.
Theorem . Letα> –,a,b∈C\{},and a+b= .Assume thatϕj∈S(+)andϕjare
distinct for j= , , and each Cϕj is bounded on Aα(+).Then T:=a(Cϕ–Cϕ) +b(Cϕ–
Cϕ)is compact on A
α(+)if and only if both Cϕ –Cϕ and Cϕ –Cϕ are compact on
A
α(+).
Proof The sufficiency is trivial, and we only prove the necessity. Assume thatT=a(Cϕ–
Cϕ) +b(Cϕ–Cϕ) is compact. We will get a contradiction if eitherCϕ–CϕorCϕ–Cϕ
is not compact. Without loss of generality, we assume thatCϕ–Cϕis not compact. Let
σj,s(z) :=|
ϕj(z)–ϕs(z)
SinceCϕ –Cϕis not compact, by (.) there are> and a sequence{zn} ⊂+such
thatzn→∂+(n→ ∞) and
Imzn
Imϕ(zn)
+ Imzn
Imϕ(zn)
σ(zn)≥. (.)
For eachj= , , , sinceCϕj is bounded onA
α(+), we haveC∗ϕjKz ≤ CϕjKzfor any z∈+, whereC∗
ϕj is the adjoint ofCϕj. Due toCϕ∗jKz=Kϕj(z), this is equivalent to
Imz
Imϕj(z)≤C for allz∈
+,
whereCis some positive constant. Duo to this and the factσ< , taking a smaller if
necessary, formula (.) gives that
σ,(zn) (.)
and
M(zn) :=
Imzn
Imϕ(zn)
or
M(zn) :=
Imzn
Imϕ(zn)
.
Without loss of generality, we assume thatM(zn)(the proof for the caseM(zn)
is similar); thus,Imzn≈Imϕ(zn). Forj= , , , we definegj,n(z) :=τϕkj(zn)(z) = (z–ϕj(zn))k
with k>α+ . Form,j= , , ,m=j, letxm j,n=
ϕm(zn)–ϕm(zn)
ϕj(zn)–ϕm(zn). Notice that
xmj,n=ϕm(zn) –ϕm(zn) ϕj(zn) –ϕm(zn)
≤ϕm(zn) –ϕj(zn)
ϕj(zn) –ϕm(zn)
+ϕj(zn) –ϕm(zn) ϕj(zn) –ϕm(zn)
=σm,j(zn) + < (.)
and
xmj,n=ϕm(zn) –ϕm(zn) ϕj(zn) –ϕm(zn)
=ϕj(zn) –ϕj(zn) ϕj(zn) –ϕm(zn)
ϕm(zn) –ϕm(zn) ϕj(zn) –ϕj(zn)
< ϕm(zn) –ϕm(zn) ϕj(zn) –ϕj(zn)
By (.) andM(zn)we have
|g,n(z)|
g,n
=C[Imϕ(zn)]
k–α–
|z–ϕ(zn)|k
(Imzn)k–α–
|z–ϕ(zn)|k
. (.)
Thus,
g,n
g,n →
uniformly on compact subsets of+asn→ ∞.
By (.) we have
Tgj,n(zn)
Tgj,n
(Imzn)α+, j= , , . (.)
Thus, using the fact thatM(zn)≥, we obtain
Tg,n
g,n
Imϕ(zn)
k–α–
(Imzn)α+Tg,n(zn)
|a| –a+b
a
ϕ(zn) –ϕ(zn)
ϕ(zn) –ϕ(zn) k
+b
a
ϕ(zn) –ϕ(zn)
ϕ(zn) –ϕ(zn) k
–a+b
a
x,nk–b
a
x,nk
– a+b
a
y,nk– b
a
y,nk
.
SinceT is compact, we have Tg,n
g,n → (n→ ∞). Therefore, at least one ofy
,nandy,n does not converge to .
Supposey,n→ buty,n. Then Imϕ(zn)
Imϕ(zn) ≥C> for some subsequence, which
we still denote by {zn}. SinceImzn≈Imϕ(zn), M(zn) := ImImzϕ(nzn) ≥C> . Therefore,
Imϕ(zn)Imzn. Similarly to (.), we have
g,n
g,n →
uniformly on compact subsets of+asn→ ∞.
