# Composition Operators on Some Banach Spaces of Harmonic Mappings

Authors: Munirah Aljuaid; Flavia Colonna
Journal: Journal of Function Spaces
 (2020)

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## Abstract

We study the composition operators on Banach spaces of harmonic mappings that extend several well-known Banach spaces of analytic functions on the open unit disk in the complex plane, including theα-Bloch spaces, the growth spaces, the Zygmund space, the analytic Besov spaces, and the space BMOA.


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## Body

## 1. Introduction

Given a simply connected regionΩ in the complex plane ℂ, a harmonic mapping with domain Ω is a complex-valued function h defined on Ω satisfying the Laplace equation:(1)Δh≔4hzz¯≡0,onΩ,where hzz¯ is the mixed complex second partial derivative of h.It is well known that a harmonic mappingh admits a representation of the form f+g¯, where f and g are analytic functions. This representation is unique if, for a fixed a base point z0 in the domain, the function g is chosen so that gz0=0.In this paper, we shall assume all the functions under consideration are defined onD=z∈ℂ:z<1. Let us denote ℋD as the set of harmonic mappings on D, HD as the set of analytic functions on D, SD as the set of analytic self-maps of D, and AutD as the group of (conformal) disk automorphisms of D.Given an analytic self-mapφ of D, the composition operator induced by φ is defined as the operator(2)Cφf=f∘φ,for all f belonging to a selected class. It is immediate to see that such an operator preserves harmonic mappings.Since analytic functions are clearly harmonic, an interesting question is how to extend to harmonic mappings Banach space structures of known spaces of analytic functionsX in such a way that the norm on the larger space agrees with the norm of X when restricting to the elements of X.An example of a space of harmonic mappings onD that extends a Banach space of analytic functions is BMOH, defined as the space of harmonic mappings on D which are Poisson integrals of functions on the unit circle ∂D belonging to BMO, which was thoroughly studied by Girela [1]. In that work, it was shown that HD∩BMOH is the space BMOA of analytic functions of bounded mean oscillation.In [2], the first author pursued this study by extending several classes of Banach spaces, including the Bloch space ℬ and its generalizations ℬα known as α-Bloch spaces introduced by Zhu in [3], the growth spaces A−α (where α>0), the Zygmund space Z, and the analytic Besov spaces Bp for p>1. In particular, the linear structure and properties of the harmonic α-Bloch spaces ℬHα, the harmonic growth spaces AH−α (for α>0), and the harmonic Zygmund space ZH were studied in [4, 5]. The harmonic Besov spaces BHp for p>1 were introduced in [2].In this work, after giving in Section2 some preliminaries on the spaces of harmonic mappings mentioned above, we introduce the harmonic Besov space BH1 and an alternative extension of BMOA to harmonic mappings denoted by BMOAH. We then analyze the composition operators acting on all such spaces. Specifically, we characterize the composition operators that are bounded, compact, or bounded below and identify the isometries for most of them. For each of these spaces, we also examine the eigenfunctions of the composition operators.Let us denoteXH as a harmonic extension of the Banach space X of analytic functions whose corresponding norms coincide on the elements of X. For the spaces treated in this work, due to simple estimates connecting the seminorm of a harmonic mapping in XH to the seminorm of the associated analytic and antianalytic (i.e., conjugate of analytic) components in X, it turns out that the composition operator Cφ acting on XH is bounded (respectively, compact, bounded below, closed range) if and only if Cφ acting on X is bounded (respectively, compact, bounded below, closed range). This result will be a consequence of a general theorem proved in Section 3.In Section4, we focus on the study of the isometries among the composition operators. Finally, in Section 5, we study the eigenfunctions of Cφ acting on the harmonic spaces ℬHα, AH−α, ZH, BHp, and BMOAH.


## 2. Harmonic Spaces Treated in This Work

### 2.1. Harmonicα-Bloch Spaces

Forα>0, the harmonic α-Bloch space ℬHα is the collection of all h∈ℋD such that(3)βhα≔supz∈D1−z2αhzz+hz¯z<∞.The mappingh⟼hℬHα≔h0+βhα defines a Banach space structure on ℬHα. This space is an extension to harmonic mappings of the (analytic) α-Bloch space ℬα introduced by Zhu in [3]. We recall that an analytic function f belongs to ℬα if and only if(4)βfα≔supz∈D1−z2αf′z<∞,with norm fℬα=f0+βfα. Thus, representing h∈ℋD as f+g¯ with f,g∈HD and g0=0, we see that hz=f′ and hz¯=g′¯. Hence, βhα=supz∈D1−z2αf′z+g′z, and(5)12βfα+βgα≤maxβfα,βgα≤βhα≤βfα+βgα.Consequently, a harmonic mappingh belongs to ℬHα if and only if the unique functions f and g analytic on D such that h=f+g¯ with g0=0 are in ℬα (for more information on the spaces ℬHα, see [4]).Forα=1, the space ℬα is the classical Bloch space ℬ, and the corresponding harmonic extension will be denoted by ℬH. The elements of this space were first studied in [6].


### 2.2. Harmonic Growth Spaces

Forα>0, the harmonic growth space AH−α is the collection of all harmonic mappings h on D such that(6)hAH−α≔supz∈D1−z2αhz<∞.The mappingh⟼hAH−α defines a Banach space structure on AH−α, which again extends that of the (analytic) growth space A−α. For α>0, the functions f,g∈A−α if and only if f+g¯∈AH−α (see [2, 4] for details; for more information on the analytic growth space, see [7]).


### 2.3. Harmonic Zygmund Space

Recall that the Zygmund space is the spaceZ consisting of the analytic functions f on D such that f′∈ℬ with norm(7)fZ≔f0+f′0+supz∈D1−z2f″z.Theharmonic Zygmund space, introduced in [2], is the collection of harmonic mappings h on D such that hz+hz¯∈ℬH. Define(8)hZH≔h0+hz0+hz¯0+βhz+hz¯=h0+hz0+hz¯0+supz∈D1−z2hzzz+hz¯z¯z.The mappingh⟼hZH defines a Banach space structure on ZH. The harmonic mapping h belongs to ZH if and only if the unique functions f and g analytic on D such that h=f+g¯ with g0=0 are in Z. Furthermore,(9)12fZ+gZ≤hZH≤fZ+ggZ.In [5], it is shown that the elements of the space ZH can be characterized in terms of the membership to the classical Zygmund class, and the corresponding norms are equivalent.


