# Weighted Composition Operators from Weighted Bergman Spaces to Weighted-Type Spaces on the Upper Half-Plane

Authors: Stevo Stević; Ajay K. Sharma; S. D. Sharma
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Letψ be a holomorphic mapping on the upper half-plane Π+={z∈ℂ:𝔍z>0} and φ be a holomorphic self-map of Π+. We characterize bounded weighted composition operators acting from the weighted Bergman space to the weighted-type space on the upper half-plane. Under a mild condition on ψ, we also characterize the compactness of these operators.


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## Body

## 1. Introduction and Preliminaries

LetΠ+={z∈ℂ:ℑz>0} be the upper half-plane, Ω a domain in ℂ or ℂn, and H(Ω) the space of all holomorphic functions on Ω. Let ψ∈H(Ω), and letφ be a holomorphic self-map of Ω. Then by (1.1)Wφ,ψ(f)(z)=ψ(f∘φ)(z),z∈Ω,
is defined a linear operator on H(Ω) which is called weighted composition operator. If ψ(z)=1, then Wφ,ψ becomes composition operator and is denoted by Cφ, and if φ(z)=z, then Wφ,ψ becomes multiplication operator and is denoted by Mψ.During the past few decades, composition operators and weighted composition operators have been studied extensively on spaces of holomorphic functions on various domains inℂ or ℂn (see, e.g., [1–22] and the references therein). There are many reasons for this interest, for example, it is well known that the surjective isometries of Hardy and Bergman spaces are certain weighted composition operators (see [23, 24]). For some other operators related to weighted composition operators, see [25–30] and the references therein.While there is a vast literature on composition and weighted composition operators between spaces of holomorphic functions on the unit disk𝔻, there are few papers on these and related operators on spaces of functions holomorphic in the upper half-plane (see, e.g., [2, 3, 5, 7–9, 11, 12, 16–18, 31] and the references therein). For related results in the setting of the complex plane see also papers [19–21].The behaviour of composition operators on spaces of functions holomorphic in the upper half-plane is considerably different from the behaviour of composition operators on spaces of functions holomorphic in the unit disk𝔻. For example, there are holomorphic self-maps of Π+ which do not induce composition operators on Hardy and Bergman spaces on the upper half-plane, whereas it is a well-known consequence of the Littlewood subordination principle that every holomorphic self-map φ of 𝔻 induces a bounded composition operator on the Hardy and weighted Bergman spaces on 𝔻. Also, Hardy and Bergman spaces on the upper half-plane do not support compact composition operators (see [3, 5]).For0<p<∞ and α∈(-1,∞), let 𝔏p(Π+,dAα) denote the collection of all Lebesgue p-integrable functions f:Π+→ℂ such that (1.2)∫Π+|f(z)|pdAα(z)<∞,
where (1.3)dAα(z)=1π(α+1)(2Iz)αdA(z),dA(z)=dxdy, and z=x+iy.Let𝒜αp(Π+)=𝔏p(Π+,dAα)∩H(Π+). For 1≤p<∞,𝒜αp(Π+) is a Banach space with the norm defined by (1.4)‖f‖Aαp(Π+)=(∫Π+|f(z)|pdAα(z))1/p<∞.
With this norm 𝒜αp(Π+) becomes a Banach space when p≥1, while for p∈(0,1) it is a Fréchet space with the translation invariant metric (1.5)d(f,g)=‖f-g‖Aαp(Π+)p,f,g∈Aαp(Π+).Recall that for everyf∈𝒜αp(Π+) the following estimate holds:  (1.6)|f(x+iy)|p≤C∥f∥Aαp(Π+)pyα+2,
where C is a positive constant independent of f.Letβ>0. The weighted-type space (or growth space) on the upper half-plane 𝒜β∞(Π+) consists of all f∈H(Π+) such that (1.7)‖f‖Aβ∞(Π+)=sup⁡z∈Π+(Iz)β|f(z)|<∞.
It is easy to check that 𝒜β∞(Π+) is a Banach space with the norm defined above. For weighted-type spaces on the unit disk, polydisk, or the unit ball see, for example, papers [10, 32, 33] and the references therein.Given two Banach spacesY and Z, we recall that a linear map T:Y→Z is bounded if T(E)⊂Z is bounded for every bounded subset E of Y. In addition, we say that T is compact if T(E)⊂Z is relatively compact for every bounded set E⊂Y.In this paper, we consider the boundedness and compactness of weighted composition operators acting from𝒜αp(Π+) to the weighted-type space 𝒜β∞(Π+). Related results on the unit disk and the unit ball can be found, for example, in [6, 13, 15].Throughout this paper, constants are denoted byC; they are positive and may differ from one occurrence to the other. The notation a⪯b means that there is a positive constant C such that a≤Cb. Moreover, if both a⪯b and b⪯a hold, then one says that a≍b.


