Weighted Composition Groups on the Little Bloch space
aa r X i v : . [ m a t h . F A ] J a n Weighted Composition Groups on the LittleBloch space
S. B. Mose and J. O. Bonyo Dedicated to Prof. Len Miller (PhD advisor to author ) and Prof. Vivien Miller ofMississippi state University on their retirement Abstract.
We determine both the semigroup and spectral properties ofa group of weighted composition operators on the Little Bloch space. Itturns out that these are strongly continuous groups of invertible isome-tries on the Bloch space. We then obtain the norm and spectra of theinfinitesimal generator as well as the resulting resolvents which are givenas integral operators. As consequences, we complete the analysis of theadjoint composition group on the predual on the nonreflexive Bergmanspace, and a group of isometries associated with a specific automorphismof the upper half plane.
Mathematics Subject Classification (2010).
Primary 47B38, 47D03, 47A10.
Keywords.
Weighted composition operator group, analytic functions,similar semigroups, spectrum, resolvent, adjoint group, nonreflexive Bergmanspace.
1. Introduction
The (open) unit disc D of the complex plane C is defined as D = { z ∈ C : | z | < } , while the upper half-plane of C , denoted by U , is given by U = { ω ∈ C : ℑ ( ω ) > } where ℑ ( ω ) stands for the imaginary part of ω . TheCayley transform ψ ( z ) := i (1+ z )1 − z maps the unit disc D conformally onto theupper half-plane U with inverse ψ − ( ω ) = ω − iω + i . For every α > −
1, we definea positive Borel measure dm α on D by dm α ( z ) = (1 − | z | ) α dA ( z ) where dA denotes the area measure on D . For an open subset Ω of C , let H (Ω) denote the Fr´echet space of analyticfunctions f : Ω → C endowed with the topology of uniform convergence oncompact subsets of Ω. Let Aut(Ω) ⊂ H (Ω) denote the group of biholomorphicmaps f : Ω → Ω. For 1 ≤ p < ∞ , α > −
1, the weighted Bergman spaces of S. B. Mose and J. O. Bonyothe unit disc D , L pa ( D , m α ), are defined by L pa ( D , m α ) := ( f ∈ H ( D ) : k f k L pa ( D ,m α ) = (cid:18)Z D | f ( z ) | p dm α ( z ) (cid:19) p < ∞ ) . Clearly L pa ( D , m α ) = L p ( D , m α ) ∩ H ( D ) where L p ( D , m α ) is the classicalLebesgue spaces. For every f ∈ L pa ( D , m α ), the growth condition is given by | f ( z ) | ≤ K k f k (1 − | z | ) γ 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 spaceof analytic functions f ∈ H ( D ) such that the seminorm k f k B ∞ , ( D ) := sup z ∈ D (cid:0) − | z | (cid:1) | f ′ ( z ) | < ∞ . Following [17, 18], B ∞ ( D ) is a Banach space with respect to the norm k f k B ∞ ( D ) := | f (0) | + k f k B ∞ , ( D ) . On the other hand, the Little Bloch space of the disc,denoted by B ∞ , ( D ), is defined to be the closed subspace of B ∞ ( D ) such that B ∞ , ( D ) := cl B ∞ C [ z ]where cl B ∞ C [ z ] denotes B ∞ closure of the set of analytic polynomials in z .Equivalently, B ∞ , ( D ) := (cid:26) f ∈ H ( D ) : lim | z |→ − , z ∈ D (cid:0) − | z | (cid:1) | f ′ ( z ) | = 0 (cid:27) , and possesses the same norm as B ∞ ( D ) . Since B ∞ , ( D ) is a closed subspaceof the Banach space B ∞ ( D ), it follows that B ∞ , ( D ) is a Banach space aswell with respect to the norm k · k B ∞ ( D ) . Note that every f ∈ B ∞ ( D ) (or f ∈ B ∞ , ( D )) satisfies the growth condition | f ( z ) | ≤ (cid:18) log (cid:18) | z | − | z | (cid:19)(cid:19) k f k B ∞ ( D ) . (1.1)See for instance [14] for details. Let 1 < p < ∞ and q be conjugate to p inthe sense that p + q = 1. If ( L pa ( D , m α )) ∗ is the dual space of L pa ( D , m α ),then ( L pa ( D , m α )) ∗ ≈ L qa ( D , m α ) , α > − , (1.2)under the integral pairing h f, g i = Z D f ( z ) g ( z ) dm α ( f ∈ L pa ( D , m α ) , g ∈ L qa ( D , m α )) . (1.3)It is well known that for 1 < p < ∞ , L pa ( D , m α ) is reflexive. The case p = 1is the nonreflexive case and the duality relations have been determined asfollows: ( L a ( D , m α )) ∗ ≈ B ∞ ( D ) , (1.4)and ( B ∞ , ( D )) ∗ ≈ L a ( D , m α ) (1.5)roups of Composition operators 3under the duality pairings given by, respectively h f, g i = Z D f ( z ) g ( z ) dm α ( f ∈ L a ( D , m α ) , g ∈ B ∞ ( D )) (1.6)and h f, g i = Z D f ( z ) g ( z ) dm α ( f ∈ B ∞ , ( D ) , g ∈ L a ( D , m α )) . (1.7)In other words, the dual and predual spaces of the nonreflexive Bergmanspace L a ( D , m α ) are the Bloch and Little Bloch spaces respectively. For acomprehensive account of the theory of Bloch and Bergman spaces, we referto [7, 10, 13, 17, 18].In [2], all the self analytic maps ( ϕ t ) t ≥ ⊆ Aut( U ) of the upper half plane U were identified and classified according to the location of their fixed pointsinto three distinct classes, namely: scaling, translation and rotation groups.For each self analytic map ϕ t , we define a corresponding group of weightedcomposition operator on H ( U ) by S ϕ t f ( z ) = ( ϕ ′ t ( z )) γ f ( ϕ t ( z )) , (1.8)for some appropriate weight γ .It is noted in [2, Section 5] that for the rotation group, we consider the cor-responding group of weighted composition operators defined on the analyticspaces of the disc H ( D ) given by T t f ( z ) = e ict f ( e ikt z ) with c, k ∈ R , k = 0 . (1.9)The study of composition operators on spaces of analytic functions still re-mains an active area of research. For Bloch spaces, most studies have onlyfocussed on the boundedness and compactness of these operators. See for in-stance [3, 9, 14, 15, 16]. In [2] and [4], both the semigroup and spectral prop-erties of the group ( T t ) t ∈ R were studied in detail on the Hardy and Bergmanspaces. 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 oflinear operators on Banach spaces to study the properties of the group ofweighted composition operators given by equation (1.9) on the little Blochspace of the disk. As a consequence, we shall complete the analysis of theadjoint group on the dual of the nonreflexive Bergman space L a ( D , m α ).The analysis of the adjoint group on the reflexive Bergman space, that is, L pa ( D , m α ) for 1 < p < ∞ , was considered exhaustively in [4]. We shall alsoconsider a specific automorphism of U and carry out an analysis of the cor-responding composition operator.If X is an arbitrary Banach space, let L ( X ) denote the algebra of boundedlinear operators on X . For a linear operator T with domain D ( T ) ⊂ X , de-note 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 spectralradius. 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, S. B. Mose and J. O. Bonyothen clearly ( A t ) t ∈ R ⊂ L ( X ) is a strongly continuous group if and only if B t := U A t U − , t ∈ R , is a strongly continuous group in L ( Y ). In this case,if ( A t ) t ∈ R has generator Γ, then ( B t ) t ∈ R has generator ∆ = U Γ U − withdomain D (∆) = U D (Γ) := { y ∈ Y : U y ∈ D (Γ) } . Moreover, σ p (∆) = σ p (Γ) , and σ (∆) = σ (Γ) , since if λ is in the resolvent set ρ (Γ) := C \ σ (Γ), wehave that R ( λ, ∆) = U R ( λ, Γ) U − . See for example [8, Chapter II] and [11,Chapter 3].
2. Groups of Composition operators on the Little Bloch space
We consider the group of weighted composition operators ( T t ) t ∈ R given byequation (1.9) and defined on the little Bloch space B ∞ , ( D ) as T t f ( z ) = e ict f ( e ikt z ) where c, k ∈ R , k = 0 and ∀ f ∈ B ∞ , ( D ). We denote the infini-tesimal generator of the group ( T t ) t ∈ R by Γ c,k and give some of its propertiesin the following Proposition, Proposition 2.1.