### 2.4. Harmonic Besov Spaces

For1<p<∞, a harmonic mapping h in D is said to belong to the harmonic Besov space BHp if the quantity(10)bph≔∫D1−z2p−2hzz+hz¯zpdAz1/p,is finite, where dAz is the normalized area measure on D such that AD=1.The mappingh⟼bph defines a Möbius-invariant seminorm, and h⟼hBHp≔h0+bph defines a Banach space structure on BHp which extends the ordinary norm on Bp. As shown in [2], for 1<p<∞, two analytic functions f and g on D belong to Bp if and only if f+g¯∈BHp. Moreover, if g0=0, then(11)12fBp+gBp≤f+g¯BHp≤2fBp+gBp.Recall that in the analytic case, the Besov spaceB2 is the Dirichlet space D, which is a Hilbert space under an equivalent norm, namely,(12)fD≔f02+∫Df′z2dAz1/2.Likewise, the harmonic Besov space forp=2, which we call the harmonic Dirichlet space and denote by DH, can be endowed with a Hilbert space structure via the inner product:(13)h1,h2DH≔h10h20¯+∫Dh1zzh2zz¯+h1z¯zh2z¯z¯dAz,whose associated norm(14)hDH=h02+∫Dhzz2+hz¯z2dAz1/2,is equivalent to hBH2. In particular, if h∈DH, h=f+g¯ with g0=0, then(15)hDH2=fD2+gD2.The analytic Besov spaceB1 is defined as the set consisting of the functions f∈HD of the form(16)fz=∑n=1∞anLwn,where an is absolutely summable, and wn∈D for n∈ℕ and for w∈D, Lw is the disk automorphism defined by(17)Lwz=w−z1−w¯z.The norm off in B1 is defined as(18)fB1≔inf∑n=1∞an,where the infimum is taken over all above representations of f. Under this norm, B1 is a Möbius-invariant Banach space contained in the disk algebra H∞∩CD¯ (see [8] for details).In [9], it was shown that B1 is the smallest Möbius-invariant space, which is why it is commonly known as the minimal Möbius-invariant space. Moreover, f∈B1 if and only if(19)∫Df″zdAz<∞.A more convenient non-Möbius-invariant norm onB1 equivalent to ⋅B1 is given by(20)f≔maxf0,f′0,∫Df″zdAz.We introduce theharmonic Besov space BH1 as the collection of harmonic mappings h on D for which(21)b1h≔∫Dhzzz+hz¯z¯zdAz<∞.Representing a harmonic mappingh as f+g¯ with f,g∈HD and g0=0, we see that hzz=f″ and hz¯z¯=g″¯. Thus, h∈BH1 if and only if f,g∈B1 and(22)12f+g≤maxf,g≤h≤f+g,where h≔maxh0,hz0+hz¯0+b1h. Consequently, the set BH1 is contained in ℋD∩CD¯, and a harmonic mapping h belongs to BH1 if and only if it admits a representation of the form(23)hz=∑n=1∞anLwn+∑n=1∞bnLζn¯,where an,bn∈ℓ1, and wn,ζn∈D for each n∈ℕ. This leads to the following equivalent Möbius-invariant norm on BH1 that extends (18):(24)hBH1≔inf∑n=1∞an+∑n=1∞bn,where the infimum is taken over all above representations of h. Under both norms, BH1 is a Banach space and for h=f+g¯ with f,g∈HD(25)12fB1+gB1≤hBH1≤fB1+gB1.


### 2.5. Harmonic Space of Bounded Mean Oscillation

Recall that the spaceBMOA of analytic functions of bounded mean oscillation is the Banach space with norm(26)fBMOA≔f0+supa∈Df∘La−faH2<∞,and the associated seminorm is Möbius invariant. We recall that the norm of a function f in the Hardy Hilbert space H2 is defined as(27)fH2=sup0<r<112π∫02πfreiθ2dθ1/2.It is well known that a functionf∈HD is in BMOA if and only if(28)supa∈D∫Df′z21−Laz2dAz<∞,and ⋅BMOA is equivalent to ⋅∗ defined by(29)f∗≔f0+supa∈D∫Df′z21−Laz2dAz1/2.see [1] for a comprehensive analysis of the functions of bounded mean oscillation.The spaceBMOH of harmonic functions of bounded mean oscillation is the collection of harmonic mappings h on D of the form h=Pf, where f∈BMO∂D and Pf is the Poisson integral of f. As shown in [1], a norm on this space whose associated seminorm yields the Möbius-invariant seminorm in BMOA when restricted to the analytic functions on D is given by(30)hBMOH=h0+supa∈Dh∘La−hah2,where for h∈ℋD,(31)hh2=sup0<r<112π∫02πhreiθ2dθ1/2.Iff is analytic, then fh2=fH2. Representing h as f+g¯ with f,g analytic and g0=0, we see that(32)h∘La−hah2≤2f∘La−faH2+g∘La−gaH2.Thus, iff,g∈BMOA, then f+g¯∈BMOH. However, this norm does not lead to a lower bound in terms of the norms of f and g. Thus, for h∈ℋD, we define the harmonic extension of the space BMOA as follows:(33)hH,∗≔h0+supa∈D∫Dhzz2+hz¯z21−Laz2dAz1/2,if this expression is finite. We see that taking h=f+g¯ with g0=0, so that hz=f′ and hz¯=g′¯, we have(34)12f∗+g∗≤f+g¯H,∗≤f∗+g∗.In particular, consistent with the notation we have been using throughout, denoting byBMOAH the collection of such harmonic mappings, we see that under ⋅H,∗, BMOAH is a Banach space that extends the norm ⋅∗ on BMOA. Moreover, h∈BMOAH if and only if the associated analytic functions f and g belong to BMOA.It is well known that for1<p≤q<∞, B1⊆Bp⊆Bq⊆BMOA⊆ℬ (e.g., see [10]), and all inclusions are proper. Due to the connection between the respective norms, we see that the same inclusion relations hold for the corresponding harmonic spaces.


