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Difference of composition operators on weighted Bergman spaces over the halfplane
Journal of Inequalities and Applications volume 2016, Article number: 206 (2016)
Abstract
Recently, the bounded, compact and HilbertSchmidt difference of composition operators on the Bergman spaces over the halfplane are characterized in (Choe et al. in Trans. Am. Math. Soc., 2016, in press). Motivated by this, we give a sufficient condition when two composition operators \(C_{\varphi}\) and \(C_{\psi}\) 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 \(C_{\varphi_{1}}\), \(C_{\varphi_{2}}\), and \(C_{\varphi_{3}}\) are distinct and bounded, then \((C_{\varphi _{1}}C_{\varphi_{2}})(C_{\varphi_{3}}C_{\varphi_{1}})\) is compact if and only if both \(C_{\varphi_{1}}C_{\varphi_{2}}\) and \(C_{\varphi _{1}}C_{\varphi_{3}}\) are compact on weighted Bergman spaces over the halfplane. Moreover, we prove the strong continuity of composition operators semigroup induced by a oneparameter semigroup of holomorphic selfmaps of halfplane.
Introduction
Let \(\Pi^{+}\) be the upper half of the complex plane, that is, \(\Pi^{+}:=\{z\in\mathbf{C}: \operatorname{Im}z>0\}\), and let \(S(\Pi^{+})\) be the set of all holomorphic selfmaps of \(\Pi ^{+}\). For \(\varphi\in S(\Pi^{+})\), the composition operator \(C_{\varphi}\) is defined by
for functions f holomorphic on \(\Pi^{+}\). It is clear that \(C_{\varphi}\) maps the space of holomorphic functions on \(\Pi^{+}\) into itself. Our purpose in this paper is to study composition operators acting on the weighted Bergman spaces over \(\Pi^{+}\). For \(\alpha>1\), let
where \(c_{\alpha}=\frac{2^{\alpha}(\alpha+1)}{\pi}\) is a normalizing constant and A is the area measure on \(\Pi^{+}\). The weighted Bergman space \(A_{\alpha}^{2}(\Pi^{+})\) consists of holomorphic functions f on \(\Pi^{+}\) such that the norm
is finite. It is well known that \(A_{\alpha}^{2}(\Pi^{+})\) is a Hilbert space with the inner product
for \(f, g\in A_{\alpha}^{2}(\Pi^{+})\).
An extensive study on the theory of composition operators has been established during the past four decades on various settings. We refer to [2–4] 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 [2], 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 \(H^{p}(\mathbf{D})\), where D is the unit disk of the complex plane, and \(0< p<\infty\). Especially, we mention that ChoeHosokawaKoo [6] studied the topological structure of the space of all composition operators under the HilbertSchmidt 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 halfplane have received more attention; for instance, refer to [10–12]. Especially, Elliott and Wynn [13] characterized bounded composition operators and showed that there is no compact composition operator on \(A_{\alpha }^{2}(\Pi^{+})\). ChoeKooSmith [1] studied the bounded and compact difference of composition operators on \(A_{\alpha}^{2}(\Pi ^{+})\). They also obtained conditions under which the difference of composition operators is HilbertSchmidt.
In this paper, we proceed along this line to give a sufficient condition when the composition operators \(C_{\varphi}\) and \(C_{\psi}\) are in the same path component under the operator norm topology. Moreover, we show that the cancellation of double difference cannot occur on \(A_{\alpha}^{2}(\Pi^{+})\). More precisely, for distinct and bounded \(C_{\varphi_{1}}\), \(C_{\varphi_{2}}\), and \(C_{\varphi_{3}}\), the difference \((C_{\varphi_{1}}C_{\varphi_{2}})(C_{\varphi _{3}}C_{\varphi_{1}})\) is compact on \(A_{\alpha}^{2}(\Pi^{+})\) if and only if both \(C_{\varphi_{1}}C_{\varphi_{2}}\) and \(C_{\varphi _{3}}C_{\varphi_{1}}\) are compact. We also study the linear sum of composition operators induced by some special classes of holomorphic selfmaps. In addition, we prove the strong continuity of composition operators semigroups induced by oneparameter semigroups of holomorphic selfmaps of \(\Pi^{+}\). Due to the unboundedness of the domain, some special techniques are needed.
In Section 2, we recall some basic facts to be used in later sections. In Section 3.1, we prove our sufficient condition for the path component of composition operators. In Section 3.2, 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 \(X\lesssim Y\) or \(Y\gtrsim X\) for nonnegative quantities X and Y to mean \(X\leq CY\) for some inessential constant \(C>0\). Similarly, we say that \(X\approx Y\) if both \(X\lesssim Y\) and \(Y\lesssim X\) hold.