1. ( T t ) t ∈ R is a strongly continuous group of isometries on B ∞ , ( D ) . The infinitesimal generator Γ c,k of ( T t ) t ∈ R on B ∞ , ( D ) is given by Γ c,k f ( z ) = i ( cf ( z ) + kzf ′ ( z )) with domain D (Γ c,k ) = { f ∈ B ∞ , ( D ) : zf ′ ∈ B ∞ , ( D ) } .Proof. To prove isometry, we have k T t f k B ∞ ( D ) = | T t f (0) | + sup z ∈ D (cid:0) − | z | (cid:1) | ( T t f ) ′ ( z ) | = | e ict f (0) | + sup z ∈ D (cid:0) − | z | (cid:1) (cid:12)(cid:12) e ict e ikt f ′ ( e ikt z ) (cid:12)(cid:12) = | f (0) | + sup z ∈ D (cid:0) − | z | (cid:1) (cid:12)(cid:12) f ′ ( e ikt z ) (cid:12)(cid:12) . By change of variables, let ω = e ikt z . Then k T t f k B ∞ ( D ) = | f (0) | + sup ω ∈ D (cid:0) − | ω | (cid:1) | f ′ ( ω ) | = k f k B ∞ ( D ) , as desired.To prove strong continuity, we shall use the density of polynomials in B ∞ , ( D ).Therefore it suffices to show that for ( z n ) n ≥ ;lim t → + k T t z n − z n k B ∞ , ( D ) = 0 . Now, T t z n − z n = e ict (cid:0) e ikt z (cid:1) n − z n = (cid:0) e i ( c + kn ) t − (cid:1) z n . Therefore,lim t → + k T t z n − z n k B ∞ , ( D ) = lim t → + (cid:18) sup z ∈ D (cid:0) − | z | (cid:1) | ( T t z n − z n ) ′ | (cid:19) = lim t → + (cid:18) sup z ∈ D (cid:0) − | z | (cid:1) (cid:12)(cid:12)(cid:12) n (cid:16) e i ( k + kn ) t − (cid:17) z n − (cid:12)(cid:12)(cid:12)(cid:19) = 0 , as claimed.roups of Composition operators 5Now, for the infinitesimal generator Γ c,k , let f ∈ D (Γ c,k ) in B ∞ , ( D ), thenthe growth condition (1.1) implies thatΓ c,k f ( z ) = lim t → + e ict f ( e ikt z ) − f ( z ) t = ∂∂t (cid:0) e ict f ( e ikt z ) (cid:1)(cid:12)(cid:12) t =0 = i ( cf ( z ) − izf ′ ( z )) . Therefore D (Γ c,k ) ⊆ { f ∈ B ∞ , ( D ) : zf ′ ∈ B ∞ , ( D ) } . Conversely, if f ∈ B ∞ , ( D )is such that zf ′ ∈ B ∞ , ( D ), then F ( z ) = i ( cf ( z ) + kzf ′ ( z )) ∈ B ∞ , ( D ) andfor all t > T t f ( z ) − f ( z ) t = 1 t Z t ∂∂s ( T s f ( z )) ds = 1 t Z t e ics (cid:0) i ( cf ( e iks z ) + k ( e iks z ) f ′ ( e iks z )) (cid:1) ds = 1 t Z t T s F ( z ) ds. Strong continuity of ( T s ) s ≥ implies that (cid:13)(cid:13)(cid:13)(cid:13) t Z t T s F ds − F (cid:13)(cid:13)(cid:13)(cid:13) ≤ t Z t k T s F − F k ds → t → + . Thus, D (Γ c,k ) ⊇ { f ∈ B ∞ , ( D ) : zf ′ ∈ B ∞ , ( D ) } . (cid:3) Define M z , Q on H ( D ) by M z f ( z ) = zf ( z ) and Qf ( z ) = f ( z ) − f (0) z ,( Qf (0) = f ′ (0)). More generally, Q m f ( z ) = P ∞ k = m f ( k ) (0) k ! z k − m , Q m f (0) = f m (0) m ! . Then M mz Q m f = P ∞ m f ( k ) (0) k ! z k and Q m M mz f = f . We now give thefollowing proposition; Proposition 2.2. M z : B ∞ ( D ) → B ∞ ( D ) is bounded M z B ∞ , ( D ) ⊆ B ∞ , ( D )3. Q : B ∞ , ( D ) → B ∞ , ( D ) is bounded For m ≥ , M mz B ∞ , ( D ) = (cid:8) f ∈ B ∞ , ( D ) : f k (0) = 0 ∀ k < m (cid:9) . In par-ticular, M z B ∞ , ( D ) is closed in B ∞ , ( D ) .Proof. If f ∈ B ∞ ( D ), then for all z ∈ D ,(1 − | z | ) | ( zf ) ′ | = (1 − | z | ) | zf ′ ( z ) + f ( z ) |≤ (1 − | z | ) | f ′ ( z ) | + (1 − | z | ) | f ( z ) |≤ (1 − | z | ) | f ′ ( z ) | + (1 − | z | ) (cid:18) log (cid:18) | z | − | z | (cid:19)(cid:19) k f k B ∞ ( D ) . S. B. Mose and J. O. BonyoTherefore assertions (1) and (2) follow. For (3), if f ∈ B ∞ , ( D ), then for | z | < − | z | ) | ( Qf ) ′ ( z ) | = (1 − | z | ) (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) − f ( z ) + f (0) z (cid:12)(cid:12)(cid:12)(cid:12) ≤ (1 − | z | | f ′ ( z ) | ) | z | + (1 − | z | ) (cid:16) log (cid:16) | z | −| z | (cid:17)(cid:17) k f k B ∞ ( D ) | z | + (1 − | z | ) k f k B ∞ ( D ) | z | → | z | → . Thus Qf ∈ B ∞ , ( D ). To prove (4), let f ∈ B ∞ , ( D ) and f (0) = 0 . Then f = M z Qf ∈ M z B ∞ , ( D ). The reverse inclusion is obvious. Therefore, theone-to-one and onto mapping M z : B ∞ , ( D ) → { f ∈ B ∞ , ( D ) : f (0) = 0 } isbounded. So the open mapping theorem implies that the inverse is bounded. Ittherefore follows that Q : span(1) ⊕ M z B ∞ , ( D ) → B ∞ , ( D ) is bounded. (cid:3) Proposition 2.3.
