Composition Operators on Some Banach Spaces of Harmonic Mappings

Aljuaid, Munirah; Colonna, Flavia

doi:https://doi.org/10.1155/2020/9034387

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Research Article | Open Access

Volume 2020 | Article ID 9034387 | https://doi.org/10.1155/2020/9034387

Composition Operators on Some Banach Spaces of Harmonic Mappings

Munirah Aljuaid¹and Flavia Colonna²

Academic Editor: Stanislav Hencl

Received26 Nov 2019

Accepted17 Dec 2019

Published19 Feb 2020

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 the α-Bloch spaces, the growth spaces, the Zygmund space, the analytic Besov spaces, and the space .

1. Introduction

Given a simply connected region in the complex plane , a harmonic mapping with domain is a complex-valued function h defined on satisfying the Laplace equation:where is the mixed complex second partial derivative of h.

It is well known that a harmonic mapping h admits a representation of the form , where f and are analytic functions. This representation is unique if, for a fixed a base point in the domain, the function is chosen so that .

In this paper, we shall assume all the functions under consideration are defined on . Let us denote as the set of harmonic mappings on , as the set of analytic functions on , as the set of analytic self-maps of , and as the group of (conformal) disk automorphisms of .

Given an analytic self-map φ of , the composition operator induced by φ is defined as the operatorfor 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 functions X 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 on that extends a Banach space of analytic functions is , defined as the space of harmonic mappings on which are Poisson integrals of functions on the unit circle belonging to , which was thoroughly studied by Girela [1]. In that work, it was shown that is the space 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 and its generalizations known as α-Bloch spaces introduced by Zhu in [3], the growth spaces (where ), the Zygmund space , and the analytic Besov spaces for . In particular, the linear structure and properties of the harmonic α-Bloch spaces , the harmonic growth spaces (for ), and the harmonic Zygmund space were studied in [4, 5]. The harmonic Besov spaces for were introduced in [2].

In this work, after giving in Section 2 some preliminaries on the spaces of harmonic mappings mentioned above, we introduce the harmonic Besov space and an alternative extension of to harmonic mappings denoted by . 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 denote 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 to the seminorm of the associated analytic and antianalytic (i.e., conjugate of analytic) components in X, it turns out that the composition operator acting on is bounded (respectively, compact, bounded below, closed range) if and only if 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 Section 4, we focus on the study of the isometries among the composition operators. Finally, in Section 5, we study the eigenfunctions of acting on the harmonic spaces , , , , and .

2. Harmonic Spaces Treated in This Work

2.1. Harmonic α-Bloch Spaces

For , the harmonic α-Bloch space is the collection of all such that

The mapping defines a Banach space structure on . This space is an extension to harmonic mappings of the (analytic) α-Bloch space introduced by Zhu in [3]. We recall that an analytic function f belongs to if and only ifwith norm . Thus, representing as with and , we see that and . Hence, , and

Consequently, a harmonic mapping h belongs to if and only if the unique functions f and analytic on such that with are in (for more information on the spaces , see [4]).

For , the space is the classical Bloch space , and the corresponding harmonic extension will be denoted by . The elements of this space were first studied in [6].

2.2. Harmonic Growth Spaces

For , the harmonic growth space is the collection of all harmonic mappings h on such that

The mapping defines a Banach space structure on , which again extends that of the (analytic) growth space . For , the functions if and only if (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 space consisting of the analytic functions f on such that with norm

The harmonic Zygmund space, introduced in [2], is the collection of harmonic mappings h on such that . Define

The mapping defines a Banach space structure on . The harmonic mapping h belongs to if and only if the unique functions f and analytic on such that with are in . Furthermore,

In [5], it is shown that the elements of the space can be characterized in terms of the membership to the classical Zygmund class, and the corresponding norms are equivalent.

2.4. Harmonic Besov Spaces

For , a harmonic mapping h in is said to belong to the harmonic Besov space if the quantityis finite, where is the normalized area measure on such that .