Preliminaries
In this section, we give some notation and wellknown results on \(A_{\alpha}^{2}(\Pi^{+})\). Recall that for a Hilbert space X, a bounded linear operator \(T: X\rightarrow X\) is said to be compact if T maps every 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 \(A_{\alpha}^{2}(\Pi^{+})\).
Lemma 2.1
Let T be a linear combination of composition operators and assume that T is bounded on \(A_{\alpha }^{2}(\Pi^{+})\). Then T is compact on \(A_{\alpha}^{2}(\Pi^{+})\) if and only if \(Tf_{n}\rightarrow0\) in \(A_{\alpha}^{2}(\Pi^{+})\) for any bounded sequence \(\{f_{n}\}\) in \(A_{\alpha}^{2}(\Pi^{+})\) satisfying \(f_{n}\rightarrow0\) uniformly on compact subsets of \(\Pi^{+}\).
A proof can be found in [2], Proposition 3.11, for composition operators on a Hardy space over the unit disk, and it can be easily modified for composition operators on \(A_{\alpha}^{2}(\Pi^{+})\).
The pseudohyperbolic distance is defined as follows:
Note that σ is invariant under dilation and horizontal translation. We know from [1] that
Given \(\varphi, \psi\in S(\Pi^{+})\), we put
For \(z\in\Pi^{+}\) and \(0<\delta<1\), let \(E_{\delta}(z)\) denote the pseudohyperbolic disk centered at z with radius δ. We may check by an elementary calculation that \(E_{\delta}(z)\) is actually a Euclidean disk centered at \(x+i\frac{1+\delta ^{2}}{1\delta^{2}}y\), of radius \(\frac{2\delta}{1\delta^{2}}y\), where \(x=\operatorname{Re}z\) and \(y=\operatorname{Im}z\).
We will often use the following submean value type inequality:
for all \(f\in A_{\alpha}^{2}(\Pi^{+})\) and some constant \(C=C(\alpha, \delta)\); see [1] for more details. In particular, we have
for \(f\in A_{\alpha}^{2}(\Pi^{+})\).
Given \(\alpha>1\), it follows from (2.2) that each point evaluation is a continuous linear functional on \(A_{\alpha}^{2}(\Pi^{+})\). Thus, for each \(z\in\Pi^{+}\), there exists a unique reproducing kernel \(K_{z}^{(\alpha)}\in A_{\alpha}^{2}(\Pi^{+})\) that has the reproducing property
for \(f\in A_{\alpha}^{2}(\Pi^{+})\). The explicit formula of \(K_{z}^{(\alpha)}\) is given as
where \(i^{2}=1\). Notice that
Thus, the normalized reproducing kernel \(\frac{K_{z}^{(\alpha)}}{\ K_{z}^{(\alpha)}\}\) converges to 0 uniformly on compact subsets of \(\Pi^{+}\) as \(z\rightarrow\partial\widehat{\Pi}^{+}\). Here, \(\widehat{\Pi}^{+}:=\overline{\Pi}^{+}\cup\{\infty\}\), and \(\partial\widehat{\Pi}^{+}\) is the boundary of \(\widehat{\Pi}^{+}\). In the sequel, we usually write \(K_{z}=K_{z}^{(\alpha)}\) and \(k_{z}=\frac{K_{z}^{(\alpha)}}{\K_{z}^{(\alpha)}\}\) for simplicity.
Before introducing angular derivatives in the halfplane setting, we first clarify the notion of nontangential limits at boundary points of \(\widehat{\Pi}^{+}\). Of course, those at a finite boundary point refer to the standard notion. Meanwhile, those at \(\infty\in\partial\widehat{\Pi}^{+}\) refer to those associated with nontangential approach regions \(\Omega_{\epsilon}\), \(\epsilon>0\), consisting of all \(z\in \mathbf{C}\) such that \(\operatorname{Im} z>\epsilon\operatorname{Re} z\). For a function \(\varphi:\Pi^{+}\to\Pi^{+}\) and \(x\in\partial\widehat{\Pi} ^{+}\), we write \(\varphi(x)=\eta\) (possibly ∞) if φ has a nontangential limit η, that is, \(\angle\lim_{z\to x}\varphi(z)=\eta\). For a holomorphic selfmap ψ of D, the angular derivative of ψ exists at \(\zeta\in\partial\mathbf{D}\) if there is \(\eta\in\partial\mathbf{D}\) such that a nontangential limit of \(\frac{\eta\psi(z)}{\zetaz}\) exists as a finite complex value as \(z\rightarrow\zeta\). Now we introduce the notion of angular derivatives on \(\Pi^{+}\) via the Caley transformation
which conformally maps D onto \(\Pi^{+}\). Note that a region Γ is contained in a nontangential approach region in D if and only if \(\gamma(\Gamma)\) is contained in some nontangential approach region in \(\Pi^{+}\).
