Difference of composition operators on weighted Bergman spaces over the half-plane

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1 Wang and Pang Journal of Inequalities and Applications (016) 016:06 DOI /s RESEARCH Open Access Difference of composition operators on weighted Bergman spaces over the half-plane Maocai Wang1* and Changbao Pang* * Correspondence: mcwang@cug.edu.cn; cbpang@whu.edu.cn 1 School of Computer, China University of Geosciences, Wuhan, , China School of Mathematics and Statistics, Wuhan University, Wuhan, 43007, China Abstract Recently, the bounded, compact and Hilbert-Schmidt difference of composition operators on the Bergman spaces over the half-plane are characterized in (Choe et al. in Trans. Am. Math. Soc., 016, in press). Motivated by this, we give a sufficient condition when two composition operators Cϕ and Cψ are 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 if Cϕ1, Cϕ, and Cϕ3 are distinct and bounded, then (Cϕ1 Cϕ ) (Cϕ3 Cϕ1 ) is compact if and only if both Cϕ1 Cϕ and Cϕ1 Cϕ3 are 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 30H0; 3A35 Keywords: Bergman space; composition operator; Hilbert-Schmidt; semigroup 1 Introduction Let + be the upper half of the complex plane, that is, + := {z C : Im z > }, and let S( + ) be the set of all holomorphic self-maps of +. For ϕ S( + ), the composition operator Cϕ is defined by Cϕ f = f ϕ for functions f holomorphic on +. It is clear that Cϕ 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 +. For α >, let daα (z) := cα (Im z)α da(z), α where cα = (α+ ) is a normalizing constant and A is the area measure on +. The weighted π Bergman space A α ( + ) consists of holomorphic functions f on + such that the norm f := + f (z) daα (z) 016 Wang and Pang. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Wangand Pang Journal of Inequalities and Applications (016) 016:06 Page of 16 is finite. It is well known that A α ( + ) is a Hilbert space with the inner product f, g = f (z)g(z) da α + for f, g A α ( + ). An extensive study on the theory of composition operators has been established during the past four decades on various settings. We refer to [ 4] for various aspects on the theory of composition operators acting on holomorphic function spaces. With the basic 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 [5] focused attention on topological structure with his isolation results on H p (D), where D is the unit disk of the complex plane, and 0 < p <. Especially,we mention that Choe-Hosokawa-Koo [6] 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 nonisolation. 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 [7 9]. Recently, composition operators on upper half-plane have received more attention; for instance, refer to [10 1]. Especially, Elliott and Wynn [13] characterizedboundedcomposition operators and showed that there is no compact composition operator on A α ( + ). Choe-Koo-Smith [1] studied the bounded and compact difference of composition operators on A α ( + ). 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 composition operators C ϕ and C ψ are in the same path component under the operator norm topology. Moreover, we show that the cancellation of double difference cannot occur on A α ( + ). More precisely, for distinct and bounded C ϕ1, C ϕ,andc ϕ3, the difference (C ϕ1 C ϕ ) (C ϕ3 C ϕ1 ) is compact on A α ( + )ifandonlyifbothc ϕ1 C ϕ and C ϕ3 C ϕ1 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 holomorphic self-maps of +. Due to the unboundedness of the domain, some special techniques are needed. In Section, werecallsomebasicfactstobeusedinlatersections.insection3.1, we prove our sufficient condition for the path component of composition operators. In Section 3., we prove that there is no cancellation property for the compactness of double difference of composition operators. The continuity of composition operators semigroup is proved in Section 3.3. In the rest of the paper, C will denote a positive constant, the exact value of which will vary from one appearance to the next. We use the notation X Y or Y X for nonnegative quantities X and Y to mean X CY for some inessential constant C > 0. Similarly, we say that X Y if both X Y and Y X hold. Preliminaries In this section, we give some notation and well-known results on A α ( + ). Recall that for a Hilbert space X, a bounded linear operator T : X X is said to be compact if T maps every

