Weighted Composition Groups on the Little Bloch Space

Mose, S. B.; Bonyo, J. O.

doi:https://doi.org/10.1155/2020/5480602

Journal of Function Spaces

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Research Article | Open Access

Volume 2020 | Article ID 5480602 | https://doi.org/10.1155/2020/5480602

Weighted Composition Groups on the Little Bloch Space

S. B. Mose¹and J. O. Bonyo¹

Academic Editor: Aurelian Gheondea

Received18 Oct 2019

Accepted05 Feb 2020

Published29 Apr 2020

Abstract

We determine both the semigroup and spectral properties of a group of weighted composition operators on the little Bloch space. It turns out that these are strongly continuous groups of invertible isometries on the Bloch space. We then obtain the norm and spectra of the infinitesimal generator as well as the resulting resolvents which are given as integral operators. As a consequence, we complete the analysis of the adjoint composition group on the predual of the nonreflexive Bergman space and a group of isometries associated with a specific automorphism of the upper half-plane.

Dedicated to Prof. Len Miller (PhD advisor to J. O. Bonyo) and Prof. Vivien Miller of Mississippi State University on their retirement

1. Introduction

The (open) unit disc of the complex plane is defined as , while the upper half-plane of , denoted by , is given by , where stands for the imaginary part of ω. The Cayley transform maps the unit disc conformally onto the upper half-plane with inverse . For every α > −1, we define a positive Borel measure dm_α on by , where dA denotes the area measure on .

For an open subset Ω of , let denote the Fréchet space of analytic functions endowed with the topology of uniform convergence on compact subsets of Ω. Let denote the group of biholomorphic maps f : Ω ⟶ Ω. For 1 ≤ p < ∞, α > −1, the weighted Bergman spaces of the unit disc , , are defined by

Clearly, , where denotes the classical Lebesgue spaces. For every , the growth condition is given bywhere K is a constant and , see, for example, [1], Theorem 4.14.

The Bloch space of the unit disc, denoted by , is defined as the space of analytic functions such that the seminorm

Following [1, 2], is a Banach space with respect to the norm . On the contrary, the little Bloch space of the disc, denoted by , is defined to be the closed subspace of such thatwhere denotes closure of the set of analytic polynomials in z. Equivalently,and possesses the same norm as . Since is a closed subspace of the Banach space , it follows that is a Banach space as well with respect to the norm . Note that every (or ) satisfies the growth condition:

See, for instance, [3] for details. Let 1 < p < ∞ and q be conjugate to p in the sense that . If is the dual space of , thenunder the integral pairing

It is well known that for 1 < p < ∞, is reflexive. The case p = 1 is the nonreflexive case and the duality relations have been determined as follows:under the duality pairings given by, respectively:

In other words, the dual and predual spaces of the nonreflexive Bergman space are the Bloch and little Bloch spaces, respectively. For a comprehensive account of the theory of Bloch and Bergman spaces, we refer to [1, 2, 4–6].

In [7], all the self analytic maps of the upper half-plane were identified and classified according to the location of their fixed points into three distinct classes, namely, scaling, translation, and rotation groups. For each self-analytic map φ_t, we define a corresponding group of weighted composition operator on byfor some appropriate weight γ.

It is noted in [7] Section 5 that for the rotation group, we consider the corresponding group of weighted composition operators defined on the analytic spaces of the disc given by

The study of composition operators on spaces of analytic functions still remains an active area of research. For Bloch spaces, most studies have only focussed on the boundedness and compactness of these operators. See, for instance, [3, 8–11]. In [7, 12], both the semigroup and spectral properties of the group were studied in detail on the Hardy and Bergman spaces. The aim of this paper is to extend the analysis of the group from the Hardy and Bergman spaces to the setting of the little Bloch space. Specifically, we apply the theory of semigroups as well as spectral theory of linear operators on Banach spaces to study the properties of the group of weighted composition operators given by equation (12) on the little Bloch space of the disk. As a consequence, we shall complete the analysis of the adjoint group on the dual of the nonreflexive Bergman space . The analysis of the adjoint group on the reflexive Bergman space, that is, for 1 < p < ∞, was considered exhaustively in [12]. We shall also consider a specific automorphism of and carry out an analysis of the corresponding composition operator.

If X is an arbitrary Banach space, let denote the algebra of bounded linear operators on X. For a linear operator T with domain , denote the spectrum and point spectrum of T by σ(T) and σ_p(T), respectively. The resolvent set of T is , while r(T) denotes its spectral radius. For a good account of the theory of spectra, see [13–15]. If X and Y are arbitrary Banach spaces and is an invertible operator, then clearly is a strongly continuous group if and only if B_t ≔ UA_tU⁻¹, , is a strongly continuous group in . In this case, if has generator Γ, then has generator Δ = UΓU⁻¹ with domain . Moreover, σ_p(Δ) = σ_p(Γ) and σ(Δ) = σ(Γ), since if λ is in the resolvent set , we have that R(λ, Δ) = UR(λ, Γ)U⁻¹. See, for example, [16], Chapter II and [15], Chapter 3.