For \(\varphi\in S(\Pi^{+})\), let
We say that φ has finite angular derivative at \(x\in\partial \widehat{\Pi}^{+}\) if \(\varphi_{\gamma}\) has a finite angular derivative at \(\widetilde{x}:=\gamma^{1}(x)\), where \(\gamma ^{1}(w)=\frac{wi}{w+i}\), \(w\in\Pi^{+}\).
The following JuliaCarathéodory theorem for the upper halfplan is proved in [13].
Proposition 2.2
For \(\varphi\in S(\Pi^{+})\), the following statements are equivalent:

(a)
\(\varphi(\infty)=\infty\), and \(\varphi'(\infty)\) exists;

(b)
\(\sup_{z\in\Pi^{+}}\frac{\operatorname{Im} z}{\operatorname{Im}\varphi{(z)}}<\infty\);

(c)
\(\limsup_{z\to\infty}\frac{\operatorname{Im} z}{\operatorname{Im}\varphi{(z)}}<\infty\).
Moreover, the quantities in (b) and (c) are equal to \(\varphi'(\infty)\).
Elliott and Wynn [13] gave the following characterization of bounded composition operators by means of angular derivatives.
Theorem 2.3
Let \(\alpha>1\) and \(\varphi\in S(\Pi^{+})\). Then \(C_{\varphi}\) is bounded on \(A_{\alpha}^{2}(\Pi^{+})\) if and only if φ has a finite angular derivative \(\varphi'(\infty)=\lambda\in(0, \infty )\). Moreover, \(\C_{\varphi}\=\lambda^{\frac{\alpha+2}{2}}\).
For \(z\in\Pi^{+}\), let \(\tau_{z}\) be the function on \(\Pi^{+}\) defined by
In [1], the authors showed that \(\tau_{z}^{s}\in A_{\alpha }^{2}(\Pi^{+})\) if and only if \(2s>\alpha+2\). In this case,
where C is a constant. Thus, \(\frac{\tau_{z}^{s}}{\\tau_{z}^{s}\}\rightarrow0\) uniformly on compact subsets of \(\Pi^{+}\) as \(z\rightarrow\partial\widehat{\Pi}^{+}\).
Main results
Path component of composition operators
In this subsection, we give a sufficient condition for composition operators \(C_{\varphi}\) and \(C_{\psi}\) to be in the same path component under the operator norm topology. To this end, we recall the definition of HilbertSchmidt operators on \(A_{\alpha}^{2}(\Pi^{+})\). A bounded linear operator T on a separable Hilbert H is HilbertSchmidt if
for any (or some) orthonormal basis \(\{e_{n}\}\) of H. As is well known, the value of the sum above does not depend on the choice of orthonormal basis \(\{e_{n}\}\) of H, and \(\T\\leq\T\_{\mathrm{HS}}\). We know that every HilbertSchmidt operator is compact; see [6] for more details. Let \(\mathcal{C}(A_{\alpha}^{2}(\Pi^{+}))\) be the space of all bounded composition operators on \(A_{\alpha}^{2}(\Pi ^{+})\) endowed with norm topology. The following theorem is due to ChoeKooSmith [1], Theorem 7.6.
Theorem 3.1
Let \(\alpha>1\) and \(\varphi, \psi\in S(\Pi^{+})\). Then \(C_{\varphi }C_{\psi}\) is HilbertSchmidt on \(A_{\alpha}^{2}(\Pi^{+})\) if and only if
where \(\sigma(z):=\vert \frac{\varphi(z)\psi(z)}{\varphi (z)\overline{\psi(z)}}\vert \).
Write \(C_{\varphi}\sim C_{\psi}\) if \(C_{\varphi}\) and \(C_{\psi }\) are in the same path component of \(\mathcal{C}(A_{\alpha}^{2}(\Pi ^{+}))\). For \(t\in[0,1]\), we put \(\varphi_{t}=(1t)\varphi+t\psi\). It is easy to see that \(\varphi_{t}\in S(\Pi^{+})\). In order to give a sufficient condition of path connected of two compositions, we need the following lemma.
Lemma 3.2
Let \(\alpha>1\) and \(\varphi, \psi\in S(\Pi^{+})\). Assume that \(C_{\varphi}C_{\psi}\) is HilbertSchmidt on \(A_{\alpha}^{2}(\Pi ^{+})\). Let \(\varphi_{t}=(1t)\varphi+t\psi\) for \(t\in[0,1]\). Then \(C_{\varphi_{s}}C_{\varphi_{t}}\) is HilbertSchmidt on \(A_{\alpha }^{2}(\Pi^{+})\) for any \(s,t\in[0,1]\).