3 Wangand Pang Journal of Inequalities and Applications (016) 016:06 Page 3 of 16 bounded set into a relatively compact set. Due to the metric topology of X, T is 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 on A α ( + ). Lemma.1 Let T be a linear combination of composition operators and assume that T is bounded on A α ( + ). Then T is compact on A α ( + ) if and only if Tf n 0 in A α ( + ) for any bounded sequence {f n } in A α ( + ) satisfying f n 0 uniformly on compact subsets of +. Aproofcanbefoundin[], Proposition 3.11, for composition operators on a Hardy space over the unit disk, and it can be easily modified for composition operators on A α ( + ). The pseudo-hyperbolic distance is defined as follows: σ (z, w):= z w z w, z, w +. Note that σ is invariant under dilation and horizontal translation. We know from [1]that σ (z, w)<1, z, w +. Given ϕ, ψ S( + ), we put σ (z):=σ ( ϕ(z), ψ(z) ). For z + and 0 < δ <1,letE δ (z) denote the pseudo-hyperbolic disk centered at z with radius δ. We may check by an elementary calculation that E δ (z) is actually a Euclidean disk centered at x + i 1+δ y, of radius δ y,wherex = Re z and y = Im z. 1 δ 1 δ We will often use the following submean value type inequality: f (z) C f (z) daα (z), z + (.1) (Im z) α+ E δ (z) for all f A α ( + ) and some constant C = C(α, δ); see [1] for more details. In particular, we have f (z) C (Im z) f α+, z + (.) for f A α ( + ). Given α > 1, it follows from (.) that each point evaluation is a continuous linear functional on A α ( + ). Thus, for each z +, there exists a unique reproducing kernel K (α) z A α ( + ) that has the reproducing property f (z)= f (w)k z (α) (w) da α (w) + for f A α ( + ). The explicit formula of K (α) z ( ) i α+ K z (α) (w)=, w z is given as

4 Wangand Pang Journal of Inequalities and Applications (016) 016:06 Page 4 of 16 where i = 1.Noticethat K z (α) = K (α) z ( ) α+ 1 (z)=. Im z K (α) z Thus, the normalized reproducing kernel converges to 0 uniformly on compact subsets of + as z +. Here, + := + { },and + is the boundary of + K z (α).inthe sequel, we usually write K z = K z (α) and k z = K(α) z for simplicity. K z (α) Before introducing angular derivatives in the half-plane setting, we first clarify the notion of nontangential limits at boundary points of +.Ofcourse,thoseatafiniteboundary point refer to the standard notion. Meanwhile, those at + refer to those associated with nontangential approach regions ɛ, ɛ >0,consistingofallz C such that Im z > ɛ Re z.forafunctionϕ : + + and x +,wewriteϕ(x)=η (possibly )if ϕ has a nontangential limit η,thatis, lim z x ϕ(z)=η. For a holomorphic self-map ψ of D, the angular derivative of ψ exists at ζ D if there is η D such that a nontangential limit of η ψ(z) exists as a finite complex value as z ζ. Now we introduce the notion of ζ z angular derivatives on + via the Caley transformation γ (z)=i 1+z 1 z, z D, which conformally maps D onto +. Note that a region Ɣ is contained in a nontangential approach region in D if and only if γ (Ɣ) is contained in some nontangential approach region in +. For ϕ S( + ), let ϕ γ = γ 1 ϕ γ. (.3) We say that ϕ has finite angular derivative at x + if ϕ γ has a finite angular derivative at x := γ 1 (x), where γ 1 (w)= w i w+i, w +. The following Julia-Carathéodory theorem for the upper half-plan is proved in [13]. Proposition. For ϕ S( + ), the following statements are equivalent: (a) ϕ( )=, and ϕ ( ) exists; Im z (b) sup z + Im ϕ(z) < ; Im z (c) lim sup z Im ϕ(z) <. Moreover, the quantities in (b) and (c) are equal to ϕ ( ). Elliott and Wynn [13] gave the following characterization of bounded composition operators by means of angular derivatives. Theorem.3 Let α > 1and ϕ S( + ). Then C ϕ is bounded on A α ( + ) if and only if ϕ has a finite angular derivative ϕ ( )=λ (0, ). Moreover, C ϕ = λ α+. For z +,letτ z be the function on + defined by τ z (w):= 1 w z.

