Composition Operators on Some Banach Spaces of Harmonic Mappings

where hzz is the mixed complex second partial derivative of h. It is well known that a harmonic mapping h admits a representation of the form f + g, where f and g are analytic functions.+is representation is unique if, for a fixed a base point z0 in the domain, the function g is chosen so that g(z0) � 0. In this paper, we shall assume all the functions under consideration are defined on D � z ∈ C: |z|< 1 { }. Let us denote H(D) as the set of harmonic mappings on D, H(D) as the set of analytic functions on D, S(D) as the set of analytic self-maps of D, and Aut(D) as the group of (conformal) disk automorphisms of D. Given an analytic self-map φ of D, the composition operator induced by φ is defined as the operator


Introduction
Given a simply connected region Ω in the complex plane C, a harmonic mapping with domain Ω is a complex-valued function h defined on Ω satisfying the Laplace equation: where h zz is the mixed complex second partial derivative of h. It is well known that a harmonic mapping h admits a representation of the form f + g, where f and g are analytic functions. is representation is unique if, for a fixed a base point z 0 in the domain, the function g is chosen so that g(z 0 ) � 0.
In this paper, we shall assume all the functions under consideration are defined on D � z ∈ C: |z| < 1 { }. Let us denote H(D) as the set of harmonic mappings on D, H(D) as the set of analytic functions on D, S(D) as the set of analytic self-maps of D, and Aut(D) as the group of (conformal) disk automorphisms of D.
Given an analytic self-map φ of D, the composition operator induced by φ is defined as the operator 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 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 D 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 zD belonging to BMO, which was thoroughly studied by Girela [1]. In that work, it was shown that H(D) ∩ 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 B α 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 B p for p > 1. In particular, the linear structure and properties of the harmonic α-Bloch spaces B α H , the harmonic growth spaces A − α H (for α > 0), and the harmonic Zygmund space Z H were studied in [4,5]. e harmonic Besov spaces B p H for p > 1 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 B 1 H and an alternative extension of BMOA to harmonic mappings denoted by BMOA H . 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 X H 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 X H to the seminorm of the associated analytic and antianalytic (i.e., conjugate of analytic) components in X, it turns out that the composition operator C φ acting on X H is bounded (respectively, compact, bounded below, closed range) if and only if C φ acting on X is bounded (respectively, compact, bounded below, closed range). is 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 C φ acting on the harmonic p H , and BMOA H .

Harmonic Spaces Treated in This Work
is space is an extension to harmonic mappings of the (analytic) α-Bloch space B α introduced by Zhu in [3]. We recall that an analytic function f belongs to B α if and only if with norm ‖f‖ B α � |f(0)| + β α f . us, representing h ∈ H(D) as f + g with f, g ∈ H(D) and g(0) � 0, we see that Consequently, a harmonic mapping h belongs to B α H if and only if the unique functions f and g analytic on D such that h � f + g with g(0) � 0 are in B α (for more information on the spaces B α H , see [4]). For α � 1, the space B α is the classical Bloch space B, and the corresponding harmonic extension will be denoted by B H . e elements of this space were first studied in [6].

Harmonic Growth Spaces.
For α > 0, the harmonic growth space A − α H is the collection of all harmonic mappings h on D such that e mapping h⟼h A − α H defines a Banach space structure on A − α H , which again extends that of the (analytic) growth [2,4] for details; for more information on the analytic growth space, see [7]).