Proof
Since \(\C_{\varphi_{s}}C_{\varphi_{t}}\_{\mathrm{HS}}\leq\C_{\varphi }C_{\varphi_{s}}\_{\mathrm{HS}}+\C_{\varphi}C_{\varphi_{t}}\_{\mathrm{HS}}\), it is enough to prove that \(C_{\varphi}C_{\varphi_{t}}\) is HilbertSchmidt on \(A_{\alpha}^{2}(\Pi^{+})\) for every \(t\in[0,1]\).
Fix \(t\in[0,1]\) and put \(\sigma_{t}(z):=\sigma(\varphi(z), \varphi _{t}(z))\). Then
for all \(z\in\Pi^{+}\).
Now, note that \(\frac{1}{\operatorname{Im}\xi}\leq\frac{1}{\operatorname {Im}z}+\frac{1}{\operatorname{Im}w}\) for any ξ on the line segment connecting z and w. Thus, we have
for all \(0\leq t\leq1\). By (3.1) and (3.2) we have
Similarly,
So, we conclude by Theorem 3.1 that \(C_{\varphi}C_{\varphi_{t}}\) is HilbertSchmidt on \(A_{\alpha}^{2}(\Pi^{+})\). □
Now, we give our first main result on sufficient conditions of path connected.
Theorem 3.3
Let \(\alpha>1\) and \(\varphi, \psi\in S(\Pi^{+})\). Assume that both \(C_{\varphi}\) and \(C_{\psi}\) are bounded on \(A_{\alpha}^{2}(\Pi ^{+})\) and \(C_{\varphi}C_{\psi}\) is HilbertSchmidt on \(A_{\alpha }^{2}(\Pi^{+})\). Then \(C_{\varphi}\sim C_{\psi}\).
Proof
Suppose that \(C_{\varphi}C_{\psi}\) is HilbertSchmidt on \(A_{\alpha }^{2}(\Pi^{+})\). Since
by Lemma 3.2 we obtain that \(C_{\varphi_{t}}\in\mathcal{C}(A_{\alpha}^{2}(\Pi ^{+}))\). We will show that \(t\in[0,1]\mapsto C_{\varphi_{t}}\) is a continuous path in \(\mathcal{C}(A_{\alpha}^{2}(\Pi^{+}))\). Since \(\C_{\varphi_{t}}C_{\varphi_{s}}\\leq\C_{\varphi _{t}}C_{\varphi_{s}}\_{\mathrm{HS}}\), it is sufficient to prove that
Given \(t, s\in[0,1]\), \(t\neq s\), put \(\sigma_{t,s}(z)=\sigma(\varphi _{t}(z),\varphi_{s}(z))\), \(t, s \in[0, 1]\) and \(z\in\Pi^{+}\). From (3.1) we have
From [1], Theorem 7.5, we know that
Put
for short. By (3.2) and (3.3) we have
Since \(C_{\varphi}C_{\psi}\) is a HilbertSchmidt operator, \(\Phi _{0,1}\) is integrable by Theorem 3.1. Again \(\sigma _{t,s}(z)\rightarrow0\) in \(\Pi^{+}\) as \(t\rightarrow s\), and we conclude by the dominated convergence theorem that
which completes the proof. □
Remark
It follows immediately from the theorem above that

Given \(C_{\varphi}\in\mathcal{C}(A_{\alpha}^{2}(\Pi^{+}))\), the set \(N(\varphi):=\{C_{\psi}:\C_{\varphi}C_{\psi}\ _{\mathrm{HS}}<\infty\}\) is the a pathconnected set in \(\mathcal{C}(A_{\alpha}^{2}(\Pi ^{+}))\) containing \(C_{\varphi}\).

The set \(N(\varphi)\) is ‘convex’ in the sense that if \(C_{\psi }\in N(\varphi)\), then \(\{C_{\varphi_{t}}\}_{t\in[0,1]}\in N(\varphi)\).
Cancellation properties of composition operators
In this subsection, we study cancellation properties of composition operators on \(A_{\alpha}^{2}(\Pi^{+})\). The following lemma is cited from [1], Corollary 4.7.
Lemma 3.4
Let \(\alpha>1\) and \(\varphi, \psi\in S(\Pi^{+})\). Assume that both \(C_{\varphi}\) and \(C_{\psi}\) are bounded on \(A_{\alpha}^{2}(\Pi ^{+})\). Then \(C_{\varphi}C_{\psi}\) is compact on \(A_{\alpha }^{2}(\Pi^{+})\) if and only if
Here \(\lim_{z\rightarrow\partial\widehat{\Pi} ^{+}}g(z)=0\) means that \(\sup_{\Pi^{+}\backslash K}g\rightarrow0\) as the compact set \(K\subset\Pi^{+}\) expands to the whole of \(\Pi^{+}\) or, equivalently, that \(g(z)\rightarrow0\) as \(\operatorname{Im} z\rightarrow0^{+}\) and \(g(z)\rightarrow0\) as \(z\rightarrow\infty\).