5 Wangand Pang Journal of Inequalities and Applications (016) 016:06 Page 5 of 16 In [1], the authors showed that τ s z A α ( + )ifandonlyifs > α +.Inthiscase, τ s C z =, (.4) (Im z) s α where C is a constant. Thus, τ s z τ s z 0 uniformly on compact subsets of + as z +. 3 Main results 3.1 Path component of composition operators In this subsection, we give a sufficient condition for composition operators C ϕ and C ψ to be in the same path component under the operator norm topology. To this end, we recall the definition of Hilbert-Schmidt operators on A α ( + ). A bounded linear operator T on a separable Hilbert H is Hilbert-Schmidt if T HS := Te j H = Tej, e n H < j=1 j,n=1 for any (or some) orthonormal basis {e n } of H. Asiswellknown,thevalueofthesum above does not depend on the choice of orthonormal basis {e n } of H, and T T HS. We know that every Hilbert-Schmidt operator is compact; see [6] for more details. Let C(A α ( + )) be the space of all bounded composition operators on A α ( + )endowedwith norm topology. The following theorem is due to Choe-Koo-Smith [1], Theorem 7.6. Theorem 3.1 Let α > 1and ϕ, ψ S( + ). Then C ϕ C ψ is Hilbert-Schmidt on A α ( + ) if and only if + [ ] 1 Im ϕ(z) + 1 α+ σ (z) da α (z)<, Im ψ(z) where σ (z):= ϕ(z) ψ(z) ϕ(z) ψ(z). Write C ϕ C ψ if C ϕ and C ψ are in the same path component of C(A α ( + )). For t [0, 1], we put ϕ t =(1 t)ϕ + tψ. Itiseasytoseethatϕ t S( + ). In order to give a sufficient condition of path connected of two compositions, we need the following lemma. Lemma 3. Let α > 1and ϕ, ψ S( + ). Assume that C ϕ C ψ is Hilbert-Schmidt on A α ( + ). Let ϕ t =(1 t)ϕ + tψ for t [0, 1]. Then C ϕs C ϕt is Hilbert-Schmidt on A α ( + ) for any s, t [0, 1]. Proof Since C ϕs C ϕt HS C ϕ C ϕs HS + C ϕ C ϕt HS,itisenoughtoprovethatC ϕ C ϕt is Hilbert-Schmidt on A α ( + )foreveryt [0, 1]. Fix t [0, 1] and put σ t (z):=σ(ϕ(z), ϕ t (z)). Then σ t (z) = ϕ(z) ϕ t (z) ϕ(z) ϕ t (z) = ϕ(z) [(1 t)ϕ(z)+tψ(z)] ϕ(z) [(1 t)ϕ(z)+tψ(z)]

6 Wangand Pang Journal of Inequalities and Applications (016) 016:06 Page 6 of 16 = t(ϕ(z) ψ(z)) ϕ(z) ψ(z) [(1 t)ϕ(z) (1 t)ψ(z)] tσ (z) < σ (z) (3.1) 1 (1 t)σ (z) for all z +. 1 Now, note that Im ξ 1 Im z + 1 for any ξ onthelinesegmentconnectingz and w.thus, Im w we have 1 Im ϕ t (z) 1 Im ϕ(z) + 1 Im ψ(z) (3.) for all 0 t 1. By (3.1)and(3.)wehave + σt (z) ( ) (Im ϕ t (z)) da 1 α(z) α+ + Im ϕ(z) + 1 α+ σ (z) da α (z). Im ψ(z) Similarly, + σt (z) ( ) (Im ϕ(z)) da 1 α(z) α+ + Im ϕ(z) + 1 α+ σ (z) da α (z). Im ψ(z) So,weconcludebyTheorem3.1 that C ϕ C ϕt is Hilbert-Schmidt on A α ( + ). Now, we give our first main result on sufficient conditions of path connected. Theorem 3.3 Let α > 1and ϕ, ψ S( + ). Assume that both C ϕ and C ψ are bounded on A α ( + ) and C ϕ C ψ is Hilbert-Schmidt on A α ( + ). Then C ϕ C ψ. Proof Suppose that C ϕ C ψ is Hilbert-Schmidt on A α ( + ). Since C ϕt C ϕt C ϕ + C ϕ C ϕt C ϕ HS + C ϕ, t [0, 1], by Lemma 3. we obtain that C ϕt C(A α ( + )). We will show that t [0, 1] C ϕt is a continuous path in C(A α ( + )). Since C ϕt C ϕs C ϕt C ϕs HS,itissufficienttoprove that lim t s C ϕt C ϕs HS =0. Given t, s [0, 1], t s, putσ t,s (z) =σ (ϕ t (z), ϕ s (z)), t, s [0, 1] and z +.From(3.1) we have σ t,s (z) σ t,0 (z)+σ s,0 (z) σ (z). (3.3) From [1], Theorem 7.5, we know that C ϕt C ϕs HS + [ ] 1 Im ϕ t (z) + 1 α+ σt,s Im ϕ s (z) (z) da α(z).