From (.) we have
x,ny,n Imzn
Imϕ(zn)≈ y,n,
which impliesx
,n→ asn→ ∞. By (.) andM(zn)≥Cwe obtain
Tg,n
g,n
Imϕ(zn)
k–α–
(Imzn)α+Tg,n(zn)
|b| –a+b
b
ϕ(zn) –ϕ(zn)
ϕ(zn) –ϕ(zn)
k +a
b
ϕ(zn) –ϕ(zn)
ϕ(zn) –ϕ(zn)
k .
SinceTis compact, we have Tg,n
g,n → asn→ ∞. Since
ϕ(zn)–ϕ(zn)
ϕ(zn)–ϕ(zn)=x
,n→, we have
+a
b
ϕ(zn) –ϕ(zn)
ϕ(zn) –ϕ(zn) k
Note that this holds for any k>α+ , which impliesa+b= . This is a contradiction to the assumptiona+b= . Therefore,y
,ndoes not converge to .
Taking a subsequence of {zn} if necessary, we have M(zn)≥C > , which implies
Imϕ(zn)Imzn. Similarly to (.), we have g,n
g,n →
uniformly on compact subsets of+asn→ ∞.
Notice that
xj,n=ϕ(zn) –ϕ(zn) ϕj(zn) –ϕ(zn)
= ϕ(zn) –ϕ(zn) ϕ(zn) –ϕ(zn) +ϕj(zn) –ϕ(zn)
=
+ϕϕj(zn)–ϕ(zn)
(zn)–ϕ(zn)
and
σ,(zn) =
ϕ(zn) –ϕ(zn)
ϕ(zn) –ϕ(zn)
=ϕ(zn) –ϕ(zn) ϕ(zn) –ϕ(zn)
ϕ(zn) –ϕ(zn)
ϕ(zn) –ϕ(zn)
=ϕ(zn) –ϕ(zn) ϕ(zn) –ϕ(zn)
x,n.
By formula (.) we have
x,n< .
Therefore, iflim supn→∞|x,n|= , then there exists some subsequence{znl}such that
ϕ(znl) –ϕ(znl)
ϕ(znl) –ϕ(znl)
→.
Thenσ,(znl)→, which contradicts (.). Hence, we havelim supn→∞|x
,n|. From (.) and the fact thatM(zn)≥we have
Imϕ(zn) k
Tg,n(zn)
Tg,n
g,n .
Thus, we get
←Tg,n
g,n
a+abϕ(zn) –ϕ(zn)
ϕ(zn) –ϕ(zn) k
+ b
a+b
ϕ(zn) –ϕ(zn)
ϕ(zn) –ϕ(zn) k
–
= a
a+b
x,nk+ b
a+b
x,nk–
for all k>α+ . Then
a a+b
x,nk+ b
a+b
x,nk– →.
Sincelim supn→∞|x,n|, we obtain ab+b– = . Namely,a= , which contradicts our assumption. Therefore, the compactness ofT =a(Cϕ–Cϕ) +b(Cϕ–Cϕ) implies that
bothCϕ–CϕandCϕ–Cϕare compact. The proof is complete.
The following theorem involves the lower estimate of the essential norm for a linear sum of some special composition operators onA
α(+). Here the essential norm of an operator
means the distance to the space of compact operators. To state the result, we need to introduce some notation. Forϕ∈S(+), ifCϕis bounded onAα(+), thenϕ(∞) =∞and
<ϕ(∞) <∞by Theorem .. Let
D[ϕ,∞] :=ϕ(∞),ϕ(∞)∈ {∞} ×(,∞).
Forε> , let
Rε,∞:=z∈+:Imz= –εRez.
Note thatRε,∞is a nontangential curve having∞as the end point. Let
S+c:=ϕ∈S+:Imz≥cImϕ(z) for eachz∈+, wherecis a constant.
The following lemma can be found in [], Lemma ..
Lemma . Letϕ,ψ∈S(+).Assume that C
ϕ,Cψ are bounded on Aα(+).Then
lim
ε→+z→∞lim,z∈Rε,∞
ϕ(z) –ϕ(z)
ϕ(z) –ψ(z)=
, ifD[ϕ,∞] =D[ψ,∞], , otherwise.