The following theorem shows that there is no cancellation property for the compactness of double difference of composition operators.
Theorem 3.5
Let \(\alpha>1\), \(a,b\in\mathbf{C}\backslash\{0\}\), and \(a+b\neq0\). Assume that \(\varphi_{j}\in S(\Pi^{+})\) and \(\varphi_{j}\) are distinct for \(j=1,2,3\) and each \(C_{\varphi_{j}}\) is bounded on \(A_{\alpha}^{2}(\Pi^{+})\). Then \(T:=a(C_{\varphi_{2}}C_{\varphi _{1}})+b(C_{\varphi_{3}}C_{\varphi_{1}})\) is compact on \(A_{\alpha }^{2}(\Pi^{+})\) if and only if both \(C_{\varphi_{2}}C_{\varphi _{1}}\) and \(C_{\varphi_{3}}C_{\varphi_{1}}\) are compact on \(A_{\alpha}^{2}(\Pi^{+})\).
Proof
The sufficiency is trivial, and we only prove the necessity. Assume that \(T=a(C_{\varphi_{2}}C_{\varphi_{1}})+b(C_{\varphi _{3}}C_{\varphi_{1}})\) is compact. We will get a contradiction if either \(C_{\varphi_{2}}C_{\varphi_{1}}\) or \(C_{\varphi _{3}}C_{\varphi_{1}}\) is not compact. Without loss of generality, we assume that \(C_{\varphi_{2}}C_{\varphi_{1}}\) is not compact. Let \(\sigma_{j,s}(z):=\vert \frac{\varphi_{j}(z)\varphi _{s}(z)}{\varphi_{j}(z)\overline{\varphi_{s}(z)}}\vert \) for \(j,s=1,2,3\) and \(s\neq j\).
Since \(C_{\varphi_{2}}C_{\varphi_{1}}\) is not compact, by (3.4) there are \(\epsilon>0\) and a sequence \(\{z_{n}\}\subset\Pi^{+}\) such that \(z_{n}\rightarrow\partial{\widehat{\Pi}^{+}}\) (\(n\rightarrow \infty\)) and
For each \(j=1,2,3\), since \(C_{\varphi_{j}}\) is bounded on \(A_{\alpha }^{2}(\Pi^{+})\), we have \(\C_{\varphi_{j}}^{*}K_{z}\\leq\ C_{\varphi_{j}}\\K_{z}\\) for any \(z\in\Pi^{+}\), where \(C_{\varphi _{j}}^{*}\) is the adjoint of \(C_{\varphi_{j}}\). Due to \(C_{\varphi _{j}}^{*}K_{z}=K_{\varphi_{j}(z)}\), this is equivalent to
where C is some positive constant. Duo to this and the fact \(\sigma _{12}<1\), taking a smaller ϵ if necessary, formula (3.5) gives that
and
or
Without loss of generality, we assume that \(M_{2}(z_{n})\gtrsim\epsilon\) (the proof for the case \(M_{1}(z_{n}) \gtrsim\epsilon\) is similar); thus, \(\operatorname{Im}z_{n}\approx \operatorname{Im}\varphi_{2}(z_{n})\). For \(j=1,2,3\), we define \(g_{j,n}(z):=\tau_{\varphi_{j}(z_{n})}^{k}(z)=\frac{1}{(z\overline {\varphi_{j}(z_{n})})^{k}}\) with \(2k>\alpha+2\). For \(m, j=1,2,3\), \(m\neq j\), let \(x_{j,n}^{m}=\frac{\varphi _{m}(z_{n})\overline{\varphi_{m}(z_{n})}}{\varphi _{j}(z_{n})\overline{\varphi_{m}(z_{n})}}\). Notice that
and
By (2.4) and \(M_{2}(z_{n})\gtrsim\epsilon\) we have
Thus,
By (2.1) we have
Thus, using the fact that \(M_{2}(z_{n})\geq\epsilon\), we obtain
Since T is compact, we have \(\frac{\Tg_{2,n}\}{\g_{2,n}\ }\rightarrow0\) (\(n\rightarrow\infty\)). Therefore, at least one of \(y_{1,n}^{2}\) and \(y_{3,n}^{2}\) does not converge to 0.