Let Γ c,k be the infinitesimal generator of the group ( T ) t ∈ R given by (1.9) on B ∞ , ( D ) , then
1. Γ c,k = ic + k Γ , with domain D (Γ c,k ) = D (Γ , ) = { f ∈ B ∞ , ( D ) : zf ′ ∈ B ∞ , ( D ) } . σ (Γ c,k ) = { ic + kσ (Γ , ) } , and σ p (Γ c,k ) = { ic + kσ p (Γ , ) } .In fact, λ ∈ ρ (Γ , ) if and only if ic + kλ ∈ ρ (Γ c,k ) , and R ( ic + kλ, Γ c,k ) = 1 k R ( λ, Γ , ) . (2.1) Proof.
See [4, Lemma 4.3]. (cid:3)
As a result of Proposition 2.3 above and without loss of generality,we restrict our attention to the generator Γ , instead of Γ c,k as the cases c = 0 and k = 1 where k = 0 can be easily obtained from Γ , . Indeed,Γ , f ( z ) = izf ′ ( z ) with domain D (Γ , ) = { f ∈ B ∞ , ( D ) : zf ′ ∈ B ∞ , ( D ) } is the infinitesimal generator of the group T t f ( z ) = f ( e it z ) which is exactlythe case when c = 0 and k = 1 in equation (1.9). We now give the spectralproperties of the generator Γ , as well as the resulting resolvents in thefollowing theorem; Theorem 2.4. σ (Γ , ) = σ p (Γ , ) = { in : n ∈ Z + } , and for each n ≥ , ker( in − Γ , ) = span ( z n ) . If λ ∈ ρ (Γ , ) , then M z B ∞ , ( D ) is R ( λ, Γ , ) - invariant ∀ m ∈ Z + , m > ℑ ( λ ) . Moreover, if h ∈ M mz B ∞ , ( D ) , then R ( λ, Γ , ) = iz − λt Z z ω iλ − h ( ω ) dω = iz m Z t m + iλ − ( Q m h ) ( tz ) dt. (2.2)3. For λ ∈ ρ (Γ , ) , the resolvent operator R ( λ, Γ , ) is compact. roups of Composition operators 74. σ ( R ( λ, Γ , )) = σ p ( R ( λ, Γ , )) = n w ∈ C : (cid:12)(cid:12)(cid:12) w − ℜ ( λ ) (cid:12)(cid:12)(cid:12) = ℜ ( λ ) o . More-over, r ( R ( λ, Γ , )) = k R ( λ, Γ , ) k = |ℜ ( λ ) | . Proof.