Now we give a lower estimate of the essential norm of a linear sum of composition op-erators induced by symbols inS(+)c.
Theorem . Letα> –andϕj∈S(+)c,j= , , . . . ,N.For each j= , , . . . ,N,assume
that Cϕjis bounded on A
α(+).Then we have the inequality
N
j= ajCϕj
e
≥
(u,v)∈{∞}×(,∞)
D[ϕj,∞]=(u,v) aj
for any a,a, . . . ,aN∈C.
Proof Leta,a, . . . ,aN∈C. SinceCϕ∗jKz=Kϕj(z), we have
N
j= ajCϕj
e =
N
j= ajC∗ϕj
e
≥ lim
ε→+z→∞lim,z∈R
ε,∞
N
j= ajCϕ∗jkz
Meanwhile, we have
N
j= ajCϕ∗jkz
=
j,k
ajak
Imz
ϕj(z) –ϕk(z)
α+
j,k
ajak
ϕj(z) –ϕj(z)
ϕj(z) –ϕk(z)
α+
.
Using Lemma ., we have
lim
ε→+z→∞lim,z∈Rε,∞
ϕj(z) –ϕj(z)
ϕj(z) –ϕk(z)=
, D[ϕj,∞] =D[ϕk,∞], , otherwise.
Thus, we obtain
N
j= ajCϕj
e
≥
D[ϕj,∞]=D[ϕk,∞] ajak,
which is the same as the desired inequality. The proof is complete.
As an immediate consequence, we have a necessary coefficient relation for the compact-ness of linear combination of composition operators.
Corollary . Letα> –andϕj∈S(+)c,j= , , . . . ,N.Assume that each Cϕjis bounded on A
α(+)and aj∈C.If N
j=ajCϕj is compact on A
α(+),then
D[ϕj,∞]=(u,v) aj= ,
(u,v)∈ {∞} ×(,∞).
Especially, we have the following useful corollary.
Corollary . Letα> –andϕ,ψ∈S(+)c.Assume that Cϕ,Cψare bounded on Aα(+) and a,b∈C\{}.If aCϕ+bCψis compact on Aα(+),then the following statements hold:
(a) a+b= ; (b) ϕ(∞) =ψ(∞).
3.3 Composition operators induced by a one-parameter semigroup
In the last subsection, we consider the strong continuity of the composition operator semi-group induced by a one-parameter semisemi-group of holomorphic self-maps of+. We first
recall some definitions and notation.
A one-parameter semigroup of holomorphic self-maps of+is a family{ϕ
t}t≥⊂S(+)
satisfying
() ϕ(z) =zfor allz∈+;
() ϕt+s(z) =ϕt◦ϕs(z)for alls,t≥andz∈+;
() (t,z)→ϕt(z)is jointly continuous on[,∞)×+.
We know that the continuity of (t,z)→ϕt(z) on [,∞)×+is equivalent to the continuity oft→ϕt(z) for eachz∈+. By [] the holomorphic functionG:+→Cgiven by
G(z) =lim
t→
is the infinitesimal generator of{ϕt}, which characterizes{ϕt}uniquely and satisfies
∂ϕt(z) ∂t =G
ϕt(z)
, z∈+,t≥.
LetGbe the infinitesimal generator of the one-parameter semigroup{(ϕt)γ}t≥, where
(ϕt)γ=γ–◦ϕt◦γ, andγ(ζ) =i–+ζζ,γ–(w) =ww+–ii,ζ∈D,w∈+. From [] we also have
G(ϕt)γ
γ–(z)=∂(ϕt)γ(γ
–(z))
∂t , z∈ +,t≥.
From [] we obtain
Gγ–(z)= i
(z+i)G(z), z∈
+. (.)