Suppose \(y_{1,n}^{2}\rightarrow0\) but \(y_{3,n}^{2}\nrightarrow0\). Then \(\frac{\operatorname{Im} \varphi_{2}(z_{n})}{\operatorname{Im}\varphi _{3}(z_{n})}\geq C>0\) for some subsequence, which we still denote by \(\{ z_{n}\}\). Since \(\operatorname{Im} z_{n}\approx\operatorname{Im}\varphi_{2}(z_{n})\), \(M_{3}(z_{n}):=\frac{\operatorname{Im} z_{n}}{\operatorname{Im}\varphi _{3}(z_{n})}\geq C>0\). Therefore, \(\operatorname{Im}\varphi _{3}(z_{n})\lesssim\operatorname{Im}z_{n}\). Similarly to (3.9), we have
From (3.8) we have
which implies \(x_{1,n}^{3}\rightarrow0\) as \(n\rightarrow\infty\). By (3.10) and \(M_{3}(z_{n})\geq C\) we obtain
Since T is compact, we have \(\frac{\Tg_{3,n}\}{\g_{3,n}\ }\rightarrow0\) as \(n\rightarrow\infty\). Since \(\frac{\varphi _{3}(z_{n})\overline{\varphi_{3}(z_{n})}}{\varphi _{1}(z_{n})\overline{\varphi_{3}(z_{n})}}=x_{1,n}^{3}\rightarrow0\), we have
Note that this holds for any \(2k>\alpha+2\), which implies \(a+b=0\). This is a contradiction to the assumption \(a+b\neq0\). Therefore, \(y_{1,n}^{2}\) does not converge to 0.
Taking a subsequence of \(\{z_{n}\}\) if necessary, we have \(M_{1}(z_{n})\geq C>0\), which implies \(\operatorname{Im}\varphi _{1}(z_{n})\lesssim\operatorname{Im} z_{n}\). Similarly to (3.9), we have
Notice that
and
By formula (3.7) we have
Therefore, if \(\limsup_{n\rightarrow\infty}x_{2,n}^{1}=1\), then there exists some subsequence \(\{z_{n_{l}}\}\) such that
Then \(\sigma_{1,2}(z_{n_{l}})\rightarrow0\), which contradicts (3.6). Hence, we have \(\limsup_{n\rightarrow\infty }x_{2,n}^{1}\lneqq1\). From (3.10) and the fact that \(M_{1}(z_{n})\geq\epsilon\) we have
Thus, we get
for all \(2k>\alpha+2\). Then
Since \(\limsup_{n\rightarrow\infty}x_{2,n}^{1}\lneqq1\), we obtain \(\frac{b}{a+b}1=0\). Namely, \(a=0\), which contradicts our assumption. Therefore, the compactness of \(T=a(C_{\varphi _{2}}C_{\varphi_{1}})+b(C_{\varphi_{3}}C_{\varphi_{1}})\) implies that both \(C_{\varphi_{2}}C_{\varphi_{1}}\) and \(C_{\varphi _{3}}C_{\varphi_{1}}\) 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 \(A_{\alpha }^{2}(\Pi^{+})\). 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 \(\varphi\in S(\Pi^{+})\), if \(C_{\varphi}\) is bounded on \(A_{\alpha}^{2}(\Pi^{+})\), then \(\varphi (\infty)=\infty\) and \(0<\varphi'(\infty)<\infty\) by Theorem 2.3. Let
For \(\varepsilon>0\), let
Note that \(R_{\varepsilon, \infty}\) is a nontangential curve having ∞ as the end point. Let
where c is a constant.
The following lemma can be found in [1], Lemma 5.2.
Lemma 3.6
Let \(\varphi, \psi\in S(\Pi^{+})\). Assume that \(C_{\varphi}\), \(C_{\psi}\) are bounded on \(A_{\alpha}^{2}(\Pi^{+})\). Then
Now we give a lower estimate of the essential norm of a linear sum of composition operators induced by symbols in \(S(\Pi^{+})_{c}\).
Theorem 3.7
Let \(\alpha>1\) and \(\varphi_{j}\in S(\Pi^{+})_{c}\), \(j=1,2,\ldots, N\). For each \(j=1, 2,\ldots, N\), assume that \(C_{\varphi_{j}}\) is bounded on \(A_{\alpha}^{2}(\Pi^{+})\). Then we have the inequality
for any \(a_{1}, a_{2}, \ldots, a_{N}\in\mathbf{C}\).
Proof
Let \(a_{1}, a_{2}, \ldots, a_{N}\in\mathbf{C}\). Since \(C_{\varphi _{j}}^{*}K_{z}=K_{\varphi_{j}(z)}\), we have
Meanwhile, we have
Using Lemma 3.6, we have
Thus, we obtain
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 \(\alpha>1\) and \(\varphi_{j}\in S(\Pi^{+})_{c}\), \(j=1, 2, \ldots , N\). Assume that each \(C_{\varphi_{j}}\) is bounded on \(A_{\alpha }^{2}(\Pi^{+})\) and \(a_{j}\in\mathbf{C}\). If \(\sum_{j=1}^{N}a_{j}C_{\varphi_{j}}\) is compact on \(A_{\alpha}^{2}(\Pi ^{+})\), then
\((u,v)\in\{\infty\}\times(0,\infty)\).
Especially, we have the following useful corollary.