## 2. Main Results

The boundedness and compactness of the weighted composition operatorWφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) are characterized in this section.Theorem 2.1.
Let1≤p<∞,α>-1,β>0,ψ∈H(Π+), and  let φ be a holomorphic self-map of  Π+. Then  Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded if and only if
(2.1)M:=sup⁡z∈Π+(Iz)β(Iφ(z))(α+2)/p|ψ(z)|<∞.
Moreover, if the operator Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded then the following asymptotic relationship holds:
(2.2)‖Wφ,ψ‖Aαp(Π+)→Aβ∞(Π+)≍M.Proof.
First suppose that (2.1) holds. Then for any z∈Π+ and f∈𝒜αp(Π+), by (1.6) we have
(2.3)(Iz)β|(Wφ,ψf)(z)|=(Iz)β|ψ(z)||f(φ(z))|⪯(Iz)β(Iφ(z))(α+2)/p|ψ(z)|‖f‖Aαp(Π+),
and so by (2.1), Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded and moreover
(2.4)‖Wφ,ψ‖Aαp(Π+)→Aβ∞(Π+)⪯M.
Conversely suppose Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded. Consider the function
(2.5)fw(z)=(Iw)(α+2)/p(z-w¯)(2α+4)/p,w∈Π+.
Then fw∈𝒜αp(Π+) and moreover supw∈Π+∥fw∥𝒜αp(Π+)⪯1 (see, e.g., Lemma 1 in [18]).
Thus the boundedness ofWφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) implies that
(2.6)(Iz)β|ψ(z)||fw(φ(z))|≤‖Wφ,ψfw‖Aβ∞(Π+)≼‖Wφ,ψ‖Aαp(Π+)→Aβ∞(Π+),
for every z,w∈Π+. In particular, if z∈Π+ is fixed then for w=φ(z), we get
(2.7)(Iz)β(Iφ(z))(α+2)/p|ψ(z)|⪯‖Wφ,ψ‖Aαp(Π+)→Aβ∞(Π+).
Since z∈Π+ is arbitrary, (2.1) follows and moreover
(2.8)M⪯‖Wφ,ψ‖Aαp(Π+)→Aβ∞(Π+).
If Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded then from (2.4) and (2.8) asymptotic relationship (2.2) follows.Corollary 2.2.
Let1≤p<∞,α>-1, and  β>0 be such that  βp≥α+2 and  ψ∈H(Π+). Then  Mψ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded if and only if  ψ∈X, where
(2.9)X={Aβ-((α+2)/p)∞(Π+)ifα+2<βp,H∞(Π+)ifα+2=βp.Example 2.3.
Let1≤p<∞,α>-1 and β>0 be such that βp≥α+2 and w∈Π+. Let ψw be a holomorphic map of Π+ defined as
(2.10)ψw(z)={1(z-w¯)β-((α+2)/p)ifα+2<βp,Iwz-w¯ifα+2=βp.
For z=x+iy and w=u+iv in Π+, we have
(2.11)sup⁡z∈Π+(Iz)β-(α+2)/p|ψw(z)|=sup⁡z=x+iy∈Π+yβ-(α+2)/p((x-u)2+(y+v)2)(βp-(α+2))/2p≤sup⁡z=x+iy∈Π+yβ-(α+2)/p(y+v)β-(α+2)/p≤1.
Thus ψw∈𝒜β-(α+2)/p∞(Π+) if α+2<βp. Similarly ψw∈H∞(Π+) if α+2=βp. By Corollary 2.2, it follows that Mψw:𝒜αp(Π+)→𝒜β∞(Π+) is bounded.Corollary 2.4.
Let1≤p<∞,α>-1,β>0, and letφ be a holomorphic self-map of Π+. Then Cφ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded if and only if
(2.12)sup⁡z∈Π+(Iz)β(Iφ(z))(α+2)/p<∞.Corollary 2.5.
Letφ be the linear fractional map
(2.13)φ(z)=az+bcz+d,a,b,c,d∈R,ad-bc>0.
Then necessary and sufficient condition that Cφ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded is that c=0 and α+2=βp.Proof.