Assume that{Cϕt}t≥is the bounded composition operator semigroup onAα(+) induced
by the one-parameter semigroup{ϕt}t≥⊂S(+). The linear operatorAdefined by
D(A) =
f ∈Aα
+:lim
t→ Cϕtf–f
t exists
,
and
Af =lim
t→ Cϕtf –f
t =
∂Cϕtf
∂t
t=
, f ∈D(A),
is the infinitesimal generator of the semigroup{Cϕt}t≥, whereD(A) is the domain ofA. If {Cϕt}t≥satisfies
lim
t→Cϕtf –f= , f ∈A
α
+,
then we say that{Cϕt}t≥is strongly continuous.
The following theorem gives some characterizations about the boundedness of{Cϕt}, t≥.
Theorem . Letα> –,and let{ϕt}t≥be a one-parameter semigroup with infinitesimal generator G,where{ϕt}t≥⊂S(+).Then the following are equivalent:
(a) For eacht> ,Cϕtis bounded onAα(+). (b) For somet> ,Cϕtis bounded onAα(+).
(c) The nontangential limitδ:=∠limz→∞G(zz) exists finitely.
Moreover,if one of these assertions holds,thenCϕt=e
(α+)δt
for each t> .
Proof (a)⇔(b). Since (a) implies (b), we only prove the converse. We now assume that there existst> such thatCϕt is bounded onAα(+). Thenϕt(∞) =∞, andϕt(∞)
exists finitely. From [], Lemma ., we know thatϕt(∞) =∞if and only if (ϕt)γ() = .
From [], Theorem , we know that all members of the semigroup{(ϕt)γ}t≥have
we have (ϕt)γ() = ((ϕt)γ())
t
t. For eacht> , since
∠lim
w→
– (ϕt)γ(w)
–w =∠w→lim
[ – (ϕt)γ(w)][ +w]
[ + (ϕt)γ(w)][ –w]
=∠lim
w→
γ(w) γ((ϕt)γ)(w)
=∠lim
z→∞
z
ϕt(z),
we have
(ϕt)γ() =ϕt(∞). (.)
Then we obtain
ϕt(∞) =(ϕt)
γ()
t t =ϕ
t(∞)
t t,
which implies thatϕt(∞) exists finitely. SoCϕt is bounded onAα(+) for eacht> by
Theorem ..
(a)⇔(c). For eacht> ,Cϕtis bounded onAα(+) if and only ifϕt(∞) =∞,ϕt(∞) exists finitely by Proposition . and Theorem .. From [], Lemma ., we have (ϕt)γ() = if
and only ifϕt(∞) =∞. Also, by (.),Cϕtis bounded onAα(+) if and only if (ϕt)γ() = ,
(ϕt)γ() exists finitely, which is equivalent to the finite existence of∠limw→G–(ww),w∈D,
by [], Theorem , whereGis the infinitesimal generator of the one-parameter semigroup
{(ϕt)γ}t≥. By (.) we have
∠lim
w→
G(w)
–w=∠w→lim
iG(γ(w))
(γ(w) +i)( –w)=∠w→lim
G(γ(w)) γ(w) +i
=∠lim
z→∞
G(z)
z+i =∠z→∞lim
G(z)
z , (.)
which implies (a)⇔(c).
For eacht> , if one of the conditions holds, we obtainϕt(∞) =eδtfrom (.), (.), and [], Theorem , whereδ=∠limw→∞Gw(w),w∈+. By Theorem . we have
Cϕt=e
(α+)δt
,
which completes the proof.
Next, we prove the strong continuity of composition operator semigroups induced by one-parameter semigroups of holomorphic self-maps of the upper half-plane.
Theorem . Letα> –,and let{ϕt}t≥⊂S(+)be a one-parameter semigroup on+. For each t≥,assume that Cϕtis bounded on Aα(+).Then{Cϕt}t≥is strongly continuous on Aα(+).
Proof Due to the denseness ofSpan{kz:z∈+}inAα(+), it is sufficient to prove
lim
By the property of reproducing kernel ofA
α(+) and the Cauchy-Schwarz inequality we
have
f(z)=f,Kz≤ fKz. (.)
Because
Cϕtkz–kz
+C
ϕtkz+kz
= C
ϕtkz
+k
z
,
we have
Cϕtkz–kz
≤ +C
ϕt
–Cϕtkz+kz .
By Theorem . we haveCϕt
=ϕ
t(∞)α+. Thus, we obtain
Cϕtkz–kz≤
+ϕt(∞)α+–Cϕtkz+kz. (.)