7 Wangand Pang Journal of Inequalities and Applications (016) 016:06 Page 7 of 16 Put ( t,s (z)= 1 Im ϕ t (z) + 1 Im ϕ s (z) ) α+ σ t,s (z), z + for short. By (3.)and(3.3)wehave ( ) 1 t,s (z) Im ϕ(z) + 1 α+ σ (z)= 0,1(z), z +. Im ψ(z) Since C ϕ C ψ is a Hilbert-Schmidt operator, 0,1 is integrable by Theorem 3.1. Again σ t,s (z) 0in + as t s, and we conclude by the dominated convergence theorem that lim t s C ϕt C ϕs HS =0, which completes the proof. Remark It follows immediately from the theorem above that Given C ϕ C(A α ( + )),thesetn(ϕ):={c ψ : C ϕ C ψ HS < } is the a path-connected set in C(A α ( + )) containing C ϕ. ThesetN(ϕ) is convex in the sense that if C ψ N(ϕ),then{C ϕt } t [0,1] N(ϕ). 3. Cancellation properties of composition operators In this subsection, we study cancellation properties of composition operators on A α ( + ). The following lemma is cited from [1], Corollary 4.7. Lemma 3.4 Let α > 1and ϕ, ψ S( + ). Assume that both C ϕ and C ψ are bounded on A α ( + ). Then C ϕ C ψ is compact on A α ( + ) if and only if [ Im z lim z + Im ϕ(z) + Im z ] σ (z)=0. (3.4) Im ψ(z) Here lim z + g(z)=0meansthatsup + \K g 0asthecompactsetK + expands to the whole of + or, equivalently, that g(z) 0asIm z 0 + and g(z) 0as z. The following theorem shows that there is no cancellation property for the compactness of double difference of composition operators. Theorem 3.5 Let α > 1,a, b C\{0}, and a + b 0.Assume that ϕ j S( + ) and ϕ j are distinct for j =1,,3and each C ϕj is bounded on A α ( + ). Then T := a(c ϕ C ϕ1 )+b(c ϕ3 C ϕ1 ) is compact on A α ( + ) ifandonlyifbothc ϕ C ϕ1 and C ϕ3 C ϕ1 are compact on A α ( + ). Proof The sufficiency is trivial, and we only prove the necessity. Assume that T = a(c ϕ C ϕ1 )+b(c ϕ3 C ϕ1 ) is compact. We will get a contradiction if either C ϕ C ϕ1 or C ϕ3 C ϕ1 is not compact. Without loss of generality, we assume that C ϕ C ϕ1 is not compact. Let σ j,s (z):= ϕ j(z) ϕ s (z) for j, s =1,,3ands j. ϕ j (z) ϕ s (z)

8 Wangand Pang Journal of Inequalities and Applications (016) 016:06 Page 8 of 16 Since C ϕ C ϕ1 is not compact, by (3.4) thereareɛ >0andasequence{z n } + such that z n + (n )and [ Im zn Im ϕ 1 (z n ) + Im z ] n σ 1 (z n ) ɛ. (3.5) Im ϕ (z n ) For each j =1,,3,sinceC ϕj is bounded on A α ( + ), we have Cϕ j K z C ϕj K z for any z +,wherecϕ j is the adjoint of C ϕj.duetocϕ j K z = K ϕj (z),thisisequivalentto Im z Im ϕ j (z) C for all z +, where C is some positive constant. Duo to this and the fact σ 1 < 1, taking a smaller ɛ if necessary, formula (3.5)givesthat σ 1, (z n ) ɛ (3.6) and or M 1 (z n ):= M (z n ):= Im z n Im ϕ 1 (z n ) ɛ Im z n ɛ. Im ϕ (z n ) Without loss of generality, we assume that M (z n ) ɛ (the proof for the case M 1 (z n ) ɛ is similar); thus, Im z n Im ϕ (z n ). For j = 1,,3, we define g j,n (z) :=τ k ϕ j (z n ) (z) = 1 (z ϕ j (z n )) k with k > α +.Form, j =1,,3,m j,letx m j,n = ϕ m(z n ) ϕ m (z n ) ϕ j (z n ) ϕ m (z n ).Noticethat x m ϕ m (z n ) ϕ m (z n ) j,n = ϕ j (z n ) ϕ m (z n ) ϕ m (z n ) ϕ j (z n ) ϕ j (z n ) ϕ m (z n ) + ϕ j (z n ) ϕ m (z n ) ϕ j (z n ) ϕ m (z n ) = σ m,j (z n )+1< (3.7) and x m j,n = ϕ m (z n ) ϕ m (z n ) ϕ j (z n ) ϕ m (z n ) [ ] = ϕ j (z n ) ϕ j (z n ) ϕm (z n ) ϕ m (z n ) ϕ j (z n ) ϕ m (z n ) ϕ j (z n ) ϕ j (z n ) < ϕ m(z n ) ϕ m (z n ) ϕ j (z n ) ϕ j (z n ) := y m j,n. (3.8)