## 2.1. Harmonicα-Bloch Spaces

Forα>0, the harmonic α-Bloch space ℬHα is the collection of all h∈ℋD such that(3)βhα≔supz∈D1−z2αhzz+hz¯z<∞.The mappingh⟼hℬHα≔h0+βhα defines a Banach space structure on ℬHα. This space is an extension to harmonic mappings of the (analytic) α-Bloch space ℬα introduced by Zhu in [3]. We recall that an analytic function f belongs to ℬα if and only if(4)βfα≔supz∈D1−z2αf′z<∞,with norm fℬα=f0+βfα. Thus, representing h∈ℋD as f+g¯ with f,g∈HD and g0=0, we see that hz=f′ and hz¯=g′¯. Hence, βhα=supz∈D1−z2αf′z+g′z, and(5)12βfα+βgα≤maxβfα,βgα≤βhα≤βfα+βgα.Consequently, a harmonic mappingh belongs to ℬHα if and only if the unique functions f and g analytic on D such that h=f+g¯ with g0=0 are in ℬα (for more information on the spaces ℬHα, see [4]).Forα=1, the space ℬα is the classical Bloch space ℬ, and the corresponding harmonic extension will be denoted by ℬH. The elements of this space were first studied in [6].


## 2.2. Harmonic Growth Spaces

Forα>0, the harmonic growth space AH−α is the collection of all harmonic mappings h on D such that(6)hAH−α≔supz∈D1−z2αhz<∞.The mappingh⟼hAH−α defines a Banach space structure on AH−α, which again extends that of the (analytic) growth space A−α. For α>0, the functions f,g∈A−α if and only if f+g¯∈AH−α (see [2, 4] for details; for more information on the analytic growth space, see [7]).


## 2.3. Harmonic Zygmund Space

Recall that the Zygmund space is the spaceZ consisting of the analytic functions f on D such that f′∈ℬ with norm(7)fZ≔f0+f′0+supz∈D1−z2f″z.Theharmonic Zygmund space, introduced in [2], is the collection of harmonic mappings h on D such that hz+hz¯∈ℬH. Define(8)hZH≔h0+hz0+hz¯0+βhz+hz¯=h0+hz0+hz¯0+supz∈D1−z2hzzz+hz¯z¯z.The mappingh⟼hZH defines a Banach space structure on ZH. The harmonic mapping h belongs to ZH if and only if the unique functions f and g analytic on D such that h=f+g¯ with g0=0 are in Z. Furthermore,(9)12fZ+gZ≤hZH≤fZ+ggZ.In [5], it is shown that the elements of the space ZH can be characterized in terms of the membership to the classical Zygmund class, and the corresponding norms are equivalent.


## 2.4. Harmonic Besov Spaces

For1<p<∞, a harmonic mapping h in D is said to belong to the harmonic Besov space BHp if the quantity(10)bph≔∫D1−z2p−2hzz+hz¯zpdAz1/p,is finite, where dAz is the normalized area measure on D such that AD=1.The mappingh⟼bph defines a Möbius-invariant seminorm, and h⟼hBHp≔h0+bph defines a Banach space structure on BHp which extends the ordinary norm on Bp. As shown in [2], for 1<p<∞, two analytic functions f and g on D belong to Bp if and only if f+g¯∈BHp. Moreover, if g0=0, then(11)12fBp+gBp≤f+g¯BHp≤2fBp+gBp.Recall that in the analytic case, the Besov spaceB2 is the Dirichlet space D, which is a Hilbert space under an equivalent norm, namely,(12)fD≔f02+∫Df′z2dAz1/2.Likewise, the harmonic Besov space forp=2, which we call the harmonic Dirichlet space and denote by DH, can be endowed with a Hilbert space structure via the inner product:(13)h1,h2DH≔h10h20¯+∫Dh1zzh2zz¯+h1z¯zh2z¯z¯dAz,whose associated norm(14)hDH=h02+∫Dhzz2+hz¯z2dAz1/2,is equivalent to hBH2. In particular, if h∈DH, h=f+g¯ with g0=0, then(15)hDH2=fD2+gD2.The analytic Besov spaceB1 is defined as the set consisting of the functions f∈HD of the form(16)fz=∑n=1∞anLwn,where an is absolutely summable, and wn∈D for n∈ℕ and for w∈D, Lw is the disk automorphism defined by(17)Lwz=w−z1−w¯z.The norm off in B1 is defined as(18)fB1≔inf∑n=1∞an,where the infimum is taken over all above representations of f. Under this norm, B1 is a Möbius-invariant Banach space contained in the disk algebra H∞∩CD¯ (see [8] for details).In [9], it was shown that B1 is the smallest Möbius-invariant space, which is why it is commonly known as the minimal Möbius-invariant space. Moreover, f∈B1 if and only if(19)∫Df″zdAz<∞.A more convenient non-Möbius-invariant norm onB1 equivalent to ⋅B1 is given by(20)f≔maxf0,f′0,∫Df″zdAz.We introduce theharmonic Besov space BH1 as the collection of harmonic mappings h on D for which(21)b1h≔∫Dhzzz+hz¯z¯zdAz<∞.Representing a harmonic mappingh as f+g¯ with f,g∈HD and g0=0, we see that hzz=f″ and hz¯z¯=g″¯. Thus, h∈BH1 if and only if f,g∈B1 and(22)12f+g≤maxf,g≤h≤f+g,where h≔maxh0,hz0+hz¯0+b1h. Consequently, the set BH1 is contained in ℋD∩CD¯, and a harmonic mapping h belongs to BH1 if and only if it admits a representation of the form(23)hz=∑n=1∞anLwn+∑n=1∞bnLζn¯,where an,bn∈ℓ1, and wn,ζn∈D for each n∈ℕ. This leads to the following equivalent Möbius-invariant norm on BH1 that extends (18):(24)hBH1≔inf∑n=1∞an+∑n=1∞bn,where the infimum is taken over all above representations of h. Under both norms, BH1 is a Banach space and for h=f+g¯ with f,g∈HD(25)12fB1+gB1≤hBH1≤fB1+gB1.