2. Groups of Composition Operators on the Little Bloch Space

We consider the group of weighted composition operators given by equation (12) and defined on the little Bloch space as T_tf(z) = e^ictf(e^iktz), where , k ≠ 0 and . We denote the infinitesimal generator of the group by Γ_{{c, k}} and give some of its properties in the following proposition.

Proposition 1. (1) is a strongly continuous group of isometries on (2)The infinitesimal generator Γ_c,k of on is given by with domain

Proof. To prove isometry, we haveBy change of variables, let ω = e^iktz. Then,To prove strong continuity, we shall use the density of polynomials in . Therefore, it suffices to show that, for ,Now, . Therefore,Now, for the infinitesimal generator Γ_c,k, let in , then the growth condition (6) implies thatTherefore, . Conversely, if is such that , then and for all t > 0,Strong continuity of implies thatThus, .

Define M_z, Q on by M_zf(z) = zf(z) and , . More generally, , . Then, and . We now give the following proposition.

Proposition 2. (1) is bounded.(2).(3) is bounded.(4)For m ≥ 1, . In particular, is closed in .

Proof. If , then for all ,Therefore, assertions (1) and (2) follow. For (3), if , then for |z| < 1,Thus, . To prove (4), let and f(0) = 0. Then, . The reverse inclusion is obvious. Therefore, the one-to-one and onto mapping is bounded. So, the open mapping theorem implies that the inverse is bounded. It therefore follows that is bounded.

Proposition 3. Let Γ_c,k be the infinitesimal generator of the group given by (12) on , then(1)Γ_c,k = ic + kΓ_0,1 with domain (2) and

In fact, λ ∈ ρ(Γ_0,1) if and only if ic + kλ ∈ ρ(Γ_c,k), and

Proof. See [12], Lemma 4.3.

As a result of Proposition 3 above and without loss of generality, we restrict our attention to the generator Γ_0,1instead of Γ_c,k as the cases c ≠ 0 and k ≠ 1 where k ≠ 0 can be easily obtained from Γ_0,1. Indeed, Γ_0,1f(z) = izf′(z) with domain is the infinitesimal generator of the group T_t = f(e^itz) which is exactly the case when c = 0 and k = 1 in equation (12). We now give the spectral properties of the generator Γ_0,1 as well as the resulting resolvents in the following theorem.

Theorem 1. (1), and for each n ≥ 0, ker(in − Γ_0,1) = span(zⁿ).(2)If λ ∈ ρ(Γ_0,1), then is R(λ, Γ_0,1), invariant , . Moreover, if , then(3)For λ ∈ ρ(Γ_0,1), the resolvent operator R(λ, Γ_0,1) is compact.(4). Moreover,

Proof. Since each T_t is an invertible isometry, its spectrum satisfies , and the spectral mapping theorem for strongly continuous groups (see, for example, [16], Theorem V.2.5 or [17]) implies that . Thus, for ω ∈ σ(Γ_0,1). It immediately follows that .
We now solve the resolvent equation: If and , (λ − Γ)f = h. This is equivalent toIn particular, (λ − Γ)f = 0 if and only f(z) = Kz^−iλ, where K is a constant. Since if and only if , it follows thatwith ker(in − Γ_0,1) = span(zⁿ). Moreover, if and λ ∈ σ_p(Γ_0,1), thenhas a unique solutionNotice that, for λ ∉ σ_p(Γ_0,1) and , (λ − Γ)f(0) = λf(0). More generally, if with , thenNote that the functions (λ − Γ)f and f have the same order of zero at 0. Thus, is invariant under λ − Γ_0,1.
Fix and let . If with , thenThus, (λ − Γ)h has a unique solution:If and 0 ≤ t < 1, thenThus, . Now, ∀m ≥ 1,Thus, λ ∉ σ_p(Γ_0,1) implying that R(λ, Γ_0,1) is bounded on . Therefore, σ(Γ_0,1) = σ_p(Γ_0,1). This proves (1) and (2).
To prove the compactness of the resolvent operator, we argue as in [7], Theorem 5.2. Fix λ ∈ ρ(Γ_0,1) and let be such that . Then, by equation (33), it suffices to show that is compact.
Let , r > 0, be the disc algebra , equipped with the supremum norm, and for each t, 0 ≤ t < 1, and , let H_tf(z) = f_t(z) = f(tz). Then, by equation (32), for every t ∈ [0, 1), H_t is a contraction on .
Now, by equation (23), with convergence in norm. Define on , for 0 < r < 1. Then,as r ⟶ 1⁻. Choosing s so that 1 < s < r⁻¹, we have that factors through . If denotes the closed unit ball of , let . Then, ∀t, 0 ≤ t ≤ r, the growth condition (6) implies that, for |z| ≤ s,Let . Thus, for |z| ≤ s,Thus, by Arzela-Ascoli, is precompact in which further implies that is precompact in by the continuous embeddedness of in . Therefore, each C_r is compact in and as a result, is compact as well.
The spectral mapping theorem for resolvents as well as assertion (1) above implies thatClearly, the spectral radius and therefore by the Hille–Yosida theorem, it follows that , as desired.