Harmonic Zygmund Space.
Recall that the Zygmund space is the space Z consisting of the analytic functions f on D such that f ′ ∈ B with norm e harmonic Zygmund space, introduced in [2], is the collection of harmonic mappings h on D such that In [5], it is shown that the elements of the space Z H can be characterized in terms of the membership to the classical Zygmund class, and the corresponding norms are equivalent.   (11) Recall that in the analytic case, the Besov space B 2 is the Dirichlet space D, which is a Hilbert space under an equivalent norm, namely, Likewise, the harmonic Besov space for p � 2, which we call the harmonic Dirichlet space and denote by D H , can be endowed with a Hilbert space structure via the inner product: whose associated norm is equivalent to e analytic Besov space B 1 is defined as the set consisting of the functions f ∈ H(D) of the form where (a n ) is absolutely summable, and w n ∈ D for n ∈ N and for w ∈ D, L w is the disk automorphism defined by e norm of f in B 1 is defined as where the infimum is taken over all above representations of f. Under this norm, B 1 is a Möbius-invariant Banach space contained in the disk algebra H ∞ ∩ C(D) (see [8] for details).
In [9], it was shown that B 1 is the smallest Möbius-invariant space, which is why it is commonly known as the minimal A more convenient non-Möbius-invariant norm on B 1 equivalent to ‖ · ‖ B 1 is given by We introduce the harmonic Besov space B 1 H as the collection of harmonic mappings h on D for which Representing a harmonic mapping h as f + g with f, g ∈ H(D) and g(0) � 0, we see that h zz � f ″ and where ‖h‖ ≔ max |h(0)|, |h where (a n ), (b n ) ∈ ℓ 1 , and w n , ζ n ∈ D for each n ∈ N. is leads to the following equivalent Möbius-invariant norm on B 1 H that extends (18): where the infimum is taken over all above representations of h. Under both norms,

Harmonic Space of Bounded Mean Oscillation.
Recall that the space BMOA of analytic functions of bounded mean oscillation is the Banach space with norm (26) and the associated seminorm is Möbius invariant. We recall that the norm of a function f in the Hardy Hilbert space H 2 is defined as and ‖ · ‖ BMOA is equivalent to ‖ · ‖ * defined by see [1] for a comprehensive analysis of the functions of bounded mean oscillation. e space BMOH of harmonic functions of bounded mean oscillation is the collection of harmonic mappings h on 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 where for h ∈ H(D),

Journal of Function Spaces
If f is analytic, then ‖f‖ h 2 � ‖f‖ H 2 .
Representing h as f + g with f, g analytic and g(0) � 0, we see that us, if f, 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. us, for h ∈ H(D), we define the harmonic extension of the space BMOA as follows: if this expression is finite. We see that taking In particular, consistent with the notation we have been using throughout, denoting by BMOA H the collection of such harmonic mappings, we see that under ‖ · ‖ H, * , BMOA H is a Banach space that extends the norm ‖ · ‖ * on BMOA. Moreover, h ∈ BMOA H if and only if the associated analytic functions f and g belong to BMOA.
It is well known that for 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.

General Theorem on Composition Operators
Let X H be a Banach space of harmonic mappings on D with seminorm ‖·‖ sX H and norm such that the point-evaluation functionals are bounded. Let X � H(D)∩X H and denote by ‖ · ‖ sX and ‖ · ‖ X the seminorm and norm induced on X, that is, for f ∈ X, let ‖f‖ X � |f(0)| + ‖f‖ sX � |f(0)| + ‖f‖ sX H � ‖f‖ X H . Assume further that for each h � f + g ∈ X H with g(0) � 0, the associated analytic functions f and g belong to X and where by A ≍ B we mean c 1 A ≤ B ≤ c 2 A for some positive constants c 1 and c 2 .
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 C φ : X ⟶ X is bounded. Let h ∈ X H and let f and g be the associated analytic functions so that h � f + g and g(0) � 0. en, f, g ∈ X, and by our assumption that C φ is bounded on X, it follows that us, h ∘ φ ∈ X H . By the Closed Graph eorem, the operator C φ is bounded on X H . Next, assume C φ : X ⟶ X is compact. Let (h n ) be a sequence in X H such that ‖h n ‖ X H ≤ 1 and for each n ∈ N, let f n , g n be the associated analytic functions so that h n � f n + g n and g n (0) � 0. By (36), ‖f n ‖ sX + ‖g n ‖ sX ≤ C‖h n ‖ X H ≤ C. Moreover, f n (0) � h n (0). us, the sequences (f n ) and (g n ) have bounded norms in X, and hence by the compactness of C φ : X ⟶ X, some subsequence (f n k ∘ φ) k∈N converges in norm to some function f ∈ X. Again, by the compactness of C φ : X ⟶ X, from the sequence (n k ) k∈N , we may then extract a subsequence (n k j ) j∈N such that (g n k j ∘ φ) j∈N converges in norm to some g ∈ X.
en, h ≔ f + g ∈ X H and the sequence (h n k j ∘ φ) j∈N converges in norm to h. erefore, C φ : X H ⟶ X H is compact.
Lastly, assume C φ : X ⟶ X is bounded below. en, there exists some constant δ > 0 such that for each f ∈ X, ‖f ∘ φ‖ X ≥ δ‖f‖ X . On the other hand, by (36), there exist constants C 1 , C 2 > 0, such that for each h � f + g ∈ X H with g(0) � 0, C 1 (‖f‖ X + ‖g‖ X ) ≤ ‖h‖ X H ≤ C 2 (‖f‖ X + ‖g‖ X ) us, (38) 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 A − α H are equivalent to B α+1 and B α+1 H , respectively, and eorem 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 eorem 3.2 of [19], we provide below a simpler characterization.