9 Wangand Pang Journal of Inequalities and Applications (016) 016:06 Page 9 of 16 By (.4)andM (z n ) ɛ we have Thus, g,n (z) g,n = C[Im ϕ (z n )] k α z ϕ (z n ) k (Im z n) k α z ϕ (z n ) k. (3.9) g,n g,n 0 uniformly on compact subsets of + as n. By (.1)wehave Tg j,n (z n ) Tg j,n, j =1,,3. (3.10) (Im z n ) α+ Thus, using the fact that M (z n ) ɛ,weobtain Tg,n g,n ( Im ϕ (z n ) ) k α (Im zn ) α+ Tg,n (z n ) a 1 a + b ( ) ϕ (z n ) ϕ (z n ) k + b ( ) ϕ (z n ) ϕ (z n ) k a ϕ 1 (z n ) ϕ (z n ) a ϕ 3 (z n ) ϕ (z n ) [ 1 a + b a x k 1,n b ] a x k 3,n [ 1 a + b ( a y ) k b ] ) 1,n a( y k 3,n. Since T is compact, we have Tg,n g,n 0(n ). Therefore, at least one of y 1,n and y 3,n does not converge to 0. Suppose y 1,n 0buty 3,n 0. Then Im ϕ (z n ) Im ϕ 3 (z n C > 0 for some subsequence, which ) we still denote by {z n }.SinceIm z n Im ϕ (z n ), M 3 (z n ):= Im z n Im ϕ 3 (z n C > 0. Therefore, ) Im ϕ 3 (z n ) Im z n. Similarly to (3.9), we have g 3,n g 3,n 0 uniformly on compact subsets of + as n. From (3.8)wehave x 3 1,n y 3 1,n Im z n Im ϕ 1 (z n ) y 1,n, which implies x 3 1,n 0asn.By(3.10)andM 3(z n ) C we obtain Tg 3,n g 3,n ( Im ϕ 3 (z n ) ) k α (Im zn ) α+ Tg3,n (z n ) b 1 a + b b ( ϕ3 (z n ) ϕ 3 (z n ) ϕ 1 (z n ) ϕ 3 (z n ) ) k + a b ( ϕ3 (z n ) ϕ 3 (z n ) ϕ (z n ) ϕ 3 (z n ) ) k. Since T is compact, we have Tg 3,n g 3,n 0asn.Since ϕ 3(z n ) ϕ 3 (z n ) = ϕ 1 (z n ) ϕ 3 (z n ) x3 1,n 0, we have ( ) ϕ3 (z n ) ϕ 3 (z n ) k a b ϕ (z n ) ϕ 3 (z n )

10 Wangand Pang Journal of Inequalities and Applications (016) 016:06 Page 10 of 16 Note that this holds for any k > α +, which implies a + b = 0. This is a contradiction to the assumption a + b 0. Therefore, y 1,n does not converge to 0. Taking a subsequence of {z n } if necessary, we have M 1 (z n ) C > 0, which implies Im ϕ 1 (z n ) Im z n. Similarly to (3.9), we have g 1,n g 1,n 0 uniformly on compact subsets of + as n. Notice that x 1 j,n = ϕ 1(z n ) ϕ 1 (z n ) ϕ j (z n ) ϕ 1 (z n ) ϕ 1 (z n ) ϕ 1 (z n ) = ϕ 1 (z n ) ϕ 1 (z n )+ϕ j (z n ) ϕ 1 (z n ) = 1 1+ ϕ j(z n ) ϕ 1 (z n ) ϕ 1 (z n ) ϕ 1 (z n ) and σ 1, (z n )= ϕ 1 (z n ) ϕ (z n ) ϕ 1 (z n ) ϕ (z n ) = ϕ 1 (z n ) ϕ (z n ) ϕ 1 (z n ) ϕ 1 (z n ) ϕ 1 (z n ) ϕ 1 (z n ) ϕ 1 (z n ) ϕ (z n ) = ϕ 1 (z n ) ϕ (z n ) ϕ 1 (z n ) ϕ 1 (z n ) x 1.,n By formula (3.7)wehave x 1,n <. Therefore, if lim sup n x 1,n = 1, then there exists some subsequence {z n l } such that ϕ (z nl ) ϕ 1 (z nl ) ϕ 1 (z nl ) ϕ 1 (z nl ) 0. Then σ 1, (z nl ) 0, which contradicts (3.6). Hence, we have lim sup n x 1,n 1. From (3.10)andthefactthatM 1 (z n ) ɛ we have ( Im ϕ1 (z n ) ) k Tg1,n (z n ) Tg 1,n g 1,n. Thus, we get 0 Tg 1,n ( ) g 1,n a ϕ1 (z n ) ϕ 1 (z n ) k + b ( ) ϕ1 (z n ) ϕ 1 (z n ) k 1 a + b ϕ (z n ) ϕ 1 (z n ) a + b ϕ 3 (z n ) ϕ 1 (z n ) = a ( ) x 1 k b ( ) a + b,n + x 1 k a + b 3,n 1