As a consequence, the properties of the general group T_t given by equation (12) is as follows.

Corollary 1. (1), and for each n ≥ 0, ker(i(c + kn) − Γ_c,k) = span(zⁿ).(2)If μ ∈ ρ(Γ_c,k), then is R(μ, Γ_c,k) -invariant , . Moreover, if , then(3)For μ ∈ ρ(Γ_c,k), the resolvent R(μ, Γ_c,k) is compact.(4).(5)

Proof. Following Proposition 3, μ ∈ ρ(Γ_c,k) if and only if . The proof now follows at once from Theorem 1. We omit the details.

3. Adjoint of the Composition Group on the Predual of Nonreflexive Bergman Space

In studying the adjoint properties of the rotation group isometries given by equation (12) on Bergman spaces , 1 ≤ p < ∞; the second author in [12] considered the reflexive case, that is, when 1 < p < ∞. This was an extension of the investigation of adjoint properties of the Cesáro operator in [18] on Hardy spaces, and later generalized to Bergman spaces in [7]. For the nonreflexive Bergman space (that is, p = 1), the analysis of the adjoint of the rotation group isometries remains open and forms the basis of this section. Specifically, we complete the analysis of the adjoint group of the group of isometries T_tf(z) = e^ictf(e^iktz), where with k ≠ 0 and .

Recall from Section 1, the duality relation under the integral pairing . In particular, the predual of is the little Bloch space . Thus, using this duality pairing, for every , we have

By a change of variables argument: Let ω = e^iktz so that z = e^−iktω andwhere for all . Thus, the adjoint group of T_t for is therefore given by

Let Γ denote the infinitesimal generator of the adjoint group . Using the results of Section 2, we easily obtain the properties of the group as we give in the following theorem.

Theorem 2. Let be the adjoint group of the group of weighted composition operators given by (41). Then, the following hold:(1) is strongly continuous group of isometries on .(2)The infinitesimal generator Γ of is given by with domain .(3), and for each n ≥ 0, (4)If μ ∈ ρ(Γ), then is R(μ, Γ)-invariant , . Moreover, if , then(5).(6)

Proof. The proof follows immediately by replacing c and k with −c and −k, respectively, in Proposition 3 and Corollary 1. We omit the details.

4. Specific Automorphism of the Half-Plane

In this section, we consider a specific automorphism group corresponding to the rotation group given by

It can be easily verified that φ_t(z) = ψ ∘ u_t ∘ ψ⁻¹(z), where u_t(z) = e^−2itz. The associated group of weighted composition operators on is given by , and by the chain rule, it follows that , where .

Now, for ,

Apparently, can be obtained as a special case of the group given by equation (12) when c = −2γ and k = −2. Let Γ = Γ_−2γ,−2 be the infinitesimal generator of the group , then the properties of Γ can be summarized by the following proposition.

Proposition 4. Let Γ be the infinitesimal generator of the group of isometries on . Then,(1) for every , with domain(2), and for each n ≥ 0,(3)If μ ∈ ρ(Γ), then is R(μ, Γ)-invariant for every , . Moreover, if , then

Proof. Take c = −2γ and k = −2 in Proposition 1 and Corollary 1. The proof follows immediately.
Now, using the similarity theory of semigroups, we detail the properties of the group of weighted composition operators associated with the automorphism group given by (43) in the following theorem.

Theorem 3. Let be given by , for all , and let be the corresponding group of isometries on . Then,(1)The infinitesimal generator Δ of the group on is given by with domain .(2), and for each n ≥ 0, .(3)If μ ∈ ρ(Δ) and if is such that . Then, if , we have(4)R(μ, Δ) is compact on .(5). Moreover

Proof. Let and . Since , it follows that , where is invertible. Let Δ be the generator of and Γ ≔ Γ_−2γ,−2 be the generator of , then with domain
Let , then f ∈ D(Γ) and define belongs to D(Δ) with . Then,As stated earlier, , implying that , and thusSince and , then we have implying that . Moreover, implying thatTherefore,As given earlier, the domain of Δ, D(Δ) is given by D(Δ) . Now h ∈ D(Δ) implies that which implies that . ButThen, we haveBy change of variables, let which implies and . Therefore,which implies that .
From Section 1, the spectrum and point spectrum of Δ are given as .
For the resolvents, if μ ∈ ρ(Δ) = ρ(Γ), then for , and if , we have and soFinally, from spectral mapping theorems it follows that, for all μ ∈ ρ(Δ), the spectrum of R(μ, Δ) is given bySimilarly, the point spectrum is given byTherefore, . Finally, we conclude this section by proving the spectral radius and .
It is clear that the spectrum of the resolvent is . Hille–Yosida theorem yields .

Data Availability

No data used in the manuscript.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was completed during the period when J. O. Bonyo was visiting the Aristotle University of Thessaloniki, Greece. He would like to sincerely thank Simon’s Foundation for funding his visit. He would also wish to thank his host Prof. Aristomenis G. Siskakis as well as the entire team at the department of Mathematics for the unmatched hospitality.

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Copyright

Copyright © 2020 S. B. Mose and J. O. Bonyo. 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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