## 2.5. Harmonic Space of Bounded Mean Oscillation

Recall that the spaceBMOA of analytic functions of bounded mean oscillation is the Banach space with norm(26)fBMOA≔f0+supa∈Df∘La−faH2<∞,and the associated seminorm is Möbius invariant. We recall that the norm of a function f in the Hardy Hilbert space H2 is defined as(27)fH2=sup0<r<112π∫02πfreiθ2dθ1/2.It is well known that a functionf∈HD is in BMOA if and only if(28)supa∈D∫Df′z21−Laz2dAz<∞,and ⋅BMOA is equivalent to ⋅∗ defined by(29)f∗≔f0+supa∈D∫Df′z21−Laz2dAz1/2.see [1] for a comprehensive analysis of the functions of bounded mean oscillation.The spaceBMOH of harmonic functions of bounded mean oscillation is the collection of harmonic mappings h on D of the form h=Pf, where f∈BMO∂D and Pf is the Poisson integral of f. As shown in [1], a norm on this space whose associated seminorm yields the Möbius-invariant seminorm in BMOA when restricted to the analytic functions on D is given by(30)hBMOH=h0+supa∈Dh∘La−hah2,where for h∈ℋD,(31)hh2=sup0<r<112π∫02πhreiθ2dθ1/2.Iff is analytic, then fh2=fH2. Representing h as f+g¯ with f,g analytic and g0=0, we see that(32)h∘La−hah2≤2f∘La−faH2+g∘La−gaH2.Thus, iff,g∈BMOA, then f+g¯∈BMOH. However, this norm does not lead to a lower bound in terms of the norms of f and g. Thus, for h∈ℋD, we define the harmonic extension of the space BMOA as follows:(33)hH,∗≔h0+supa∈D∫Dhzz2+hz¯z21−Laz2dAz1/2,if this expression is finite. We see that taking h=f+g¯ with g0=0, so that hz=f′ and hz¯=g′¯, we have(34)12f∗+g∗≤f+g¯H,∗≤f∗+g∗.In particular, consistent with the notation we have been using throughout, denoting byBMOAH the collection of such harmonic mappings, we see that under ⋅H,∗, BMOAH is a Banach space that extends the norm ⋅∗ on BMOA. Moreover, h∈BMOAH if and only if the associated analytic functions f and g belong to BMOA.It is well known that for1<p≤q<∞, B1⊆Bp⊆Bq⊆BMOA⊆ℬ (e.g., see [10]), and all inclusions are proper. Due to the connection between the respective norms, we see that the same inclusion relations hold for the corresponding harmonic spaces.


## 3. General Theorem on Composition Operators

LetXH be a Banach space of harmonic mappings on D with seminorm ⋅sXH and norm(35)hXH=h0+hsXH,such that the point-evaluation functionals are bounded. Let X=HD∩XH and denote by ⋅sX and ⋅X the seminorm and norm induced on X, that is, for f∈X, let fX=f0+fsX=f0+fsXH=fXH. Assume further that for each h=f+g¯∈XH with g0=0, the associated analytic functions f and g belong to X and(36)f+g¯sXH≍fsX+gsX,where by A≍B we mean c1A≤B≤c2A for some positive constants c1 and c2.Theorem 1.
Ifφ∈SD, then(a)
Cφ:XH⟶XH is bounded if and only if Cφ:X⟶X is bounded(b)
Cφ:XH⟶XH is compact if and only if Cφ:X⟶X is compact(c)
Cφ:XH⟶XH is bounded below if and only if Cφ:X⟶X is bounded below(d)
Cφ:XH⟶XH is closed range if and only if Cφ:X⟶X is closed rangeProof.
Note that (c) and (d) are equivalent since composition operators are injective. For statements (a), (b), and (c), the implication⟹ is obvious. To prove that the converse statements hold, assume Cφ:X⟶X is bounded. Let h∈XH and let f and g be the associated analytic functions so that h=f+g¯ and g0=0. Then, f,g∈X, and by our assumption that Cφ is bounded on X, it follows that f∘φ,g∘φ∈X. Thus, h∘φ∈XH. By the Closed Graph Theorem, the operator Cφ is bounded on XH.
Next, assumeCφ:X⟶X is compact. Let hn be a sequence in XH such that hnXH≤1 and for each n∈ℕ, let fn,gn be the associated analytic functions so that hn=fn+gn¯ and gn0=0. By (36), fnsX+gnsX≤ChnXH≤C. Moreover, fn0=hn0. Thus, the sequences fn and gn have bounded norms in X, and hence by the compactness of Cφ:X⟶X, some subsequence fnk∘φk∈ℕ converges in norm to some function f∈X. Again, by the compactness of Cφ:X⟶X, from the sequence nkk∈ℕ, we may then extract a subsequence nkjj∈ℕ such that gnkj∘φj∈ℕ converges in norm to some g∈X. Then, h≔f+g¯∈XH and the sequence hnkj∘φj∈ℕ converges in norm to h. Therefore, Cφ:XH⟶XH is compact.
Lastly, assumeCφ:X⟶X is bounded below. Then, there exists some constant δ>0 such that for each f∈X, f∘φX≥δfX. On the other hand, by (36), there exist constants C1,C2>0, such that for each h=f+g¯∈XH with g0=0, C1fX+gX≤hXH≤C2fX+gX Thus,(37)h∘φXH≥C1f∘φX+g∘φX≥C1δfX+gX≥C1δC2hXH,proving that Cφ:XH⟶XH is bounded below.Of course, the conclusion of Theorem1 also holds if condition (36) holds for the norms of X and XH.Focusing on the compactness ofCφ acting on the spaces ℬHα, ZH, BHp, and BMOAH, which satisfy along with the corresponding analytic counterparts ℬα, Z, Bp, and BMOA, the conditions of Theorem 1, from Theorem 2 of [11], Theorem 1.4 of [12], Theorem 4.3 of [13], Theorem 3.7 of [14], Corollary 5 of [15], and Corollary 3.2 and Remark 3.3 of [16] (see also Theorem 1.1 of [17] and Theorem 1 of [18] for alternative characterizations), we deduce the following corollaries.Corollary 1.
Letφ∈SD and α>0. Then, Cφ is compact on ℬHα if and only if φ∈ℬα and(38)limφz⟶11−z2αφ′z1−φz2α=0.Corollary 2.
Forφ∈SD, a bounded operator Cφ on ZH (equivalently on Z) is compact if and only if(39)limφz⟶11−z2φ′z21−φz2=0,limφz⟶11−z2φ″zlog11−φz2=0.Corollary 3.
Letφ∈SD, 1<p≤q, 1<r<q. If Cφ is bounded on BHr (equivalently on Br), then Cφ:BHp⟶BHq is compact if and only if Cφ:ℬ⟶ℬ is compact if and only if(40)limφz⟶11−z2φ′z1−φz2=0.Corollary 4.
Forφ∈SD, if Cφ is bounded on BH1 (equivalently on B1), then Cφ:BH1⟶BH1 is compact if and only if Cφ:ℬ⟶ℬ is compact if and only if(41)limφz⟶11−z2φ′z1−φz2=0.Corollary 5.
The composition operatorCφ is compact on BMOAH if and only if(42)limφa⟶1Lφa∘φ∘LaH2=0,and,for allR∈0,1,limt⟶1supφa≤Rmζ∈∂D:φ∘Laζ>t=0,where m is the Lebesgue measure.
Since by [3] and its extension to the corresponding harmonic spaces provided in [4], for α>0, the spaces A−α and AH−α are equivalent to ℬα+1 and ℬHα+1, respectively, and Theorem 1 can be applied to the growth spaces as well. In particular, with the appropriate modification of the parameter, Corollary 1 yields a characterization of compactness on the harmonic growth spaces. Using the following result, which is a special case of Theorem 3.2 of [19], we provide below a simpler characterization.Theorem 2.
Letφ be an analytic self-map of D, X a Banach space of analytic functions with norm ⋅ satisfying the following conditions:(a)
X contains the constants.(b)
For eachz∈D, the point-evaluation functional δzf≔fz is in X∗. Thus,(43)δz=supfz:f∈X,f≤1<∞.(c)
The unit ball ofX is relatively compact with respect to the topology of uniform convergence on compact subsets of D.(d)
There is a constantC>0 such that for all S∈AutD and f∈X,(44)Sf≤Cf.
Letα>0 and assume Cφ:X⟶A−α is bounded. Then, Cφ:X⟶A−α is compact if and only if(45)limφz⟶11−z2αδφz=0.
In the special case, whenX=A−α,(46)fz≤11−z2αfA−α,for each z∈D and f∈A−α. Therefore,(47)δz≤11−z2α.
Next, note that the constant function 1 has norm 1 inA−α. Thus, in (47), equality holds at z=0. On the other hand, for z=reiθ, for 0<r<1, and θ∈ℝ, taking as test function(48)fzw=11−e−2iθw2α,we see that fz∈A−α, fzA−α=1, and(49)11−r2α=fzz≤δz.
Hence,(50)δz=11−z2α.
Since the above conditions (a)–(d) clearly hold forA−α, applying Theorem 2 to X=A−α, using (50), and then taking the αth root, we deduce the following result.Corollary 6.
Letφ∈SD, α>0 and assume Cφ is bounded on AH−α (equivalently on A−α). Then, Cφ is compact if and only if(51)limφz⟶11−z21−φz2=0.