Since each T t is an invertible isometry, its spectrum satisfies σ ( T t ) ⊆ ∂ D , and the spectral mapping theorem for strongly continuous groups (seefor example [8, Theorem V.2.5] or [12]) implies that e tσ (Γ , ) ⊆ σ ( T t ) . Thus, e tσ (Γ , ) ⊆ ∂ D ⇒ | e tσ (Γ , ) | = 1 ⇒ e t ℜ ( ω ) = 1 ⇒ ℜ ( ω ) = 0 for ω ∈ σ (Γ , ). It immediately follows that σ (Γ , ) ⊆ i R .We now solve the resolvent equation: If λ ∈ C and h ∈ H ( D ), ( λ − Γ) f = h .This is equivalent to f ′ ( z ) + iλz f ( z ) = iz h ( z ) , ( z = 0)or (cid:0) z iλ f ( z ) = iz iλ − h ( z ) , ( z ∈ D \ ( − , . (cid:1) In particular, ( λ − Γ) f = 0 if and only f ( z ) = Kz − iλ , where K is a constant.Since z − iλ ∈ H ( D ) if and only if − iλ ∈ Z + , it follows that σ p (Γ , ) = { in : n ∈ Z + } with ker( in − Γ , ) = span( z n ). Moreover, if n ∈ Z + and λ ∈ σ p (Γ , ), then( λ − Γ) f = z n has a unique solution f ( z ) = 1 λ − in z n . Notice that for λ / ∈ σ p (Γ , ) and f ∈ D (Γ , ), ( λ − Γ) f (0) = λf (0). Moregenerally, if f ( z ) = z n g ( z ) with g (0) = 0, then( λ − Γ) f = λf − z ( z m g ) ′ = z m (cid:0) λg − mz m g − z m +1 g ′ (cid:1) . Note that the functions ( λ − Γ) f and f have the same order of zero at 0.Thus M mz B ∞ , ( D ) is invariant under λ − Γ , .Fix λ ∈ C \ σ p (Γ , ) and let m > ℑ ( λ ). If h = z m g with g ∈ B ∞ , ( D ), then i Z z ω iλ − h ( ω ) dω = iz m + iλ Z t m + iλ − g ( tz ) dt. Thus ( λ − Γ) h has a unique solution f ( z ) = iz m Z t m + iλ − ( Q m h )( tz ) dt. If u ∈ B ∞ ( D ) and 0 ≤ t < , then k u ( tz ) k B ∞ ( D ) = sup | z | < (cid:0) − | z | (cid:1) t | u ′ ( tz ) |≤ sup | z | < (cid:0) − t | z | (cid:1) | u ′ ( tz ) |≤ k u k B ∞ ( D ) . (2.3) S. B. Mose and J. O. BonyoThus k f k ≤ m −ℑ ( λ ) k M mz kk Q m kk h k . Now, ∀ m ≥ , B ∞ , ( D ) = span( z n ) ⊕ M mz B ∞ , ( D ) (2.4)and R ( λ, Γ , ) | span( z n ) ≤ n
0, be the disc algebra A ( r D ) = C ( r D ) ∩H ( r D ), equippedwith the supremum norm, and for each t , 0 ≤ t <
1, and f ∈ H ( D ), let H t f ( z ) = f t ( z ) = f ( tz ). Then by equation (2.3), for every t ∈ [0 , H t is acontraction on B ∞ , ( D ).Now, by equation (2.2), R m ( λ, Γ , ) = iM mz R t m + iλ − H t Q m dt withconvergence in norm. Define C r = iM mz R r t m + iλ − H t Q m dt on M mz B ∞ , ( D )for 0 < r <
1. Then k R m − C r k ≤ Z r t m −ℑ ( λ ) − k Q k m dt = k Q k m m − ℑ ( λ ) (1 − r m −ℑ ( λ ) ) → r → − . Choosing s so that 1 < s < r − , we have that C r : M mz B ∞ , ( D ) → M mz B ∞ , ( D ) factors through A ( s D ). If B denotes the closed unit ball of M mz B ∞ , ( D ), let h = Q m f ( f ∈ M mz B ∞ , ( D )). Then ∀ t , 0 ≤ t ≤ r , thegrowth condition (1.1) implies that for | z | ≤ s , | h ( tz ) | ≤ (cid:18) (cid:18) rs − rs (cid:19)(cid:19) k h k B ∞ , ( D ) and (cid:12)(cid:12)(cid:12)(cid:12) ddt h ( tz ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k h k B ∞ , ( D ) − rs . Let K = (cid:16) log (cid:16) rs − rs (cid:17)(cid:17) k h k B ∞ , ( D ) . Thus for | z | ≤ s , | C r f ( z ) | ≤ K s m r m −ℑ ( λ ) m − ℑ ( λ ) , and (cid:12)(cid:12)(cid:12)(cid:12) ddz C r f ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ K ms m − r m −ℑ ( λ ) m − ℑ ( λ ) + s m r m −ℑ ( λ ) m − ℑ ( λ ) k h k B ∞ , ( D ) − rs . roups of Composition operators 9Thus by Arzela-Ascoli, C r B is pre-compact in A ( s D ) which further impliesthat C r B is pre-compact in B ∞ , ( D ) by the continuous embeddedness of A ( s D ) in B ∞ , ( D ). Therefore each C r is