The mapping defines a Möbius-invariant seminorm, and defines a Banach space structure on which extends the ordinary norm on . As shown in [2], for , two analytic functions f and on belong to if and only if . Moreover, if , then

Recall that in the analytic case, the Besov space is the Dirichlet space , which is a Hilbert space under an equivalent norm, namely,

Likewise, the harmonic Besov space for , which we call the harmonic Dirichlet space and denote by , can be endowed with a Hilbert space structure via the inner product:whose associated normis equivalent to . In particular, if , with , then

The analytic Besov space is defined as the set consisting of the functions of the formwhere is absolutely summable, and for and for , is the disk automorphism defined by

The norm of f in is defined aswhere the infimum is taken over all above representations of f. Under this norm, is a Möbius-invariant Banach space contained in the disk algebra (see [8] for details).

In [9], it was shown that is the smallest Möbius-invariant space, which is why it is commonly known as the minimal Möbius-invariant space. Moreover, if and only if

A more convenient non-Möbius-invariant norm on equivalent to is given by

We introduce the harmonic Besov space as the collection of harmonic mappings h on for which

Representing a harmonic mapping h as with and , we see that and . Thus, if and only if andwhere . Consequently, the set is contained in , and a harmonic mapping h belongs to if and only if it admits a representation of the formwhere , and for each . This leads to the following equivalent Möbius-invariant norm on that extends (18):where the infimum is taken over all above representations of h. Under both norms, is a Banach space and for with

2.5. Harmonic Space of Bounded Mean Oscillation

Recall that the space of analytic functions of bounded mean oscillation is the Banach space with normand the associated seminorm is Möbius invariant. We recall that the norm of a function f in the Hardy Hilbert space is defined as

It is well known that a function is in if and only ifand is equivalent to defined bysee [1] for a comprehensive analysis of the functions of bounded mean oscillation.

The space of harmonic functions of bounded mean oscillation is the collection of harmonic mappings h on of the form , where and is the Poisson integral of f. As shown in [1], a norm on this space whose associated seminorm yields the Möbius-invariant seminorm in when restricted to the analytic functions on is given bywhere for ,

If f is analytic, then . Representing h as with analytic and , we see that

Thus, if , then . However, this norm does not lead to a lower bound in terms of the norms of f and . Thus, for , we define the harmonic extension of the space as follows:if this expression is finite. We see that taking with , so that and , we have

In particular, consistent with the notation we have been using throughout, denoting by the collection of such harmonic mappings, we see that under , is a Banach space that extends the norm on . Moreover, if and only if the associated analytic functions f and belong to .

It is well known that for , (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

Let be a Banach space of harmonic mappings on with seminorm and normsuch that the point-evaluation functionals are bounded. Let and denote by and the seminorm and norm induced on X, that is, for , let . Assume further that for each with , the associated analytic functions f and belong to X andwhere by we mean for some positive constants and .

Theorem 1. If , then(a) is bounded if and only if is bounded(b) is compact if and only if is compact(c) is bounded below if and only if is bounded below(d) is closed range if and only if is closed range

Proof. 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 is bounded. Let and let f and be the associated analytic functions so that and . Then, , and by our assumption that is bounded on X, it follows that . Thus, . By the Closed Graph Theorem, the operator is bounded on .
Next, assume is compact. Let be a sequence in such that and for each , let be the associated analytic functions so that and . By (36), . Moreover, . Thus, the sequences and have bounded norms in X, and hence by the compactness of , some subsequence converges in norm to some function . Again, by the compactness of , from the sequence , we may then extract a subsequence such that converges in norm to some . Then, and the sequence converges in norm to h. Therefore, is compact.
Lastly, assume is bounded below. Then, there exists some constant such that for each , . On the other hand, by (36), there exist constants , such that for each with , Thus,proving that is bounded below.

Of course, the conclusion of Theorem 1 also holds if condition (36) holds for the norms of X and .