Assume thatCφ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded. Then
(2.14)sup⁡z∈Π+(Iz)β(Iφ(z))(α+2)/p=sup⁡z=x+iy∈Π+((cx+d)2+c2y2)(α+2)/pyβ(ad-bc)(α+2)/py(α+2)/p,
which is finite only if c=0 and α+2=βp.
Conversely, ifc=0 and α+2=βp, then from (2.13) we get a≠0, and by some calculation
(2.15)sup⁡z∈Π+(Iz)β(Iφ(z))(α+2)/p=(da)β<∞.
Hence Cφ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded.Corollary 2.6.
Let1≤p<∞,α>-1, and  β>0 be such that  βp=α+2. Let  φ be a holomorphic self-map of  Π+ and  ψ=(φ′)β. Then the weighted composition operator  Wφ,ψ acts boundedly from  𝒜αp(Π+) to  𝒜β∞(Π+).Proof.
By Theorem2.1, Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded if and only if
(2.16)sup⁡z∈Π+(Iz)β(Iφ(z))β|φ′(z)|β<∞.
By the Schwarz-Pick theorem on the upper half-plane we have that for every holomorphic self-map φ of Π+ and all z∈Π+(2.17)|φ′(z)|Iφ(z)≤1Iz,
where the equality holds when φ is a Möbius transformation given by (2.13). From (2.17), condition (2.16) follows and consequently the boundedness of the operator Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+).Corollary2.6 enables us to show that there exist 1≤p<∞,α>-1,β>0, and holomorphic maps φ and ψ of the upper half-plane Π+ such that neither Cφ:𝒜αp(Π+)→𝒜β∞(Π+) nor Mψ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded, but Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded.Example 2.7.
Let1≤p<∞,α>-1, and β>0 be such that βp=α+2. Let φ(z)=(az+b)/(cz+d),a,b,c,d∈ℝ,ad-bc>0, and c≠0. Then by Corollary 2.5, Cφ:𝒜αp(Π+)→𝒜β∞(Π+) is not bounded. On the other hand, if
(2.18)ψ(z)=(φ′(z))β=(ad-bc(cz+d)2)β,
then ψ∉H∞(Π+) and so by Corollary 2.2, Mψ:𝒜αp(Π+)→𝒜β∞(Π+) is not bounded. However, by Corollary 2.6, we have that Wφ,(φ′)β:𝒜αp(Π+)→𝒜β∞(Π+) is bounded.
The next Schwartz-type lemma characterizes compact weighted composition operatorsWφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) and it follows from standard arguments ([4]).Lemma 2.8.
Let1≤p<∞,α>-1,β>0,ψ∈H(Π+), and letφ be a holomorphic self-map of  Π+. Then Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is compact if and only if, for any bounded sequence (fn)n∈ℕ⊂𝒜αp(Π+) converging to zero on compacts of  Π+, one has
(2.19)lim⁡n→∞‖Wφ,ψfn‖Aβ∞(Π+)=0.Theorem 2.9.
Let1≤p<∞,α>-1,β>0,ψ∈H(Π+) and  φ be a holomorphic self-map of   Π+. If  Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is compact, then
(2.20)lim⁡r→0sup⁡Iφ(z)<r(Iz)β(Iφ(z))(α+2)/p|ψ(z)|=0.Proof.
SupposeWφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is compact and (2.20) does not hold. Then there is a δ>0 and a sequence (zn)n∈ℕ⊂Π+ such that ℑφ(zn)→0 and
(2.21)(Izn)β(Iφ(zn))(α+2)/p|ψ(zn)|>δ
for all n∈ℕ. Let wn=φ(zn), n∈ℕ, and