(c) e unit ball of X is relatively compact with respect to the topology of uniform convergence on compact subsets of D. (d) ere is a constant C > 0 such that for all S ∈ Aut(D)
and f ∈ X, Let α > 0 and assume C φ : X ⟶ A − α is bounded. en, In the special case, when X � A − α , for each z ∈ D and f ∈ A − α . erefore, Next, note that the constant function 1 has norm 1 in A − α . us, in (47), equality holds at z � 0. On the other hand, for z � re iθ , for 0 < r < 1, and θ ∈ R, taking as test function we see that f z ∈ A − α , ‖f z ‖ A − α � 1, and Hence, Since the above conditions (a)-(d) clearly hold for A − α , applying eorem 2 to X � A − α , using (50), and then taking the αth root, we deduce the following result.

Isometries of C φ
In this section, we wish to characterize the isometries on the harmonic spaces B α H , A − α H , Z H , and B p H . For most of the corresponding analytic counterparts, namely, B α for α ≠ 1, A − α , Z, and B p for p > 1 and p ≠ 2, the only composition operators that are isometries are those induced by rotations [20][21][22][23] (see also [24]). Since an isometry on a harmonic space X H that extends an analytic space X is also an isometry on X,  Aut(D). Of course, these are also isometries when regarding C φ as an operator on B 1 H . To the best of our knowledge, it is not known whether linear isometries among the composition operators on B 1 other than those induced by disk automorphisms exist. e following result partially answers this question.

is an isometry if and only if φ ∈ Aut(D).
Proof.
e sufficiency is clear. Assume p ∈ D is a fixed point of φ, C φ : B 1 ⟶ B 1 is an isometry, and φ is not a disk automorphism. First, assume p � 0. en, φ is not a rotation, and the nth iterate φ n of φ is also an isometry of B 1 for any n ∈ N. By the Grand Iteration eorem (see [26], p. 78), the sequence (φ n ) converges to 0 uniformly on compact subsets of D. us, lim n⟶∞ ‖φ n ‖ ∞ � 0. erefore, for all n sufficiently large, ‖φ n ‖ ∞ < 1/2. Hence, for such an n, C φ n is a compact operator on B, and by Corollary 4, it is also compact on B 1 . By Lemma 3.7 of [14], since the sequence (p k ) ∞ k�2 defined by p k (z) � (1/(k − 1))z k is bounded in B 1 and converges to 0 uniformly on compact subsets of D, On the other hand, since φ n is an isometry of B 1 , ‖C φ n p k ‖ B 1 � ‖p k ‖ B 1 is bounded away from 0, as due to the equivalence of the two norms on B 1 , We have reached a contradiction. erefore, φ must be a rotation in this case.
Next, assume φ(p) � p for some p ∈ D. Due to the Möbius invariance of ‖ · ‖ B 1 , letting L p denote the disk automorphism interchanging 0 and p, the function ψ � L p ∘ φ ∘ L p induces an isometry on B 1 and has 0 as a fixed point. By the previous case, ψ must be a rotation. en, φ � L p ∘ ψ ∘ L p is a disk automorphism.