compact in L ( M mz B ∞ , ( D )) and asa result, R m ( λ, Γ , ) = (norm) lim r → − C r is compact as well.The spectral mapping theorem for resolvents as well as assertion (1) aboveimplies that σ ( R ( λ, Γ , )) = σ p ( R ( λ, Γ , )) = { λ − im : m ∈ Z + } ∪ { } = n ω ∈ C : | ω − ℜ ( λ ) | = |ℜ ( λ ) | o . Clearly the spectral radius r ( R ( λ, Γ , )) = |ℜ ( λ ) | and therefore by the Hille-Yosida theorem, it follows that |ℜ ( λ ) | = r ( R ( λ, Γ , )) ≤ k R ( λ, Γ , ) k ≤ |ℜ ( λ ) | ,as desired. (cid:3) As a consequence, the properties of the general group T t given by equa-tion (1.9) is the following Corollary 2.5. σ (Γ c,k ) = σ p (Γ c,k ) = { i ( c + kn ) : n ∈ Z + } , and for each n ≥ , ker( i ( c + kn ) − Γ c,k ) = span ( z n ) . If µ ∈ ρ (Γ c,k ) , then M z B ∞ , ( D ) is R ( µ, Γ c,k ) -invariant ∀ m ∈ Z + , m > ℑ (cid:0) µ − ick (cid:1) . Moreover, if h ∈ M mz B ∞ , ( D ) , then R ( µ, Γ c,k ) = ik z − ( µ − ick ) t Z z ω i ( µ − ick ) − h ( ω ) dω = ik z m Z t m + i ( µ − ick ) − ( Q m h ) ( tz ) dt. (2.5)3. For µ ∈ ρ (Γ c,k ) , the resolvent R ( µ, Γ c,k ) is compact. σ ( R ( µ, Γ c,k )) = σ p ( R ( µ, Γ c,k )) = n w ∈ C : (cid:12)(cid:12)(cid:12) w − ℜ ( µ ) (cid:12)(cid:12)(cid:12) = ℜ ( µ ) o . r ( R ( µ, Γ c,k )) = k R ( µ, Γ c,k ) k = |ℜ ( µ ) | . Proof.
Following proposition 2.3, µ ∈ ρ (Γ c,k ) if and only if µ − ick ∈ ρ (Γ , ).The proof now follows at once from Theorem 2.4. We omit the details. (cid:3)
3. Adjoint of the Composition group on the predual ofnonreflexive Bergman space L a ( D , m α ) In studying the adjoint properties of the rotation group isometries given byequation (1.9) on Bergman spaces L pa ( D , m α ), 1 ≤ p < ∞ , the second authorin [4] considered the reflexive case, that is when 1 < p < ∞ . This was anextension of the investigation of adjoint properties of the Ces´aro operatorin [1] on Hardy spaces, and later generalized to Bergman spaces in [2]. Forthe nonreflexive Bergman space L a ( D , m α ) (that is, p = 1), the analysis ofthe adjoint of rotation group isometries remains open and forms the basisof this section. Specifically, we complete the analysis of the adjoint group ofthe group of isometries T t f ( z ) = e ict f ( e ikt z ) where c, k ∈ R with k = 0 and ∀ f ∈ L a ( D , m α ).0 S. B. Mose and J. O. BonyoRecall from section 1 the duality relation ( B ∞ , ( D )) ∗ ≈ L a ( D , m α ) under theintegral pairing h g, f i = R D g ( z ) f ( z ) dm α ( g ∈ B ∞ , ( D ) , g ∈ L a ( D , m α )) . Inparticular, the predual of L a ( D , m α ) is the Little Bloch space B ∞ , ( D ). Thus,using this duality pairing, for every g ∈ B ∞ , ( D ), we have h g, T t f i = Z D g ( z ) e ict f ( e ikt z ) dm α ( z )= Z D e − ict g ( z ) f ( e ikt z )(1 − | z | ) α dA ( z ) . By a change of variables argument: Let ω = e ikt z so that z = e − ikt ω and h g, T t f i = Z D e − ict g ( e − ikt ω ) f ( ω )(1 − | e − ikt ω | ) α dA ( ω )= Z D e − ict g ( e − ikt ω ) f ( ω ) dm α ( ω )= Z D T − t g ( ω ) f ( ω ) dm α ( ω ) = h T t g, f i , where T − t g ( ω ) = e − ict g ( e − ikt ω ) for all g ∈ B ∞ , ( D ). Thus, the adjoint group T ∗ t of T t for t ∈ R is therefore given by T ∗ t g ( ω ) := T − t g ( ω ) = e − ict g ( e − ikt ω ) , for all g ∈ B ∞ , ( D ) . (3.1)Let Γ denotes the infinitesimal generator of the adjoint group T ∗ t . Using theresults of Section 2, we easily obtain the properties of the group ( T ∗ t ) t ∈ R aswe give in the following theorem; Theorem 3.1.