Focusing on the compactness of acting on the spaces , , , and , which satisfy along with the corresponding analytic counterparts , , , and , 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. Let and . Then, is compact on if and only if and

Corollary 2. For , a bounded operator on (equivalently on ) is compact if and only if

Corollary 3. Let , , . If is bounded on (equivalently on ), then is compact if and only if is compact if and only if

Corollary 4. For , if is bounded on (equivalently on ), then is compact if and only if is compact if and only if

Corollary 5. The composition operator is compact on if and only ifwhere m is the Lebesgue measure.
Since by [3] and its extension to the corresponding harmonic spaces provided in [4], for , the spaces and are equivalent to and , 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. Let φ be an analytic self-map of , X a Banach space of analytic functions with norm satisfying the following conditions:(a)X contains the constants.(b)For each , the point-evaluation functional is in . Thus,(c)The unit ball of X is relatively compact with respect to the topology of uniform convergence on compact subsets of .(d)There is a constant such that for all and ,Let and assume is bounded. Then, is compact if and only ifIn the special case, when ,for each and . Therefore,Next, note that the constant function 1 has norm 1 in . Thus, in (47), equality holds at . On the other hand, for , for , and , taking as test functionwe see that , , andHence,Since the above conditions (a)–(d) clearly hold for , applying Theorem 2 to , using (50), and then taking the αth root, we deduce the following result.

Corollary 6. Let , and assume is bounded on (equivalently on ). Then, is compact if and only if

4. Isometries of

In this section, we wish to characterize the isometries on the harmonic spaces , , , and . For most of the corresponding analytic counterparts, namely, for , , , and for , the only composition operators that are isometries are those induced by rotations [20–23] (see also [24]). Since an isometry on a harmonic space that extends an analytic space X is also an isometry on X, is an isometry on for , for , and for , , if and only if φ is a rotation. In the case of the minimal Möbius-invariant space 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 are all of the form , where λ is a unimodular constant and . Of course, these are also isometries when regarding as an operator on . To the best of our knowledge, it is not known whether linear isometries among the composition operators on other than those induced by disk automorphisms exist. The following result partially answers this question.

Theorem 3. If has a fixed point in , then is an isometry if and only if .

Proof. The sufficiency is clear. Assume is a fixed point of φ, is an isometry, and φ is not a disk automorphism. First, assume . Then, φ is not a rotation, and the nth iterate of φ is also an isometry of for any . By the Grand Iteration Theorem (see [26], p. 78), the sequence converges to 0 uniformly on compact subsets of . Thus, . Therefore, for all n sufficiently large, . Hence, for such an n, is a compact operator on , and by Corollary 4, it is also compact on . By Lemma 3.7 of [14], since the sequence defined by is bounded in and converges to 0 uniformly on compact subsets of ,On the other hand, since is an isometry of , is bounded away from 0, as due to the equivalence of the two norms on ,We have reached a contradiction. Therefore, φ must be a rotation in this case.
Next, assume for some . Due to the Möbius invariance of , letting denote the disk automorphism interchanging 0 and p, the function induces an isometry on and has 0 as a fixed point. By the previous case, ψ must be a rotation. Then, is a disk automorphism.

We conjecture that the conclusion of Theorem 3 also holds if φ does not have fixed points in . In this case, if , then φ has a boundary fixed point ω (i.e., the Denjoy–Wolff point of φ) such that 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 space 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 on the Bloch space are also isometries on . However, the inclusion is proper. For example, finite Blaschke products induce isometries on but not on unless they are rotations. We are not aware of results of this type on 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 on with respect to the norm other than those induced by rotations?(2)If this question has a positive answer, are all nontrivial isometries on also isometries on ?

To complete the study of the isometries on the harmonic spaces under consideration in our work, it remains to analyze the cases when acts on the harmonic Bloch space and on the harmonic Dirichlet space to determine whether the nontrivial isometries on and are also isometries on the larger spaces and . 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). Let analytic function, then the following are equivalent:(a) is an isometry.(b) and .(c)Either for some , , or , where is nonvanishing and B is an infinite Blaschke product whose zeros form a sequence containing 0 and a subsequence such that and

Theorem 5 (see [29], Theorem 2.7). For , is an isometry on if and only if and either φ is a rotation or for every there exists a sequence in such that , andWe can now prove our main result in this section.