11 Wangand Pang Journal of Inequalities and Applications (016) 016:06 Page 11 of 16 for all k > α +.Then a ( ) x 1 k b ( ) a + b,n + x 1 k a + b 3,n 1 0. Since lim sup n x 1,n 1, we obtain b 1=0.Namely,a = 0, which contradicts our a+b assumption. Therefore, the compactness of T = a(c ϕ C ϕ1 )+b(c ϕ3 C ϕ1 )impliesthat both C ϕ C ϕ1 and C ϕ3 C ϕ1 are 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 on A 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. For ϕ S( + ), if C ϕ is bounded on A α ( + ), then ϕ( )= and 0<ϕ ( )< by Theorem.3.Let D[ϕ, ]:= ( ϕ( ), ϕ ( ) ) { } (0, ). For ε >0,let R ε, := { z + : Im z = ε Re z }. Note that R ε, is a nontangential curve having as the end point. Let S ( +) c := { ϕ S ( +) : Im z c Im ϕ(z)foreachz +}, where c is a constant. The following lemma can be found in [1], Lemma 5.. Lemma 3.6 Let ϕ, ψ S( + ). Assume that C ϕ, C ψ are bounded on A α ( + ). Then ϕ(z) ϕ(z) lim lim z,z R ε, ϕ(z) ψ(z) = ε 0 + { 1, if D[ϕ, ]=D[ψ, ], 0, otherwise. Now we give a lower estimate of the essential norm of a linear sum of composition operators induced by symbols in S( + ) c. Theorem 3.7 Let α > 1and ϕ j S( + ) c, j =1,,...,N. For each j =1,,...,N, assume that C ϕj is bounded on A α ( + ). Then we have the inequality N a j C ϕj j=1 for any a 1, a,...,a N C. e (u,v) { } (0, ) D[ϕ j, ]=(u,v) a j Proof Let a 1, a,...,a N C.SinceC ϕ j K z = K ϕj (z),wehave N N a j C ϕj = a j Cϕ j lim j=1 e j=1 e lim ε 0 + z,z R ε, N a j Cϕ j k z. j=1

12 Wangand Pang Journal of Inequalities and Applications (016) 016:06 Page 1 of 16 Meanwhile, we have N a j Cϕ j k z = j=1 j,k a j a k ( Im z ϕ j (z) ϕ k (z) ) α+ j,k ( ) ϕj (z) ϕ j (z) α+ a j a k. ϕ j (z) ϕ k (z) Using Lemma 3.6,wehave ϕ j (z) ϕ j (z) lim lim z,z R ε, ϕ j (z) ϕ k (z) = ε 0 + Thus, we obtain N a j C ϕj j=1 e D[ϕ j, ]=D[ϕ k, ] { 1, D[ϕ j, ]=D[ϕ k, ], 0, otherwise. a j a k, which is the same as the desired inequality. The proof is complete. As an immediate consequence, we have a necessary coefficient relation for the compactness of linear combination of composition operators. Corollary 3.8 Let α > 1and ϕ j S( + ) c, j =1,,...,N. Assume that each C ϕj is bounded on A α ( + ) and a j C. If N j=1 a jc ϕj is compact on A α ( + ), then a j =0, D[ϕ j, ]=(u,v) (u, v) { } (0, ). Especially, we have the following useful corollary. Corollary 3.9 Let α > 1and ϕ, ψ S( + ) c. Assume that C ϕ, C ψ are bounded on A α ( + ) and a, b C\{0}. If ac ϕ + bc ψ is compact on A α ( + ), then the following statements hold: (a) a + b =0; (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 semigroup induced by a one-parameter semigroup of holomorphic self-maps of +.Wefirst recall some definitions and notation. A one-parameter semigroup of holomorphic self-maps of + is a family {ϕ t } t 0 S( + ) satisfying (1) ϕ 0 (z)=z for all z + ; () ϕ t+s (z)=ϕ t ϕ s (z) for all s, t 0 and z + ; (3) (t, z) ϕ t (z) is jointly continuous on [0, ) +. We know that the continuity of (t, z) ϕ t (z)on[0, ) + is equivalent to the continuity of t ϕ t (z)foreachz +.By[14] the holomorphic function G : + C given by G(z)=lim t 0 ϕ t (z) t