Corollary 3.9
Let \(\alpha>1\) and \(\varphi, \psi\in S(\Pi^{+})_{c}\). Assume that \(C_{\varphi}\), \(C_{\psi}\) are bounded on \(A_{\alpha}^{2}(\Pi^{+})\) and \(a,b\in\mathbf{C}\backslash\{0\}\). If \(aC_{\varphi}+bC_{\psi}\) is compact on \(A_{\alpha}^{2}(\Pi^{+})\), then the following statements hold:

(a)
\(a+b=0\);

(b)
\(\varphi'(\infty)=\psi'(\infty)\).
Composition operators induced by a oneparameter semigroup
In the last subsection, we consider the strong continuity of the composition operator semigroup induced by a oneparameter semigroup of holomorphic selfmaps of \(\Pi^{+}\). We first recall some definitions and notation.
A oneparameter semigroup of holomorphic selfmaps of \(\Pi^{+}\) is a family \(\{\varphi_{t}\}_{t\geq0}\subset S(\Pi^{+})\) satisfying

(1)
\(\varphi_{0}(z)=z\) for all \(z\in\Pi^{+}\);

(2)
\(\varphi_{t+s}(z)=\varphi_{t}\circ\varphi_{s}(z)\) for all \(s, t\geq0\) and \(z\in\Pi^{+}\);

(3)
\((t,z)\mapsto\varphi_{t}(z)\) is jointly continuous on \([0,\infty)\times\Pi^{+}\).
We know that the continuity of \((t,z)\mapsto\varphi_{t}(z)\) on \([0,\infty)\times\Pi^{+}\) is equivalent to the continuity of \(t\mapsto\varphi_{t}(z)\) for each \(z\in\Pi^{+}\). By [14] the holomorphic function \(G: \Pi^{+}\rightarrow\mathbf{C}\) given by
is the infinitesimal generator of \(\{\varphi_{t}\}\), which characterizes \(\{\varphi_{t}\}\) uniquely and satisfies
Let G̃ be the infinitesimal generator of the oneparameter semigroup \(\{(\varphi_{t})_{\gamma}\}_{t\geq0}\), where \((\varphi_{t})_{\gamma}=\gamma^{1}\circ\varphi_{t}\circ\gamma\), and \(\gamma(\zeta)=i\frac{1+\zeta}{1\zeta}\), \(\gamma^{1}(w)=\frac {wi}{w+i}\), \(\zeta\in\mathbf{D}\), \(w\in\Pi^{+}\). From [14] we also have
From [15] we obtain
Assume that \(\{C_{\varphi_{t}}\}_{t\geq0}\) is the bounded composition operator semigroup on \(A_{\alpha}^{2}(\Pi^{+})\) induced by the oneparameter semigroup \(\{\varphi_{t}\}_{t\geq0}\subset S(\Pi ^{+})\). The linear operator A defined by
and
is the infinitesimal generator of the semigroup \(\{C_{\varphi_{t}}\} _{t\geq0}\), where \(D(A)\) is the domain of A. If \(\{C_{\varphi_{t}}\} _{t\geq0}\) satisfies
then we say that \(\{C_{\varphi_{t}}\}_{t\geq0}\) is strongly continuous.
The following theorem gives some characterizations about the boundedness of \(\{C_{\varphi_{t}}\}\), \(t\ge0\).
Theorem 3.10
Let \(\alpha>1\), and let \(\{\varphi_{t}\}_{t\geq0}\) be a oneparameter semigroup with infinitesimal generator G, where \(\{ \varphi_{t}\}_{t\geq0}\subset S(\Pi^{+})\). Then the following are equivalent:

(a)
For each \(t>0\), \(C_{\varphi_{t}}\) is bounded on \(A_{\alpha }^{2}(\Pi^{+})\).

(b)
For some \(t>0\), \(C_{\varphi_{t}}\) is bounded on \(A_{\alpha }^{2}(\Pi^{+})\).

(c)
The nontangential limit \(\delta:=\angle\lim_{z\rightarrow\infty}\frac{G(z)}{z}\) exists finitely.
Moreover, if one of these assertions holds, then \(\C_{\varphi_{t}}\=e^{\frac{(\alpha+2)\delta t}{2}}\) for each \(t>0\).
Proof
(a) ⇔ (b). Since (a) implies (b), we only prove the converse. We now assume that there exists \(t_{0}>0\) such that \(C_{\varphi_{t_{0}}}\) is bounded on \(A_{\alpha}^{2}(\Pi^{+})\). Then \(\varphi_{t_{0}}(\infty)=\infty\), and \(\varphi_{t_{0}}'(\infty )\) exists finitely. From [15], Lemma 2.1, we know that \(\varphi _{t_{0}}(\infty)=\infty\) if and only if \((\varphi_{t_{0}})_{\gamma }(1)=1\). From [16], Theorem 5, we know that all members of the semigroup \(\{(\varphi_{t})_{\gamma}\}_{t\geq0}\) have common boundary fixed points, that is, \((\varphi_{t})_{\gamma}(1)=1\) for each \(t\geq 0\). Then, by [17], Theorem 1, we have \((\varphi_{t})_{\gamma }'(1)=((\varphi_{t_{0}})_{\gamma}'(1))^{\frac{t}{t_{0}}}\). For each \(t>0\), since
we have
Then we obtain
which implies that \(\varphi_{t}'(\infty)\) exists finitely. So \(C_{\varphi_{t}}\) is bounded on \(A_{\alpha}^{2}(\Pi^{+})\) for each \(t>0\) by Theorem 2.3.