## 4. Isometries ofCφ

In this section, we wish to characterize the isometries on the harmonic spacesℬHα, AH−α, ZH, and BHp. For most of the corresponding analytic counterparts, namely, ℬα for α≠1, A−α, Z, and Bp for p>1andp≠2, the only composition operators that are isometries are those induced by rotations [20–23] (see also [24]). Since an isometry on a harmonic space XH that extends an analytic space X is also an isometry on X, Cφ is an isometry on ℬHα for α≠1, AH−α for α>0, ZH and BHp for p>1, p≠2, if and only if φ is a rotation. In the case of the minimal Möbius-invariant space B1 under the Möbius-invariant norm (18), composition operators induced by a disk automorphism are clearly isometries. In fact, Bao and Wulan proved in Theorem 5.2 of [25] that the isometric isomorphisms on B1 are all of the form λCφ, where λ is a unimodular constant and φ∈AutD. Of course, these are also isometries when regarding Cφ as an operator on BH1. To the best of our knowledge, it is not known whether linear isometries among the composition operators on B1 other than those induced by disk automorphisms exist. The following result partially answers this question.Theorem 3.
Ifφ∈SD has a fixed point in D, then Cφ:B1⟶B1 is an isometry if and only if φ∈AutD.Proof.
The sufficiency is clear. Assumep∈D is a fixed point of φ, Cφ:B1⟶B1 is an isometry, and φ is not a disk automorphism. First, assume p=0. Then, φ is not a rotation, and the nth iterate φn of φ is also an isometry of B1 for any n∈ℕ. By the Grand Iteration Theorem (see [26], p. 78), the sequence φn converges to 0 uniformly on compact subsets of D. Thus, limn⟶∞φn∞=0. Therefore, for all n sufficiently large, φn∞<1/2. Hence, for such an n, Cφn is a compact operator on ℬ, and by Corollary 4, it is also compact on B1. By Lemma 3.7 of [14], since the sequence pkk=2∞ defined by pkz=1/k−1zk is bounded in B1 and converges to 0 uniformly on compact subsets of D,(52)limk⟶∞CφnpkB1=0.
On the other hand, sinceφn is an isometry of B1, CφnpkB1=pkB1 is bounded away from 0, as due to the equivalence of the two norms on B1,(53)pk=∫Dpk″zdAz=2∫01krk−1dr=2.
We have reached a contradiction. Therefore,φ must be a rotation in this case.
Next, assumeφp=p for some p∈D. Due to the Möbius invariance of ⋅B1, letting Lp denote the disk automorphism interchanging 0 and p, the function ψ=Lp∘φ∘Lp induces an isometry on B1 and has 0 as a fixed point. By the previous case, ψ must be a rotation. Then, φ=Lp∘ψ∘Lp is a disk automorphism.We conjecture that the conclusion of Theorem3 also holds if φ does not have fixed points in D. In this case, if φ∉AutD, then φ has a boundary fixed point ω (i.e., the Denjoy–Wolff point of φ) such that φn⟶ω locally uniformly, and φ has an angular derivative at ω (see [26], p. 78).We shall see that by contrast, the Bloch space has a very rich set of isometries among the composition operators. This feature is shared by the spaceBMOA with respect to the Möbius-invariant seminorm. As noted in Proposition 2.1 of [27], all such isometries must be induced by a symbol that fixes 0. Laitila observed in Corollary 2 of [27] that the isometries Cφ on the Bloch space are also isometries on BMOA. However, the inclusion is proper. For example, finite Blaschke products induce isometries Cφ on BMOA but not on ℬ unless they are rotations. We are not aware of results of this type on BMOA with respect to the equivalent norm ⋅∗. Thus, we leave to the reader the following problem for future investigation.Open questions:(1)
Are there any isometries among the composition operators onBMOA with respect to the norm ⋅∗ other than those induced by rotations?(2)
If this question has a positive answer, are all nontrivial isometries onBMOA also isometries on BMOAH?To complete the study of the isometries on the harmonic spaces under consideration in our work, it remains to analyze the cases whenCφ acts on the harmonic Bloch space ℬH and on the harmonic Dirichlet space DH to determine whether the nontrivial isometries Cφ on ℬ and D are also isometries on the larger spaces ℬH and DH. We shall prove that this is indeed the case.We shall make use of the following results.Theorem 4 (see [28], Theorem 5 and Corollary 2).
