Weighted Composition Groups on the Little Bloch space

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 consequences, we complete the analysis of the adjoint composition group on the predual on the nonreflexive Bergman space, and a group of isometries associated with a specific automorphism of the upper half plane.


Introduction
The (open) unit disc D of the complex plane C is defined as D = {z ∈ C : |z| < 1}, while the upper half-plane of C, denoted by U, is given by U = {ω ∈ C : ℑ(ω) > 0} where ℑ(ω) stands for the imaginary part of ω.
where K is a constant and γ = α+2 p , see for example [18,Theorem 4.14]. The Bloch space of the unit disc, denoted by B ∞ (D), is defined as the space of analytic functions f ∈ H(D) such that the seminorm Following [17,18] where cl B∞ C[z] denotes B ∞ closure of the set of analytic polynomials in z. Equivalently, and possesses the same norm as B ∞ (D). Since B ∞,0 (D) is a closed subspace of the Banach space B ∞ (D), it follows that B ∞,0 (D) is a Banach space as well with respect to the norm · B∞(D) . Note that every f ∈ B ∞ (D) (or f ∈ B ∞,0 (D)) satisfies the growth condition See for instance [14] for details. Let 1 < p < ∞ and q be conjugate to p in the sense that 1 under the integral pairing It is well known that for 1 < p < ∞, L p a (D, m α ) is reflexive. The case p = 1 is the nonreflexive case and the duality relations have been determined as follows: and under the duality pairings given by, respectively In other words, the dual and predual spaces of the nonreflexive Bergman space L 1 a (D, m α ) are the Bloch and Little Bloch spaces respectively. For a comprehensive account of the theory of Bloch and Bergman spaces, we refer to [7,10,13,17,18]. In [2], all the self analytic maps (ϕ t ) t≥0 ⊆ Aut(U) of the upper half plane U 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 H(U) by for some appropriate weight γ.
It is noted in [2, Section 5] that for the rotation group, we consider the corresponding group of weighted composition operators defined on the analytic spaces of the disc H(D) 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 [3,9,14,15,16]. In [2] and [4], both the semigroup and spectral properties of the group (T t ) t∈R were studied in detail on the Hardy and Bergman spaces. The aim of this paper is to extend the analysis of the group (T t ) t∈R 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 (1.9) 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 L 1 a (D, m α ). The analysis of the adjoint group on the reflexive Bergman space, that is, L p a (D, m α ) for 1 < p < ∞, was considered exhaustively in [4]. We shall also consider a specific automorphism of U and carry out an analysis of the corresponding composition operator. If X is an arbitrary Banach space, let L(X) denote the algebra of bounded linear operators on X. For a linear operator T with domain D(T ) ⊂ X, denote the spectrum and point spectrum of T by σ(T ) and σ p (T ) respectively. The resolvent set of T is ρ(T ) = C \ σ(T ) while r(T ) denotes its spectral radius. For a good account of the theory of spectra, see [6,5,11]. If X and Y are arbitrary Banach spaces and U ∈ L(X, Y ) is an invertible operator, then clearly (A t ) t∈R ⊂ L(X) is a strongly continuous group if and only if

Groups of Composition operators on the Little Bloch space
We consider the group of weighted composition operators (T t ) t∈R given by equation (1.9) and defined on the little Bloch space . We denote the infinitesimal generator of the group (T t ) t∈R by Γ c,k and give some of its properties in the following Proposition, Proof. To prove isometry, we have By change of variables, let ω = e ikt z. Then To prove strong continuity, we shall use the density of polynomials in B ∞,0 (D). Therefore it suffices to show that for (z n ) n≥0 ; Strong continuity of (T s ) s≥0 implies that k! z k and Q m M m z f = f . We now give the following proposition; Proof. If f ∈ B ∞ (D), then for all z ∈ D, Therefore assertions (1) and (2) follow. For (3), if f ∈ B ∞,0 (D), then for |z| < 1, Thus Qf ∈ B ∞,0 (D). To prove (4) In fact, λ ∈ ρ(Γ 0,1 ) if and only if ic + kλ ∈ ρ(Γ c,k ), and As a result of Proposition 2.3 above and without loss of generality, we restrict our attention to the generator Γ 0,1 instead of Γ c,k as the cases c = 0 and k = 1 where k = 0 can be easily obtained from Γ 0,1 . Indeed, which is exactly the case when c = 0 and k = 1 in equation (1.9). We now give the spectral properties of the generator Γ 0,1 as well as the resulting resolvents in the following theorem; Theorem 2.4. 1. σ(Γ 0,1 ) = σ p (Γ 0,1 ) = {in : n ∈ Z + }, and for each n ≥ 0, ker(in − Γ 0,1 ) = span(z n ).
As a consequence, the properties of the general group T t given by equation (1.9) is the following 3. For µ ∈ ρ(Γ c,k ), the resolvent R(µ, Γ c,k ) is compact.
3. Adjoint of the Composition group on the predual of nonreflexive Bergman space L 1 a (D, m α ) In studying the adjoint properties of the rotation group isometries given by equation (1.9) on Bergman spaces L p a (D, m α ), 1 ≤ p < ∞, the second author in [4] considered the reflexive case, that is when 1 < p < ∞. This was an extension of the investigation of adjoint properties of the Cesáro operator in [1] on Hardy spaces, and later generalized to Bergman spaces in [2]. For the nonreflexive Bergman space L 1 a (D, m α ) (that is, p = 1), the analysis of the adjoint of 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 t f (z) = e ict f (e ikt z) where c, k ∈ R with k = 0 and ∀ f ∈ L 1 a (D, m α ).