Takingf=kz◦ϕt+kzin (.), we obtain
Cϕtkz+kz
≥|Kz◦ϕt(z) +Kz|
Kz
. (.)
Combining (.) and (.), we obtain
Cϕtkz–kz≤
+ϕt(∞)α+–|Kz◦ϕt(z) +Kz
|
Kz .
Since ( + (ϕt(∞))α+)→ and|Kz◦ϕt(z)+Kz|
Kz → ast→, we obtain
lim
t→Cϕtkz–kz= , z∈ +.
The proof is complete.
As an application, we have the following corollary by using standard arguments as in [], Theorem .. We omit its proof to the reader.
Corollary . Letα> –,and let{ϕt}t≥⊂S(+)be a one-parameter semigroup on+. Assume that each Cϕtis bounded on Aα(+).If G is the infinitesimal generator of{ϕt},then
the infinitesimal generator A of{Cϕt}has the domain of definition
D(A) =f ∈Aα+:Gf∈Aα+
and is given by
4 Conclusion
This paper studied the path component of composition operators spaces and the continu-ity of composition operator semigroups. In addition, the paper showed that the cancella-tion of double difference cannot occur in our settings. The results obtained extend some classical results on the unit disk to the upper half-plane. Due to the unboundedness of the half-plane, some special new techniques are used to overcome obstacles.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final version of this paper.
Acknowledgements
The authors thank the anonymous referees for their comments and suggestions, which improved our final manuscript. In addition, the first author is supported by Natural Science Foundation of China (41571403), and the second author is supported by Natural Science Foundation of China (11271293).
Received: 8 April 2016 Accepted: 16 August 2016
References
1. Choe, B, Koo, H, Smith, W: Difference of composition operators over the half-plane. Trans. Am. Math. Soc. (2016, in press)
2. Cowen, C, MacCluer, B: Composition Operators on Spaces of Analytic Functions. CRC Press, New York (1995) 3. Shapiro, J: Composition Operators and Classical Function Theory. Springer, New York (1993)
4. Zhu, K: Operator Theory in Function Spaces, 2nd edn. Mathematical Surveys and Monographs, vol. 138. Am. Math. Soc., Providence (2007)
5. Berkson, E: Composition operators isolated in the uniform topology. Proc. Am. Math. Soc.81, 230-232 (1981) 6. Choe, B, Hosokawa, T, Koo, H: Hilbert-Schmidt differences of composition operators on the Bergman space. Math. Z.
269, 751-775 (2011)
7. Choe, B, Koo, H, Wang, M, Yang, J: Compact linear combinations of composition operators induced by linear fractional maps. Math. Z.280, 807-824 (2015)
8. Koo, H, Wang, M: Joint Carleson measure and the difference of composition operators onApα(Bn). J. Math. Anal. Appl. 419, 1119-1142 (2014)
9. Moorhouse, J: Compact differences of composition operators. J. Funct. Anal.219, 70-92 (2005)
10. Stevi´c, S: Composition operators from the Hardy space to thenth weighted-type space on the unit disk and the half-plane. Appl. Math. Comput.215, 3950-3955 (2010)
11. Stevi´c, S: Composition operators from the weighted Bergman space to thenth weighted-type space on the upper half-plane. Appl. Math. Comput.217, 3379-3384 (2010)
12. Stevi´c, S, Sharma, A: Weighted composition operators between growth spaces of the upper half-plane. Util. Math.84, 265-272 (2011)
13. Elliiott, S, Wynn, A: Composition operators on weighted Bergman spaces of a half-plane. Proc. Edinb. Math. Soc.54, 373-379 (2011)
14. Berkson, E, Porta, H: Semigroups of analytic functions and composition operators. Mich. Math. J.25, 101-115 (1978) 15. Arvanitidis, A: Semigroup of composition operators on Hardy spaces of the half-plane. Acta Sci. Math.81, 293-308
(2015)
16. Contreras, M, Díaz-Madrigal, S, Pommerenke, C: Fixed points and boundary behaviour of the Koenigs function. Ann. Acad. Sci. Fenn., Math.29, 471-488 (2004)