Letφ:D⟶D analytic function, then the following are equivalent:(a)
Cφ:ℬ⟶ℬ is an isometry.(b)
φ0=0 and φℬ=1.(c)
Eitherφz=λz for some λ∈ℂ, λ=1, or φ=gB, where g∈SD is nonvanishing and B is an infinite Blaschke product whose zeros form a sequence znn∈ℕ containing 0 and a subsequence znjj∈ℕ such that gznj⟶1 and(54)limj⟶∞∏k≠njznj−zk1−znj¯zk=1.Theorem 5 (see [29], Theorem 2.7).
Forφ∈SD, Cφ is an isometry on ℬ if and only if φ0=0 and either φ is a rotation or for every a∈D there exists a sequence zn in D such that zn⟶1, φzn=a and(55)limn⟶∞1−zn2φ′zn=1−a2.
We can now prove our main result in this section.Theorem 6.
Letφ be an analytic self-map of D. Then, the following statements are equivalent.(a)
Cφ is an isometry on ℬH.(b)
Cφ is an isometry on ℬ.(c)
φ0=0 and φℬ=1.Proof.
Since by Theorem4, (b) is equivalent to (c), and as observed above, (a) clearly implies (b); we only need to show that (c) implies (a).
Assumeφ0=0 and φℬ=1. Then, Cφ=1. If φ is a rotation, the result is clear. So, assume φ is not a rotation.
Observe that sinceφ0=0, it suffices to show that Cφ is seminorm preserving on ℬH. Equivalently, h∘φℬH=hℬH for all h∈ℬH such that h0=0. Also, by dividing by the norm, we only need to prove that h∘φℬH=1 for hℬH=1 with h0=0.
So, assumeh satisfies the conditions hℬH=1 and h0=0. Then,(56)supz∈D1−z2hzz+hz¯z=1.
Thus, either this supremum is attained at some point inside the disk or it is a limit along a sequence of points inD approaching the unit circle. Specifically, one of the following two cases must hold:(i)
There existsa∈D such that(57)1−a2hza+hz¯a=1.(ii)
There exists a sequenceak is D such that ak⟶1 and(58)limk⟶∞1−ak2hzak+hz¯ak=1.
Assume first (i) holds. Then, by Theorem5, there exists a sequence zn in D such that zn⟶1 as n⟶∞, φzn=a for each n∈ℕ, and(59)limn⟶∞1−zn2φ′zn=1−a2.
Therefore,(60)1−zn2h∘φzzn+h∘φz¯zn=1−zn2hzφznφ′zn+hz¯φznφ′zn¯=1−zn2φ′znhza+hz¯a⟶1−a2hza+hz¯a,asn⟶∞.
Using (i), it follows that(61)1=limn⟶∞1−zn2h∘φzzn+h∘φz¯zn≤supz∈D1−z2h∘φzz+h∘φz¯z=CφhℬH.
Since, as observed above, the operatorCφ has norm 1, CφhℬH≤1. It follows that CφhℬH=1.
Next, assume (ii) holds. Then, again by Theorem5, for each k∈ℕ, there exists a sequence zn,kn∈ℕ in D such that zn,k⟶1 as n⟶∞, φzn,k=ak and(62)limn⟶∞1−zn,k2φ′zn,k=1−ak2.
Proceeding as above, for eachk∈ℕ, we have(63)1−zn,k2h∘φzzn,k+h∘φz¯zn,k=1−zn,k2φ′zn,khzak+hz¯ak⟶1−ak2hzak+hz¯ak,asn⟶∞.
Hence, taking the limit ask⟶∞ and using (ii), we deduce that(64)1=limk⟶∞1−ak2hzak+hz¯ak=limk⟶∞limn⟶∞1−zn,k2h∘φzzn,k+h∘φz¯zn,k≤supz∈D1−z2h∘φzz+h∘φz¯z=CφhℬH.
Thus,CφhℬH=1 also in this case. Therefore, Cφ is an isometry on ℬH.We now turn our attention to the identification of the isometries onDH.A functionφ∈SD is said to be a univalent full map if it is one-to-one, and the complement of the range of φ has null area measure. In [30], the authors characterize the isometries among the composition operators on the Dirichlet space D.Theorem 7 (see [30], p.1703).
A composition operatorCφ acting on the Dirichlet space D is an isometry if and only if φ is a univalent full map of D that fixes 0.
We are ready to characterize the isometry onDH.Theorem 8.
The bounded composition operatorCφ on DH is an isometry if and only if φ is a univalent full map fixing 0.Proof.
AssumeCφ is an isometry on DH. Since the Dirichlet space D is a subspace of DH and the norm of DH equals the norm of D, the operator Cφ is also an isometry on D. Thus, by Theorem 7, φ is a univalent full map fixing the origin.
Conversely, assumeφ is a univalent full map fixing the origin. Let h∈DH. Then, h=f+g¯ with f,g∈HD and g0=0. So, f,g∈D, and(65)Cφh=Cφf+Cφg¯=Cφf+Cφg¯.
Thus, by Theorem7, CφfD=fD, CφgD=gD. Therefore, by (15), we have(66)CφhDH2=CφfD2+CφgD2=fD2+gD2=hDH2,proving that Cφ is an isometry on DH.