13 Wangand Pang Journal of Inequalities and Applications (016) 016:06 Page 13 of 16 is the infinitesimal generator of {ϕ t }, which characterizes {ϕ t } uniquely and satisfies ϕ t (z) t = G ( ϕ t (z) ), z +, t 0. Let G be the infinitesimal generator of the one-parameter semigroup {(ϕ t ) γ } t 0,where (ϕ t ) γ = γ 1 ϕ t γ,andγ(ζ)=i 1+ζ 1 ζ, γ 1 (w)= w i w+i, ζ D, w +.From[14]wealsohave G ( ( (ϕ t ) γ γ 1 (z) )) = (ϕ t) γ (γ 1 (z)), z +, t 0. t From [15]we obtain G ( γ 1 (z) ) = i (z + i) G(z), z +. (3.11) Assume that {C ϕt } t 0 is the bounded composition operator semigroup on A α ( + )induced by the one-parameter semigroup {ϕ t } t 0 S( + ). The linear operator A defined by { D(A)= f A α( + ) C ϕt f f : lim t 0 t } exists, and Af = lim t 0 C ϕt f f t = C ϕ t f t, t=0 f D(A), is the infinitesimal generator of the semigroup {C ϕt } t 0,whereD(A) is the domain of A.If {C ϕt } t 0 satisfies lim t 0 C ϕ t f f =0, f A ( α + ), then we say that {C ϕt } t 0 is strongly continuous. The following theorem gives some characterizations about the boundedness of {C ϕt }, t 0. Theorem 3.10 Let α > 1,and let {ϕ t } t 0 be a one-parameter semigroup with infinitesimal generator G, where {ϕ t } t 0 S( + ). Then the following are equivalent: (a) For each t >0, C ϕt is bounded on A α ( + ). (b) For some t >0, C ϕt is bounded on A α ( + ). G(z) (c) The nontangential limit δ := lim z exists finitely. z Moreover, if one of these assertions holds, then C ϕt = e (α+)δt for each t >0. Proof (a) (b).since(a)implies(b),weonlyprovetheconverse.wenowassumethat there exists t 0 >0suchthatC ϕt0 is bounded on A α ( + ). Then ϕ t0 ( ) =, andϕ t 0 ( ) exists finitely. From [15], Lemma.1, we know that ϕ t0 ( )= if and only if (ϕ t0 ) γ (1) = 1. From [16], Theorem 5, we know that all members of the semigroup {(ϕ t ) γ } t 0 have common boundary fixed points, that is, (ϕ t ) γ (1) = 1 for each t 0. Then, by [17], Theorem 1,

14 Wangand Pang Journal of Inequalities and Applications (016) 016:06 Page 14 of 16 we have (ϕ t ) γ (1) = ((ϕ t 0 ) γ (1)) t t 0.Foreacht >0,since we have 1 (ϕ t ) γ (w) [1 (ϕ t ) γ (w)][1 + w] lim = lim w 1 1 w w 1 [1 + (ϕ t ) γ (w)][1 w] = lim w 1 γ (w) γ ((ϕ t ) γ )(w) = lim z z ϕ t (z), (ϕ t ) γ (1) = ϕ t ( ). (3.1) Then we obtain ϕ t ( )=( (ϕ t0 ) γ (1)) t t 0 = ( ϕ t 0 ( ) ) t t 0, which implies that ϕ t ( ) existsfinitely.soc ϕ t is bounded on A α ( + )foreacht >0by Theorem.3. (a) (c). For each t >0,C ϕt is bounded on A α ( + )ifandonlyifϕ t ( )=, ϕ t ( )exists finitelybyproposition. and Theorem.3.From[15], Lemma.1, we have (ϕ t ) γ (1) = 1 if and only if ϕ t ( )=.Also,by(3.1), C ϕt is bounded on A α ( + )ifandonlyif(ϕ t ) γ (1) = 1, (ϕ t ) γ (1) exists finitely, which is equivalent to the finite existence of lim G(w) w 1 1 w, w D, by [17],Theorem1,where G is the infinitesimal generator of the one-parameter semigroup {(ϕ t ) γ } t 0.By(3.11)wehave G(w) lim w 1 1 w = lim ig(γ (w)) w 1 (γ (w)+i) (1 w) = lim G(γ (w)) w 1 γ (w)+i G(z) = lim z z + i = lim G(z) z z, (3.13) which implies (a) (c). For each t > 0, if one of the conditions holds, we obtain ϕ t ( )=eδt from (3.1), (3.13), G(w) and [17], Theorem 1, where δ = lim w w, w +.ByTheorem.3 we have C ϕt = 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 3.11 Let α > 1,and let {ϕ t } t 0 S( + ) be a one-parameter semigroup on +. For each t 0, assume that C ϕt is bounded on A α ( + ). Then {C ϕt } t 0 is strongly continuous on A α ( + ). Proof Due to the denseness of Span{k z : z + } in A α ( + ), it is sufficient to prove lim C ϕ t k z k z =0, z +. t 0