(2.22)fn(z)=(Iwn)(α+2)/p(z-w¯n)(2α+4)/p,n∈N.
Then fn is a norm bounded sequence and fn→0 on compacts of Π+ as ℑφ(zn)→0. By Lemma 2.8 it follows that
(2.23)lim⁡n→∞‖Wφ,ψfn‖Aβ∞(Π+)=0.
On the other hand,
(2.24)‖Wφ,ψfn‖Aβ∞(Π+)≥(Izn)β|(Wφ,ψfn)(zn)|=(Izn)β|ψ(zn)||fn(φ(zn))|=(Izn)β2(2α+4)/p(Iφ(zn))(α+2)/p|ψ(zn)|>δ2(2α+4)/p,
which is a contradiction. Hence (2.20) must hold, as claimed.Before we formulate and prove a converse of Theorem2.9, we define, for every a,b∈(0,∞) such that a<b, the following subset of Π+: (2.25)Γa,b={z∈Π+:a≤Iz≤b}.Theorem 2.10.
Let1≤p<∞,α>-1,β>0,ψ∈H(Π+),  and letφ be a holomorphic self-map of  Π+ and  Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) be bounded. Suppose that  ψ∈𝒜β∞(Π+) and  (ℑz)β|ψ(z)|→0 as  |ℜφ(z)|→∞ within  Γa,b for all  a and  b,0<a<b<∞. Then  Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is compact if condition (2.20) holds.Proof.
Assume (2.20) holds. Then for each ɛ>0, there is an M1>0 such that
(2.26)(Iz)β(Iφ(z))(α+2)/p|ψ(z)|<ɛ,wheneverIφ(z)<M1.
Let (fn)n∈ℕ be a sequence in 𝒜αp(Π+) such that supn∈ℕ∥fn∥𝒜αp(Π+)≤M and fn→0 uniformly on compact subsets of Π+ as n→∞. Thus for z∈Π+ such that ℑφ(z)<M1 and each n∈ℕ, we have
(2.27)(Iz)β|ψ(z)||fn(φ(z))|⪯(Iz)β(Iφ(z))(α+2)/p|ψ(z)|‖fn‖Aαp(Π+)<ɛM.
From estimate (1.6) we have
(2.28)|fn(z)|⪯‖fn‖Aαp(Π+)(Iz)(α+2)/p⪯M(Iz)(α+2)/p.
Thus there is an M2>M1 such that
(2.29)|fn(φ(z))|<ɛ,
whenever ℑφ(z)>M2. Hence for z∈Π+ such that ℑφ(z)>M2 and each n∈ℕ we have
(2.30)(Iz)β|ψ(z)||fn(φ(z))|<ɛ‖ψ‖Aβ∞(Π+).
If M1≤ℑφ(z)≤M2, then by the assumption there is an M3>0 such that (ℑz)β|ψ(z)|<ɛ, whenever |ℜφ(z)|>M3. Therefore, for each n∈ℕ we have
(2.31)(Iz)β|ψ(z)||fn(φ(z))|⪯ɛ‖fn‖Aαp(Π+)(Iφ(z))(α+2)/p≤ɛMM1(α+2)/p,
whenever M1≤ℑφ(z)≤M2 and |ℜφ(z)|>M3.
IfM1≤ℑφ(z)≤M2 and |ℜφ(z)|≤M3, then there exists some n0∈ℕ such that |fn(φ(z))|<ɛ for all n≥n0, and so
(2.32)(Iz)β|ψ(z)||fn(φ(z))|<ɛ‖ψ‖Aβ∞(Π+).
Combining (2.27)–(2.32), we have that
(2.33)‖Wφ,ψfn‖Aβ∞(Π+)<ɛC,
for n≥n0 and some C>0 independent of n. Since ɛ is an arbitrary positive number, by Lemma 2.8, it follows that Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is compact.Example 2.11.
Let1≤p<∞,α>-1, and β>0 be such that α+2=βp. Let φ(z)=z+i and ψ(z)=1/(z+i)β, then ℜφ(z)=x and ℑφ(z)=y+1. It is easy to see that ψ∈𝒜β∞(Π+). Beside this, for z∈Γa,b, we have
(2.34)(Iz)β|ψ(z)|=yβ(x2+(y+1)2)β/2≤bβ(x2+a2)β/2⟶0asRφ(z)=x⟶∞.
Also(2.35)sup⁡z∈Π+(Iz)β(Iφ(z))β|ψ(z)|=sup⁡z=x+iy∈Π+yβ(y+1)β1(x2+(y+1)2)β/2≤1<∞,
and the set {z:ℑφ(z)<1} is empty. Thus φ and ψ satisfy all the assumptions of Theorem 2.10, and so Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is compact.



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