## 5. Eigenfunctions ofCφ

In the early 1870s, Ernst Schröder introduced the eigenvalue equation for composition operators. Supposeφ is an analytic self-map of D, and λ is complex constant. The functional equation(67)f∘φ=λf,forf∈HD,is known as Schröder’s equation.Lemma 1.
Assumeφ∈SD, φ nonconstant, p∈D with φp=p. Let λ≠1, and suppose f is an analytic function such that for each z∈D(68)f∘φz=λfz.
Ifφ′p=0, then f≡0.Proof.
Assume an analytic functionf satisfying equation (68) exists with f not identically zero. Since φp=p and φ′p=0, there is a natural number k>1, such that for z sufficiently near p(69)φz−p=z−pkψz,for some analytic function ψ with ψp≠0. Evaluating equation (68) at p, we get(70)fp=fφp=λfp.
Sinceλ≠1, it follows that fp=0. Thus, in some neighborhood of p, f admits the representation of the form(71)fz=z−pmgz,for some analytic function g such that gp≠0 and m∈ℕ. Then, substituting this expression into equation (68), we obtain(72)z−pkmψzmgφz=φz−pmgφz=λz−pmgz,for z near p. Since km>m, we have reached a contradiction.Lemma 2.
Assumeφ∈SD fixes 0 and that φ is not a rotation. Then, the only eigenfunctions of Cφ relative to the eigenvalue 1 are the nonzero constants.Proof.
Assume 1 is an eigenvalue ofCφ, and let f be a corresponding eigenfunction. Then, f∘φ=f implies that for all n∈ℕ(73)f∘φn=f,where φn is the nth iterate of φ.
Fixa∈D,a≠0. Let an=φna for all n∈ℕ. Then, the sequence an must converge to 0. Thus,(74)fan=fφna=fa,for alln∈ℕ.
By continuityfa=f0. Therefore, f is constant.Forφ∈SD which fixes a point p∈D and such that 0<φ′p<1, the unique solution σ of Schröder’s equation corresponding to the eigenvalue φ′p such that σ′p=1 is called the Königs function of φ or principal eigenfunction of Cφ.We state below Königs’ theorem, which gives a description of the eigenvalues and corresponding eigenfunctions of the composition operatorCφ, considered as a linear transformation on HD, when the symbol is nonautomorphic with a fixed point in D.Theorem 9 (see [26]).
Letφ∈SD∖AutD with a fixed point p∈D.(a)
Ifφ′p=0, then 1 is the only eigenvalue of Cφ.(b)
Ifφ′p≠0, then the set of eigenvalues of Cφ is given by(75)φ′pn:n∈ℕ∪0.Each eigenvalue has multiplicity 1, and the functionσn spans the eigenspace for φ′pnn∈ℕ.(c)
Ifφ is univalent, then so is σ.
The following result is a special case of a theorem proved by Hammond in his doctoral dissertation that extends to a general Banach space of analytic functions onD the Eigenfunction Theorem valid for the Hardy Hilbert space (see [26], p.94).Theorem 10 (see [31]).
IfCφ is compact on a Banach space X of analytic functions on D, then the eigenfunctions σn belong to the space X for each n∈ℕ.
Using Theorem10, Paudyal in [32] obtained the following sufficient condition that ensures that all the powers of the Königs function belong to the α-Bloch space.Theorem 11.
(see [32], Theorem 2.1.9). Suppose φ∈SD with φ0=0, 0<φ′0<1, Cφ is bounded on ℬα for some α>0, and there exists k∈ℕ such that φk∞<1. Then, σn∈ℬα for all n∈ℕ.
Since forα>0, the growth space A−α can be identified with the Bloch-type space ℬα+1, the conclusion of Theorem 11 also holds when Cφ is bounded on A−α. Arguing as in the proof of Theorem 11 provided in [32] since composition operators whose symbol has supremum norm smaller than 1 are compact on the Zygmund space, the Besov spaces, and BMOA, using Theorem 10, we obtain the following extension to the other spaces treated in this paper.Corollary 7.
Supposeφ∈SD with φ0=0, 0<φ′0<1, Cφ is bounded on Z, respectively, on Bp for p≥1, respectively, on BMOA, and there exists k∈ℕ such that φk∞<1. Then, σn∈Z, respectively, Bp, respectively, BMOA, for all n∈ℕ.
As observed above, ifφ∈SD with φ0=0 and 0<φ′0<1, then the only eigenvalues of Cφ must be of the form φ′0n for some n∈ℕ∪0, and the corresponding eigenfunctions must be constant multiples of σn.
We are now interested in determining the eigenfunctions ofCφ as an operator acting on spaces of harmonic mappings with domain D. The following result shows that these eigenfunctions are closely related to the eigenfunctions of the analytic counterparts.Theorem 12.
Letφ∈SD with φ0=0 and 0<φ′0<1. Suppose λ∈ℂ and h is a harmonic mapping on D such that(76)h∘φ=λh.
Then, eitherλ=1, in which case h is constant, or there exists n∈ℕ such that λ=φ′0n. In the latter case, h is a linear combination of σn and its conjugate. Moreover, the argument of λ must be a rational multiple of π. When reduced to lowest terms, the denominator of this rational factor of π must be a divisor of n.Proof.
Letf,g∈HD with g0=0 and h=f+g¯. Then, by (76),(77)f∘φ+g∘φ¯=λf+g¯,so rearranging the terms, we obtain(78)f∘φ−λf=λg¯−g∘φ¯.
Since the left-hand side of (78) is analytic, and the right-hand side is antianalytic, their common value must be a constant η. Thus,(79)f∘φ−λf=η,g∘φ¯−λg¯=η.
Sinceφ0=0 and g0=0, from (79), we obtain η=0. Hence,(80)f∘φ=λf,g∘φ¯=λg¯.
Ifλ=1, then by Lemma 2, the only analytic functions F satisfying the equation F∘φ=F are the constant functions. Thus, g is the constant 0, and h=f is constant.
Next assumeλ≠1 and φ′0=reiθ with 0<r<1 and θ∈ℝ. From Königs’ Theorem applied to the equation f∘φ=λf, it follows that λ must be a power of φ′0, and f is a multiple of the corresponding power of σ, that is, there exist an n∈ℕ and α∈ℂ such that λ=φ′0n and f=ασn.