Let ( T ∗ t ) t ∈ R ⊆ L ( B ∞ , ( D )) be the adjoint group of the group ofweighted composition operators ( T t ) t ∈ R ⊆ L ( L a ( D , m α )) given by (3.1) . Thenthe following hold:
1. ( T ∗ t ) t ∈ R is strongly continuous group of isometries on B ∞ , ( D ) . The infinitesimal generator Γ of ( T ∗ t ) t ≥ is given by Γ g ( ω ) = − i ( cg ( ω ) + kωg ′ ( ω )) with domain D (Γ) = { g ∈ B ∞ , ( D ) : ωg ′ ∈ B ∞ , ( D ) } . σ (Γ) = σ p (Γ) = {− i ( c + kn ) : n ∈ Z + } , and for each n ≥ , ker ( − i ( c + kn ) − Γ) =span( ω n )4. If µ ∈ ρ (Γ) , then M ω B ∞ , ( D ) is R ( µ, Γ) -invariant ∀ m ∈ Z + , m > ℑ (cid:0) − µ − ick (cid:1) . Moreover, if h ∈ M mω B ∞ , ( D ) , then R ( µ, Γ) = − ik ω ( µ + ick ) t Z ω ζ − i ( µ + ick ) − h ( z ) dz = − ik ω m Z t m − i ( µ + ick ) − ( Q m h ) ( tω ) dt. σ ( R ( µ, Γ)) = σ p ( R ( µ, Γ)) = n w ∈ C : (cid:12)(cid:12)(cid:12) w − ℜ ( µ ) (cid:12)(cid:12)(cid:12) = ℜ ( µ ) o . r ( R ( µ, Γ)) = k R ( µ, Γ) k = |ℜ ( µ ) | . Proof.
The proof follows immediately by replacing c and k with − c and − k respectively in Proposition 2.3 and Corollary 2.5. We omit the details. (cid:3) roups of Composition operators 11
4. Specific Automorphism of the half-plane
In this section, we consider a specific automorphism group ( ϕ t ) t ∈ R ⊂ Aut( U )corresponding to the rotation group given by ϕ t ( z ) = z cos t − sin tz sin t + cos t . (4.1)It can be easily verified that ϕ t ( z ) = ψ ◦ u t ◦ ψ − ( z ), where u t ( z ) = e − it z .The associated group of weighted composition operators on H ( D ) is given by S ϕ t and by chain rule, it follows that S ϕ t = S ψ − S u t S ψ , where S ψ − = S − ψ .Now, for f ∈ B ∞ , ( D ), S u t f ( z ) = ( u ′ t ( z )) γ f ( u t ( z ))= e − iγt f ( e − it z ) . Apparently, S u t can be obtained as a special case of the group ( T t ) t ≥ givenby equation (1.9) when c = − γ and k = −
2. Let Γ = Γ − γ, − be theinfinitesimal generator of the group S u t , then the properties of Γ can besummarized by the following proposition; Proposition 4.1.
Let Γ be the infinitesimal generator of the group of isometries S u t on B ∞ , ( D ) . Then
1. Γ f ( z ) = i ( − γf ( z ) − zf ′ ( z )) for every f ∈ B ∞ , ( D ) , with domain D (Γ) = { f ∈ B ∞ , ( D ) : f ′ ∈ B ∞ , ( D ) } . σ (Γ) = σ p (Γ) = {− γ + n ) i : n ∈ Z + } , and for each n ≥ , ker ( − γ + n ) i − Γ) = span ( z n )3. If µ ∈ ρ (Γ) , then R ( M mz ) is R ( µ, Γ) -invariant for every m ∈ Z + , m > ℑ ( − ( µ + 2 γi ) / . Moreover, if h ∈ R ( M mz ) , then R ( µ, Γ) h ( z ) = − i z ( µ +2 iγ ) i Z z ω − ( µ − iγ ) i − h ( ω ) dω := R µ h ( z ) . Proof.
Take c = − γ and k = − (cid:3) Now, using the similarity theory of semigroups, we detail the propertiesof the group of weighted composition operators associated with the automor-phism group ( ϕ t ) t ≥ given by (4.1) in the following theorem; Theorem 4.2.