□
We conjecture that the conclusion of eorem 3 also holds if φ does not have fixed points in D. In this case, if φ ∉ Aut(D), then φ has a boundary fixed point ω (i.e., the Denjoy-Wolff point of φ ) such that φ n ⟶ 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. is feature is shared by the space BMOA 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 C φ on the Bloch space are also isometries on BMOA. However, the inclusion is proper. For example, finite Blaschke products induce isometries C φ 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 ‖ · ‖ * . us, we leave to the reader the following problem for future investigation.
Open questions: (1) Are there any isometries among the composition operators on BMOA with respect to the norm ‖ · ‖ * other than those induced by rotations? (2) If this question has a positive answer, are all nontrivial isometries on BMOA also isometries on BMOA H ?
To complete the study of the isometries on the harmonic spaces under consideration in our work, it remains to analyze the cases when C φ acts on the harmonic Bloch space B H and on the harmonic Dirichlet space D H to determine whether the nontrivial isometries C φ on B and D are also isometries on the larger spaces B H and D H . We shall prove that this is indeed the case.
We shall make use of the following results.
Theorem 4 (see [28], eorem 5 and Corollary 2). Let φ: D ⟶ D analytic function, then the following are equivalent: where g ∈ S(D) is nonvanishing and B is an infinite Blaschke product whose zeros form a sequence (z n ) n∈N containing 0 and a subsequence (z n j ) j∈N such that |g(z n j )| ⟶ 1 and Theorem 5 (see [29], eorem 2.7). For φ ∈ S(D), C φ is an isometry on B if and only if φ(0) � 0 and either φ is a rotation or for every a ∈ D there exists a sequence (z n ) in D such that |z n | ⟶ 1, φ(z n ) � a and We can now prove our main result in this section.
Theorem 6. Let φ be an analytic self-map of D. en, the following statements are equivalent.
(a) C φ is an isometry on B H .
Observe that since φ(0) � 0, it suffices to show that C φ is seminorm preserving on B H . Equivalently, ||h ∘ φ|| B H � ‖h‖ B H for all h ∈ B H such that h(0) � 0. Also, by dividing by the norm, we only need to prove that ‖h ∘ φ‖ B H � 1 for So, assume h satisfies the conditions ‖h‖ B H � 1 and h(0) � 0. en, us, either this supremum is attained at some point inside the disk or it is a limit along a sequence of points in D approaching the unit circle. Specifically, one of the following two cases must hold: (ii) ere exists a sequence (a k ) is D such that |a k | ⟶ 1 and Assume first (i) holds. en, by eorem 5, there exists a sequence (z n ) in D such that |z n | ⟶ 1 as n ⟶ ∞, φ(z n ) � a for each n ∈ N, and erefore, Using (i), it follows that Since, as observed above, the operator C φ has norm 1, Next, assume (ii) holds. en, again by eorem 5, for each k ∈ N, there exists a sequence (z n,k ) n∈N in D such that |z n,k | ⟶ 1 as n ⟶ ∞, φ(z n,k ) � a k and Proceeding as above, for each k ∈ N, we have h z a k + h z a k , as n ⟶ ∞. (63) Hence, taking the limit as k ⟶ ∞ and using (ii), we deduce that us, ‖C φ h‖ B H � 1 also in this case. erefore, C φ is an isometry on B H .

□
We now turn our attention to the identification of the isometries on D H .
A function φ ∈ S(D) 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 operator C φ 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 on D H .