15 Wangand Pang Journal of Inequalities and Applications (016) 016:06 Page 15 of 16 By the property of reproducing kernel of A α ( + ) and the Cauchy-Schwarz inequality we have f (z) = f, K z f K z. (3.14) Because C ϕt k z k z + C ϕt k z + k z = ( C ϕt k z + k z ), we have C ϕt k z k z ( 1+ C ϕt ) C ϕt k z + k z. By Theorem.3 we have C ϕt = ϕ t ( )α+.thus,weobtain C ϕt k z k z ( 1+ ( ϕ t ( )) α+) Cϕt k z + k z. (3.15) Taking f = k z ϕ t + k z in (3.14), we obtain C ϕt k z + k z K z ϕ t (z)+ K z K z 4. (3.16) Combining (3.15)and(3.16), we obtain C ϕt k z k z ( 1+ ( ϕ t ( )) α+) K z ϕ t (z)+ K z K z 4. Since (1 + (ϕ t ( ))α+ ) 4and K z ϕ t (z)+ K z K z 4 4ast 0, we obtain lim C ϕ t k z k z =0, z +. t 0 The proof is complete. As an application, we have the following corollary by using standard arguments as in [15], Theorem 3.3. We omit its proof to the reader. Corollary 3.1 Let α > 1,and let {ϕ t } t 0 S( + ) be a one-parameter semigroup on +. Assume that each C ϕt is bounded on A α ( + ). If G is the infinitesimal generator of {ϕ t }, then the infinitesimal generator A of {C ϕt } has the domain of definition D(A)= { f A α( + ) : Gf A ( α + )} and is given by Af = Gf.

16 Wangand Pang Journal of Inequalities and Applications (016) 016:06 Page 16 of 16 4 Conclusion This paper studied the path component of composition operators spaces and the continuity of composition operator semigroups. In addition, the paper showed that the cancellation 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 ( ), and the second author is supported by Natural Science Foundation of China (117193). Received: 8 April 016 Accepted: 16 August 016 References 1. Choe, B, Koo, H, Smith, W: Difference of composition operators over the half-plane. Trans. Am. Math. Soc. (016, in press). 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, nd edn. Mathematical Surveys and Monographs, vol Am. Math. Soc., Providence (007) 5. Berkson, E: Composition operators isolated in the uniform topology. Proc. Am. Math. Soc. 81, 30-3 (1981) 6. Choe, B, Hosokawa, T, Koo, H: Hilbert-Schmidt differences of composition operators on the Bergman space. Math. Z. 69, (011) 7. Choe, B, Koo, H, Wang, M, Yang, J: Compact linear combinations of composition operators induced by linear fractional maps. Math. Z. 80, (015) 8. Koo, H, Wang, M: Joint Carleson measure and the difference of composition operators on A p α(b n ). J. Math. Anal. Appl. 419, (014) 9. Moorhouse, J: Compact differences of composition operators. J. Funct. Anal. 19, 70-9 (005) 10. Stević, S: Composition operators from the Hardy space to the nth weighted-type space on the unit disk and the half-plane. Appl. Math. Comput. 15, (010) 11. Stević, S: Composition operators from the weighted Bergman space to the nth weighted-type space on the upper half-plane. Appl. Math. Comput. 17, (010) 1. Stević, S, Sharma, A: Weighted composition operators between growth spaces of the upper half-plane. Util. Math. 84, 65-7 (011) 13. Elliiott, S, Wynn, A: Composition operators on weighted Bergman spaces of a half-plane. Proc. Edinb. Math. Soc. 54, (011) 14. Berkson, E, Porta, H: Semigroups of analytic functions and composition operators. Mich. Math. J. 5, (1978) 15. Arvanitidis, A: Semigroup of composition operators on Hardy spaces of the half-plane. Acta Sci. Math. 81, (015) 16. Contreras, M, Díaz-Madrigal, S, Pommerenke, C: Fixed points and boundary behaviour of the Koenigs function. Ann. Acad. Sci. Fenn., Math. 9, (004) 17. Contreras, M, Díaz-Madrigal, S, Pommerenke, C: On boundary critical points for semigroups of analytic functions. Math. Scand. 98, (006)