Theorem 6. Let φ be an analytic self-map of . Then, the following statements are equivalent.(a) is an isometry on (b) is an isometry on (c) and

Proof. Since by Theorem 4, (b) is equivalent to (c), and as observed above, (a) clearly implies (b); we only need to show that (c) implies (a).
Assume and . Then, . If φ is a rotation, the result is clear. So, assume φ is not a rotation.
Observe that since , it suffices to show that is seminorm preserving on . Equivalently, for all such that . Also, by dividing by the norm, we only need to prove that for with .
So, assume h satisfies the conditions and . Then,Thus, either this supremum is attained at some point inside the disk or it is a limit along a sequence of points in approaching the unit circle. Specifically, one of the following two cases must hold:(i)There exists such that(ii)There exists a sequence is such that andAssume first (i) holds. Then, by Theorem 5, there exists a sequence in such that as , for each , andTherefore,Using (i), it follows thatSince, as observed above, the operator has norm 1, . It follows that .
Next, assume (ii) holds. Then, again by Theorem 5, for each , there exists a sequence in such that as , andProceeding as above, for each , we haveHence, taking the limit as and using (ii), we deduce thatThus, also in this case. Therefore, is an isometry on .

We now turn our attention to the identification of the isometries on .

A function 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 .

Theorem 7 (see [30], p.1703). A composition operator acting on the Dirichlet space is an isometry if and only if φ is a univalent full map of that fixes 0.
We are ready to characterize the isometry on .

Theorem 8. The bounded composition operator on is an isometry if and only if φ is a univalent full map fixing 0.

Proof. Assume is an isometry on . Since the Dirichlet space is a subspace of and the norm of equals the norm of , the operator is also an isometry on . Thus, by Theorem 7, φ is a univalent full map fixing the origin.
Conversely, assume φ is a univalent full map fixing the origin. Let . Then, with and . So, , andThus, by Theorem 7, , . Therefore, by (15), we haveproving that is an isometry on .

5. Eigenfunctions of

In the early 1870s, Ernst Schröder introduced the eigenvalue equation for composition operators. Suppose φ is an analytic self-map of , and λ is complex constant. The functional equationis known as Schröder’s equation.

Lemma 1. Assume , φ nonconstant, with Let , and suppose f is an analytic function such that for each If , then .

Proof. Assume an analytic function f satisfying equation (68) exists with f not identically zero. Since and , there is a natural number , such that for z sufficiently near for some analytic function with . Evaluating equation (68) at , we getSince , it follows that . Thus, in some neighborhood of , f admits the representation of the formfor some analytic function such that and . Then, substituting this expression into equation (68), we obtainfor z near . Since , we have reached a contradiction.

Lemma 2. Assume fixes 0 and that φ is not a rotation. Then, the only eigenfunctions of relative to the eigenvalue 1 are the nonzero constants.

Proof. Assume 1 is an eigenvalue of , and let f be a corresponding eigenfunction. Then, implies that for all where is the iterate of φ.
Fix . Let for all . Then, the sequence must converge to 0. Thus,By continuity . Therefore, f is constant.

For which fixes a point and such that , the unique solution σ of Schröder’s equation corresponding to the eigenvalue such that is called the Königs function of φ or principal eigenfunction of .

We state below Königs’ theorem, which gives a description of the eigenvalues and corresponding eigenfunctions of the composition operator , considered as a linear transformation on , when the symbol is nonautomorphic with a fixed point in .

Theorem 9 (see [26]). Let with a fixed point .(a)If , then 1 is the only eigenvalue of .(b)If , then the set of eigenvalues of is given by Each eigenvalue has multiplicity 1, and the function spans the eigenspace for .(c)If φ 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 on the Eigenfunction Theorem valid for the Hardy Hilbert space (see [26], p.94).

Theorem 10 (see [31]). If is compact on a Banach space X of analytic functions on , then the eigenfunctions belong to the space X for each .
Using Theorem 10, 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 with , , is bounded on for some , and there exists such that . Then, for all .
Since for , the growth space can be identified with the Bloch-type space , the conclusion of Theorem 11 also holds when is bounded on . 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 , using Theorem 10, we obtain the following extension to the other spaces treated in this paper.