Theorem 8. e bounded composition operator C φ on D H is an isometry if and only if φ is a univalent full map fixing 0.
Proof. Assume C φ is an isometry on D H . Since the Dirichlet space D is a subspace of D H and the norm of D H equals the norm of D, the operator C φ is also an isometry on D. us, by eorem 7, φ is a univalent full map fixing the origin.
Conversely, assume φ is a univalent full map fixing the origin. Let h ∈ D H . en, h � f + g with f, g ∈ H(D) and g(0) � 0. So, f, g ∈ D, and us, by eorem 7, ‖C φ f‖ D � ‖f‖ D , ‖C φ g‖ D � ‖g‖ D . erefore, by (15), we have proving that C φ is an isometry on D H .

Eigenfunctions of C φ
In the early 1870s, Ernst Schröder introduced the eigenvalue equation for composition operators. Suppose φ is an analytic self-map of D, and λ is complex constant. e functional equation is known as Schröder's equation.
, and suppose f is an analytic function such that for each z ∈ D If φ ′ (p) � 0, then f ≡ 0.
Proof. Assume an analytic function f satisfying equation (68) exists with f not identically zero. Since φ(p) � p and φ ′ (p) � 0, there is a natural number k > 1, such that for z sufficiently near p for some analytic function ψ with ψ(p) ≠ 0. Evaluating equation (68) at p, we get Since λ ≠ 1, it follows that f(p) � 0. us, in some neighborhood of p, f admits the representation of the form for some analytic function g such that g(p) ≠ 0 and m ∈ N. en, substituting this expression into equation (68), we obtain for z near p. Since km > m, we have reached a contradiction.
□ Lemma 2. Assume φ ∈ S(D) fixes 0 and that φ is not a rotation. en, the only eigenfunctions of C φ relative to the eigenvalue 1 are the nonzero constants.
Proof. Assume 1 is an eigenvalue of C φ , and let f be a corresponding eigenfunction. en, where φ n is the n th iterate of φ. Fix a ∈ D, a ≠ 0. Let a n � φ n (a) for all n ∈ N. en, the sequence (a n ) must converge to 0. us, f a n � f φ n (a) � f(a), for all n ∈ N. (74) By continuity f(a) � f(0). erefore, f is constant. □ For φ ∈ S(D) 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 C φ .
We state below Königs' theorem, which gives a description of the eigenvalues and corresponding eigenfunctions of the composition operator C φ , considered as a linear transformation on H(D), when the symbol is nonautomorphic with a fixed point in D.
(a) If φ ′ (p) � 0, then 1 is the only eigenvalue of C φ . (b) If φ ′ (p) ≠ 0, then the set of eigenvalues of C φ is given by Each eigenvalue has multiplicity 1, and the function σ n spans the eigenspace for φ ′ (p) n (n ∈ N). (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 D the Eigenfunction eorem valid for the Hardy Hilbert space (see [26], p.94).
Theorem 10 (see [31]). If C φ is compact on a Banach space X of analytic functions on D, then the eigenfunctions σ n belong to the space X for each n ∈ N.
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.
Since for α > 0, the growth space A − α can be identified with the Bloch-type space B α+1 , the conclusion of eorem 11 also holds when C φ is bounded on A − α . Arguing as in the proof of eorem 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 eorem 10, we obtain the following extension to the other spaces treated in this paper.
As observed above, if φ ∈ S(D) with φ(0) � 0 and 0 < |φ ′ (0)| < 1, then the only eigenvalues of C φ must be of the form φ ′ (0) n for some n ∈ N ∪ 0 { }, and the corresponding eigenfunctions must be constant multiples of σ n . 8 Journal of Function Spaces We are now interested in determining the eigenfunctions of C φ as an operator acting on spaces of harmonic mappings with domain D.
e following result shows that these eigenfunctions are closely related to the eigenfunctions of the analytic counterparts.
Theorem 12. Let φ ∈ S(D) with φ(0) � 0 and 0 < |φ′(0)| < 1. Suppose λ ∈ C and h is a harmonic mapping on D such that Then, either λ � 1, in which case h is constant, or there exists n ∈ N such that λ � φ ′ (0) n . In the latter case, h is a linear combination of σ n 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.