On the other hand,g∘φ=λ¯g, so, again by Königs’ Theorem, there exist m∈ℕ and β∈ℂ such that λ¯=φ′0m and g=βσm. Then, rneinθ=rme−imθ. Thus, n=m and e2inθ=1. Therefore,(81)h=ασn+β¯σ¯n,and θ=πk/n, for some integer k, completing the proof.Remark 1.
From the above, we see that ifh satisfies (76), where λ=φ′0 (which corresponds to the special case n=1), then φ′0 must be real.
In the context of the composition operator acting on the harmonic spacesXH treated in this work, equivalently, on X=XH∩HD, a natural question that arises is as follows.
For which symbolsφ fixing 0 is the set of eigenvalues of Cφ regarded as an operator on XH given by φ′0n:n∈ℕ∪0?
By Theorem12, the required conditions are that φ′0 must be real, and all the powers of the Königs function σ must be in X.
In particular, every compact operatorCφ on X (equivalently, on XH) satisfies the condition σn∈X for all n∈ℕ. Since Cφ is a compact operator whenever φ∞<1, we are interested in the case when the closure of the range of φ intersects the unit circle.
As a consequence of Theorem11 and Corollary 7, we can now give a simple example of a nonunivalent function φ∈SD with supremum norm 1 for which the operator Cφ:X⟶X is noncompact and has φ′0n:n∈ℕ∪0 as full set of eigenvalues for any of the spaces X treated in this work.
Example: consider the polynomial:(82)φz=12z2−z.
Then,φ maps D into itself, has 0 as a fixed point, φ′0=−1/2 and is not univalent since φ1/4=φ3/4. Moreover, regarding φ as a function on D, φ∞=1 since φ−1=1. In fact, φ−11=−1, and a straightforward computation shows that φD¯ touches the unit circle only at the point 1.
Let us first focus on the case whenX is ℬα. Since(83)supz∈D1−z2αφ′z1−φz2α<∞,by Theorem 1.4 of [12], the operator Cφ is bounded on ℬα or ℬHα. Moreover, for z∈−1,0,(84)1−14z2−z2=144−z2z−12=142+z1−z2+z+z2.
Thus, asz⟶−1+ along the real axis,(85)1−z2α1/22z−11−1/4z2−z2α=22α−11+zα2z−12+zα2+z+z2α⟶23α−1.
Hence,(86)limφz⟶11−z2αφ′z1−φz2α=limz⟶−1+1−z2α1/22z−11−1/4z2−z2α≠0,where the last limit is taken along the real axis.
Therefore, by Corollary1, the operator Cφ is not compact on ℬα or ℬHα.
Since the only point of contact ofφ with the unit circle occurs at −1 and φ2−1=φ1=0, we see that φ2∞<1. Thus, by Theorem 11, σn∈ℬα for all n∈ℕ.
Of course, the same conclusion holds for the growth spacesA−α since, as observed in the paragraph before Theorem 2, they are isomorphic to the ℬα+1.
LetX be the Zygmund space. Then, by Theorem 3.1 of [13], the conditions that guarantee the boundedness of Cφ as an operator on Z (or ZH) are(87)supz∈D1−z2φ′z21−φz2<∞,supz∈D1−z2φ″zlog11−φz2<∞,which are both satisfied by the polynomial φ. By Theorem 2, Cφ is not compact on Z or ZH since(88)limφz⟶11−z2φ′z21−φz2=limz⟶−1+1−z21/42z−121−1/4z2−z2=limz⟶−1+1+z2z+122+z2+z+z2=32,where the last two limits are taken along the real axis.
By Corollary7, σn∈Z for each n∈ℕ.
Assume nextX is the Besov space Bp. By Theorem 18 of [9], since φ″ is constant, Cφ is bounded as an operator on the minimal Möbius-invariant space B1. Since the Besov spaces Bp are interpolating spaces between B1 and ℬ and Cφ is bounded on ℬ, Cφ is bounded on Bp for any p>1. By Corollaries 3.3 and 3.4, and having shown above that Cφ is not compact on ℬ, this operator is not compact on Bp either. By Corollary 7, σn∈Bp for each n∈ℕ and p≥1.
Finally, assumeX is the space BMOA. It is well known that every composition operator on BMOA is bounded. Thus, by Corollary 7, σn∈BMOA for each n∈ℕ. By Theorem 4.1 of [33], if Cφ:BMOA⟶BMOA is compact, then Cφ:H2⟶H2. On the other hand, if φ was the symbol of a compact composition operator on H2, then (see [26], p. 43)(89)limz⟶11−φz1−z=∞.
However, the above calculations show that(90)limz⟶11−φz1−z≤limz⟶11−φz21−z=limr⟶1−r2+r+2r+2=12.
Thus,Cφ is not compact as an operator on BMOA.
By Theorem12, for each n∈ℕ∪0, φ′0n is an eigenvalue of Cφ as an operator acting on any of the spaces ℬHα, AH−α, ZH, BHp, and BMOAH, and the corresponding eigenfunctions are given by ασn+β¯σ¯n, where α and β are arbitrary complex constants, not both zero.


### 5.1. Concluding Remarks

We wish to point out that since harmonicity is not preserved under multiplication unless the multiplier is constant, multiplication operators are of no interest for the operator theory of spaces of harmonic mappings.Composition operators on spaces of harmonic mappings could be expanded to include those induced by antianalytic self-maps ofD. We did not pursue this study as we expect that results for this case can easily be obtained from the analytic symbol case using appropriate transformations.


## 5.1. Concluding Remarks

We wish to point out that since harmonicity is not preserved under multiplication unless the multiplier is constant, multiplication operators are of no interest for the operator theory of spaces of harmonic mappings.Composition operators on spaces of harmonic mappings could be expanded to include those induced by antianalytic self-maps ofD. We did not pursue this study as we expect that results for this case can easily be obtained from the analytic symbol case using appropriate transformations.



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