Corollary 7. Suppose with , , is bounded on , respectively, on for , respectively, on , and there exists such that . Then, , respectively, , respectively, , for all .
As observed above, if with and , then the only eigenvalues of must be of the form for some , and the corresponding eigenfunctions must be constant multiples of .
We are now interested in determining the eigenfunctions of as an operator acting on spaces of harmonic mappings with domain . The following result shows that these eigenfunctions are closely related to the eigenfunctions of the analytic counterparts.

Theorem 12. Let with and . Suppose and h is a harmonic mapping on such thatThen, either , in which case h is constant, or there exists such that . In the latter case, h is a linear combination of 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. Let with and . Then, by (76),so rearranging the terms, we obtainSince the left-hand side of (78) is analytic, and the right-hand side is antianalytic, their common value must be a constant η. Thus,Since and , from (79), we obtain . Hence,If , then by Lemma 2, the only analytic functions F satisfying the equation are the constant functions. Thus, is the constant 0, and is constant.
Next assume and with and . From Königs’ Theorem applied to the equation , it follows that λ must be a power of , and f is a multiple of the corresponding power of σ, that is, there exist an and such that and .
On the other hand, , so, again by Königs’ Theorem, there exist and such that and . Then, . Thus, and . Therefore,and , for some integer k, completing the proof.

Remark 1. From the above, we see that if h satisfies (76), where (which corresponds to the special case ), then must be real.
In the context of the composition operator acting on the harmonic spaces treated in this work, equivalently, on , a natural question that arises is as follows.
For which symbols φ fixing 0 is the set of eigenvalues of regarded as an operator on given by ?
By Theorem 12, the required conditions are that must be real, and all the powers of the Königs function σ must be in X.
In particular, every compact operator on X (equivalently, on ) satisfies the condition for all . Since is a compact operator whenever , we are interested in the case when the closure of the range of φ intersects the unit circle.
As a consequence of Theorem 11 and Corollary 7, we can now give a simple example of a nonunivalent function with supremum norm 1 for which the operator is noncompact and has as full set of eigenvalues for any of the spaces X treated in this work.
Example: consider the polynomial:Then, φ maps into itself, has 0 as a fixed point, and is not univalent since . Moreover, regarding φ as a function on , since . In fact, , and a straightforward computation shows that touches the unit circle only at the point 1.
Let us first focus on the case when X is . Sinceby Theorem 1.4 of [12], the operator is bounded on or . Moreover, for ,Thus, as along the real axis,Hence,where the last limit is taken along the real axis.
Therefore, by Corollary 1, the operator is not compact on or .
Since the only point of contact of φ with the unit circle occurs at and , we see that . Thus, by Theorem 11, for all .
Of course, the same conclusion holds for the growth spaces since, as observed in the paragraph before Theorem 2, they are isomorphic to the .
Let X be the Zygmund space. Then, by Theorem 3.1 of [13], the conditions that guarantee the boundedness of as an operator on (or ) arewhich are both satisfied by the polynomial φ. By Theorem 2, is not compact on or sincewhere the last two limits are taken along the real axis.
By Corollary 7, for each .
Assume next X is the Besov space . By Theorem 18 of [9], since is constant, is bounded as an operator on the minimal Möbius-invariant space . Since the Besov spaces are interpolating spaces between and and is bounded on , is bounded on for any . By Corollaries 3.3 and 3.4, and having shown above that is not compact on , this operator is not compact on either. By Corollary 7, for each and .
Finally, assume X is the space . It is well known that every composition operator on is bounded. Thus, by Corollary 7, for each . By Theorem 4.1 of [33], if is compact, then . On the other hand, if φ was the symbol of a compact composition operator on , then (see [26], p. 43)However, the above calculations show thatThus, is not compact as an operator on BMOA.
By Theorem 12, for each , is an eigenvalue of as an operator acting on any of the spaces , , , , and , and the corresponding eigenfunctions are given by , 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 of . 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.

Data Availability

The findings in this research do not make use of data.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2020 Munirah Aljuaid and Flavia Colonna. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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