Let ϕ t ∈ Aut( U ) be given by ϕ t ( z ) = z cos t − sin tz sin t +cos t , for all t ∈ R , z ∈ U , and let S ϕ t f ( z ) := ( ϕ ′ t ) γ f ( ϕ t ( z )) be the corresponding group ofisometries on B ∞ , ( D ) . Then The infinitesimal generator ∆ of the group S ϕ t on B ∞ , ( D ) is given by ∆( h ( z )) = − γzh ( z ) − (1 + z ) h ′ ( z ) , with domain D (∆) = { h ∈ B ∞ , ( D ) : 2 γ ( ω + i ) h +( ω + i ) h ′ ∈ B ∞ , ( D ) } . σ p (∆) = σ (∆) = {− γ + n ) i : n ∈ Z + } , and for each n ≥ , ker( − γ + n ) i − ∆) = span ( S − ψ z n ) . If µ ∈ ρ (∆) and if m ∈ Z + is such that m > ℑ ( − µ/ − iγ ) . Then, if h ∈ R ( M mz ) , we have R ( µ, ∆) h ( z ) = ( z − i ) µ +2 iγ i ( z + i ) − ( µ +2 iγ i +2 γ ) Z z ( ω − i ) − ( µ +2 iγ ) i − ( ω + i ) µ +2 iγ i +2 γ − h ( ω ) dω. (4.2)4. R ( µ, ∆) is compact on B ∞ , ( D ) . σ ( R ( µ, ∆)) = σ p ( R ( µ, ∆)) = n w ∈ C : (cid:12)(cid:12)(cid:12) w − ℜ ( µ ) (cid:12)(cid:12)(cid:12) = |ℜ ( µ ) | o . Morover, r ( R ( µ, ∆)) = k R ( µ, ∆) k = ℜ ( µ ) . Proof.
Since ϕ t ( z ) = ψ ◦ u t ◦ ψ − ( z ), it follows that S ϕ t = S ψ − S u t S ψ = S − ψ S u t S ψ , where S ψ is invertible. Let ∆ be the generator of S ϕ t and Γ :=Γ − γ, − be the generator of S u t . Then using similarity theory as presentedin section 1 of this paper, we have that:(a) ∆ = S g Γ S − g with domain D (∆) = S g D (Γ)(b) σ (∆) = σ (Γ) and σ p (∆) = σ p (Γ)(c) If µ ∈ ρ (∆), then R ( µ, ∆) = S − ψ R ( µ, Γ) S ψ .With relations (a)-(c) above, and using Proposition 4.1, a direct computationyields assertions 1-3. We omit the details and instead refer to [4, Theorem4.4] for a similar computation. Assertion 4 follows from the compactness of R ( µ, Γ), while assertion 5 is immediate from Corollary 2.5(4) as well as theHille - Yosida theorem. (cid:3)
References [1] A. G. Arvanitidis, A. G. Siskakis,
Ces`aro operators on the Hardy spaces of thehalf plane.
Canadian Math. Bull. (2013), 229–240.[2] S. Ballamoole, J. O. Bonyo, T. L. Miller, V. G. Miller, Ces`aro operators on theHardy and Bergman spaces of the half plane.
Complex Anal. Oper. Theory (2016), 187–203.[3] S. Ballamoole, T. L. Miller, V. G. Miller, Extensions of spaces of analytic func-tions via pointwise limits of bounded sequences and two integral operators ongeneralized Bloch spaces
Arch. Math. (2013), 269–283.[4] J. O. Bonyo,
Spectral Analysis of certain groups of isometries on Hardy andBergman spaces.
J. Math. Anal. Appl. (2017), 1470–1481.[5] J. B. Conway,
A course in functional analysis.
Springer - Verlag, New York,1985.[6] N. Dunford, J. T. Schwartz,
Linear Operators Part I.
Interscience Publishers,New York, 1958.[7] P. Duren, A. Schuster,
Bergman spaces.
Mathematical Surveys and Monographs , Amer. Math. Soc., Providence, RI, 2004.[8] K.-J. Engel, R. Nagel,
A short course on operator semigroups.
Universitext,Springer, New York, 2006. roups of Composition operators 13 [9] X. Fu, J. Zhang,
Bloch - type spaces on the upper half - plane
Bull. KoreanMath. Soc. (2017), 1337–1346.[10] J. B. Garnett, Bounded Analytic Functions.
Graduate Texts in Mathematics,Revised First Edition, Springer, Berlin, 2010.[11] K. B. Laursen, M. M. Neumann,
An introduction to local spectral theory.
Clarendon Press, Oxford, 2000.[12] A. Pazy,
Semigroups of linear operators and applications to partial differentialequations.
Applied Mathematical Sciences , Springer, New York, 1983.[13] M. M. Peloso, Classical spaces of Holomorphic functions.
Technical report,Universi`t di Milano, 2014.[14] S. Ohno, R. Zhao,
Weighted composition operators on the Bloch space
Bull.Austral. Math. Soc. (2001), 177–185.[15] Y. Shi, S. Li, Differences of Composition operators on Bloch type spaces
Com-plex Anal. Oper. Theory (2017), 227–242.[16] H. Wulan, D. Zheng, K. Zhu, Composition operators on BMOA and the Blochspace
Proc. Amer. Math. Soc. (2009), 3861–3868.[17] K. Zhu,
Bloch type spaces of analytic functions.
Rocky Mountain J. Math. (1993), 1143–1177.[18] K. Zhu, Operator theory in function spaces.