(a) ⇔ (c). For each \(t>0\), \(C_{\varphi_{t}}\) is bounded on \(A_{\alpha}^{2}(\Pi^{+})\) if and only if \(\varphi_{t}(\infty)=\infty\), \(\varphi_{t}'(\infty)\) exists finitely by Proposition 2.2 and Theorem 2.3. From [15], Lemma 2.1, we have \((\varphi_{t})_{\gamma}(1)=1\) if and only if \(\varphi_{t}(\infty )=\infty\). Also, by (3.12), \(C_{\varphi_{t}}\) is bounded on \(A_{\alpha}^{2}(\Pi ^{+})\) if and only if \((\varphi_{t})_{\gamma}(1)=1\), \((\varphi _{t})_{\gamma}'(1)\) exists finitely, which is equivalent to the finite existence of \(\angle\lim_{w\rightarrow1}\frac{\widetilde {G}(w)}{1w}\), \(w\in\mathbf{D}\), by [17], Theorem 1, where G̃ is the infinitesimal generator of the oneparameter semigroup \(\{(\varphi_{t})_{\gamma}\}_{t\geq0}\). By (3.11) we have
which implies (a) ⇔ (c).
For each \(t>0\), if one of the conditions holds, we obtain \(\varphi _{t}'(\infty)=e^{\delta t}\) from (3.12), (3.13), and [17], Theorem 1, where \(\delta=\angle\lim_{w\rightarrow\infty }\frac{G(w)}{w}\), \(w\in\Pi^{+}\). By Theorem 2.3 we have
which completes the proof. □
Next, we prove the strong continuity of composition operator semigroups induced by oneparameter semigroups of holomorphic selfmaps of the upper halfplane.
Theorem 3.11
Let \(\alpha>1\), and let \(\{\varphi_{t}\}_{t\geq0}\subset S(\Pi ^{+})\) be a oneparameter semigroup on \(\Pi^{+}\). For each \(t\geq0\), assume that \(C_{\varphi_{t}}\) is bounded on \(A_{\alpha}^{2}(\Pi ^{+})\). Then \(\{C_{\varphi_{t}}\}_{t\geq0}\) is strongly continuous on \(A_{\alpha}^{2}(\Pi^{+})\).
Proof
Due to the denseness of \(Span\{k_{z}: z\in\Pi^{+}\}\) in \(A_{\alpha }^{2}(\Pi^{+})\), it is sufficient to prove
By the property of reproducing kernel of \(A_{\alpha}^{2}(\Pi ^{+})\) and the CauchySchwarz inequality we have
Because
we have
By Theorem 2.3 we have \(\C_{\varphi_{t}}\^{2}=\varphi_{t}'(\infty )^{\alpha+2}\). Thus, we obtain
Taking \(f=k_{z}\circ\varphi_{t}+k_{z}\) in (3.14), we obtain
Combining (3.15) and (3.16), we obtain
Since \(2(1+(\varphi_{t}'(\infty))^{\alpha+2})\rightarrow4\) and \(\frac{K_{z}\circ\varphi_{t}(z)+\K_{z}\^{2}^{2}}{\K_{z}\ ^{4}}\rightarrow4\) as \(t\rightarrow0\), we obtain
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 3.12
Let \(\alpha>1\), and let \(\{\varphi_{t}\}_{t\geq0}\subset S(\Pi ^{+})\) be a oneparameter semigroup on \(\Pi^{+}\). Assume that each \(C_{\varphi_{t}}\) is bounded on \(A_{\alpha}^{2}(\Pi^{+})\). If G is the infinitesimal generator of \(\{\varphi_{t}\}\), then the infinitesimal generator A of \(\{C_{\varphi_{t}}\}\) has the domain of definition
and is given by
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 halfplane. Due to the unboundedness of the halfplane, some special new techniques are used to overcome obstacles.
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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).
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Wang, M., Pang, C. Difference of composition operators on weighted Bergman spaces over the halfplane. J Inequal Appl 2016, 206 (2016). https://doi.org/10.1186/s1366001611492
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MSC
 47B33
 30H20
 32A35
Keywords
 Bergman space
 composition operator
 HilbertSchmidt
 semigroup