Semigroups of composition operators and integral operators in spaces of analytic functions

Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 38, 2013, 67–89 SEMIGROUPS OF COMPOSITION OPERATORS AND INTEGRAL OPERATORS IN SPACES OF ANALYTIC FUNCTIONS Oscar Blasco, Manuel D. Contreras, Santiago Díaz-Madrigal, Josep Martínez, Michael Papadimitrakis and Aristomenis G. Siskakis Universidad de Valencia, Departamento de Análisis Matemático 46100 Burjassot, Valencia, Spain; [email protected] Universidad de Sevilla, Departamento de Matemática Aplicada II Camino de los Descubrimientos, s/n, 41092, Sevilla, Spain; con[email protected] Universidad de Sevilla, Departamento de Matemática Aplicada II Camino de los Descubrimientos, s/n, 41092, Sevilla, Spain; [email protected] Universidad de Valencia, Departamento de Análisis Matemático 46100 Burjassot, Valencia, Spain; [email protected] University of Crete, Department of Mathematics Knossos Avenue 714 09 Iraklion–Crete, Greece; [email protected] Aristotle University of Thessaloniki, Department of Mathematics 54124 Thessaloniki, Greece; sisk[email protected] Abstract. We study the maximal spaces of strong continuity on BMOA and the Bloch space Bfor semigroups of composition operators. Characterizations are given for the cases when these maximal spaces are V MOA or the little Bloch B0. These characterizations are in terms of the weak compactness of the resolvent function or in terms of a specially chosen symbol gof an integral operator Tg. For the second characterization we prove and use an independent result, namely that the operators Tgare weakly compact on the above mentioned spaces if and only if they are compact. 1. Introduction Let H(D)be the Fréchet space of all analytic functions in the unit disk endowed with the topology of uniform convergence on compact subsets of D. We say that a Banach space Xis a Banach space of analytic functions if consists of functions of H(D)such that the inclusion i(f) = f:X→ H(D)is continuous. A (one-parameter) semigroup of analytic functions is any continuous homomorphism Φ: t7→ Φ(t) = ϕtfrom the additive semigroup of nonnegative real numbers into the composition semigroup of all analytic functions which map Dinto D. In doi:10.5186/aasfm.2013.3806 2010 Mathematics Subject Classification: Primary 30H05, 32A37, 47B33, 47D06; Secondary 46E15. Key words: Composition semigroups, BMOA, Bloch spaces, integration operators, weak compactness. The first and fourth authors were partially supported by the Ministerio de Ciencia e Innovación MTM2008-04594/MTM. The second and third author were partially supported by the Ministerio de Ciencia e Innovación and the European Union (FEDER), project MTM2009-14694-C02-02, the ESF Networking Programme “Harmonic and Complex Analysis and its Applications” and by La Consejería de Economía, Innovación y Ciencia de la Junta de Andalucía (research group FQM-133). The last author has received partial technical support by a grant from Empeirikion Foundation. 68 O. Blasco, M. D. Contreras, S. Díaz-Madrigal, J. Martínez, M. Papadimitrakis and A. G. Siskakis other words, Φ = (ϕt)consists of analytic functions on Dwith ϕt(D)⊂Dand for which the following three conditions hold: (1) ϕ0is the identity in D, (2) ϕt+s=ϕt◦ϕs,for all t, s ≥0, (3) ϕt→ϕ0,as t→0, uniformly on compact subsets of D. It is well known that condition (3) above can be replaced by (30) For each z∈D,ϕt(z)→z, as t→0. Each such semigroup gives rise to a semigroup (Ct)consisting of composition operators on H(D), Ct(f) := f◦ϕt, f ∈ H(D). We are going to be interested in the restriction of (Ct)to certain linear subspaces H(D). Given a Banach space of analytic functions and a semigroup (ϕt),we say that (ϕt)generates a semigroup of operators on Xif (Ct)is a well-defined strongly continuous semigroup of bounded operators in X. This exactly means that for every f∈X, we have Ct(f)∈Xfor all t≥0and lim t→0+kCt(f)−fkX= 0. Thus the crucial step to show that (ϕt)generates a semigroup of operators in Xis to pass from the pointwise convergence limt→0+f◦ϕt(z) = f(z)on Dto the convergence in the norm of X. This connection between composition operators and operator semigroups opens the possibility of studying spectral properties, operator ideal properties or dynamical properties of the semigroup of operators (Ct)in terms of the theory of functions. The paper [3] can be considered as the starting point in this direction. Classical choices of Xtreated in the literature are the Hardy spaces Hp, the disk algebra A(D), the Bergman spaces Ap, the Dirichlet space Dand the chain of spaces Qpand Qp,0which have been introduced recently and which include the spaces BMOA, Bloch as well as their “little oh” analogues. See [26] and [28] for definitions and basic facts of the spaces and [22], [23] and [25] for composition semigroups on these spaces. Very briefly, the state of the art is the following: (i) Every semigroup of analytic functions generates a semigroup of operators on the Hardy spaces Hp(1 ≤p < ∞), the Bergman spaces Ap(1 ≤p < ∞), the Dirichlet space, and on the spaces VMOA and little Bloch. (ii) No non-trivial semigroup generates a semigroup of operators in the space H∞of bounded analytic functions. (iii) There are plenty of semigroups (but not all) which generate semigroups of operators in the disk algebra. Indeed, they can be well characterized in different analytical terms [6]. Recently, in [4], the study of semigroups of composition operators in the framework of the space BMOA was initiated. The present paper can be considered as a sequel of [4]. In section 3 we state some general facts about the maximal subspace of strong continuity [ϕt, X]for the composition semigroup induced by (ϕt)on an abstract Banach space Xof analytic functions satisfying certain conditions. In section 4 we consider the maximal subspaces [ϕt, BMOA]and [ϕt,B]. In particular, for the case of the Bloch space we show that no non-trivial composition semigroup is strongly continuous on the whole space Bthus [ϕt,B]is strictly contained in B. We also show that that the equality [ϕt, BMOA] = V MOA or [ϕt,B] = B0is equivalent to the Semigroups of composition operators and integral operators in spaces of analytic functions 69 weak compactness of the resolvent operator R(λ, Γ) for the composition semigroup {Ct}acting on V MOA (respectively on B0). Here Γdenotes the infinitesimal generator of {Ct}and for each λin the resolvent set of Γ,R(λ, Γ) denotes the (bounded) resolvent operator R(λ, Γ) = (λ−Γ)−1. In section 5 we study an integral operator Tgon BMOA and Band we show in particular that its compactness and weak compactness are equivalent (Theorem 6). This result for the case of BMOA was also obtained independently by different methods in [13]. In section 6 we apply these results for Tgfor a special choice of the symbol g=γ(defined later in Definition 4) to obtain a characterization of the cases of equality [ϕt, BMOA] = V MOA and [ϕt,B] = B0in terms of γ(Corollary 2). In the final section 7 we make some additional observations concerning the Koenigs function of the semigroup in relation to space of strong continuity, and state some related open questions. 2. Background If (ϕt)is a semigroup, then each map ϕtis univalent. The infinitesimal generator of (ϕt)is the function G(z) = lim t→0+ ϕt(z)−z t, z ∈D. This convergence holds uniformly on compact subsets of Dso G∈ H(D). Moreover Gsatisfies (1) G(ϕt(z)) = ∂ϕt(z) ∂t =G(z)∂ϕt(z) ∂z , z ∈D, t ≥0. Further Ghas a representation (2) G(z) = (bz −1)(z−b)P(z), z ∈D, where b∈Dand P∈ H(D)with Re P(z)≥0for all z∈D. If Gis not identically null, the couple (b, P)is uniquely determined from (ϕt)and the point bis called the Denjoy–Wolff point of the semigroup. We want to mention that this point plays a crucial role in the dynamical behavior of the semigroup (see [23, 7]). Recall also the notion of Koenigs function associated with a semigroup. For every non-trivial semigroup (ϕt)with generator G, there exists a unique univalent function h:D→C, called the Koenigs function of (ϕt), such that 1. If the Denjoy–Wolff point bof (ϕt)is in Dthen h(b) = 0,h0(b) = 1 and (3) h(ϕt(z)) = eG0(b)th(z)for all z∈Dand t≥0. Moreover, (4) h0(z)G(z) = G0(b)h(z), z ∈D. 2. If the Denjoy–Wolff point bof (ϕt)is on ∂Dthen h(0) = 0 and (5) h(ϕt(z)) = h(z) + tfor all z∈Dand t≥0. Moreover, (6) h0(z)G(z) = 1, z ∈D. 70 O. Blasco, M. D. Contreras, S. Díaz-Madrigal, J. Martínez, M. Papadimitrakis and A. G. Siskakis For the sake of completeness and to fix notations, we present a quick review of basic properties of BMOA,V MOA, the Bloch space B, and the little Bloch space B0. BMOA is the Banach space of all analytic functions in the Hardy space H2whose boundary values have bounded mean oscillation. There are many characterizations of this space but we will use the one in terms of Carleson measures (see [28, 12]). Namely, a function f∈H2belongs to BMOA if and only if there exists a constant C > 0such that ˆR(I)|f0(z)|2(1 −|z|2)dA(z)≤C|I|, for any arc I⊂∂D, where R(I)is the Carleson rectangle determined by I, that is, R(I) := ½reiθ ∈D: 1 −|I| 2π< r < 1and eiθ ∈I¾. As usual, |I|denotes the length of Iand dA(z)the normalized Lebesgue measure on D. The corresponding BMOA norm is kfkBMOA := |f(0)|+ sup I⊂∂Dµ1 |I|ˆR(I)|f0(z)|2(1 −|z|2)dA(z)¶1/2 . Trivially, each polynomial belongs to BMOA. The closure of all polynomials in BMOA is denoted by V MOA. Alternatively, V MOA is the subspace of BMOA formed by those f∈BMOA such that lim |I|→0 1 |I|ˆR(I)|f0(z)|2(1 −|z|2)dA(z) = 0. Particular and quite interesting examples of members of V MOA are provided by functions in the Dirichlet space D, which is the space of those functions f∈ H(D) such that f0∈L2(D, dA)with norm kfkD=|f(0)|2+kf0k2 L2(D, dA). In fact, for every f∈ D, 1 |I|ˆR(I)|f0(z)|2(1 −|z|2)dA(z)≤2ˆR(I)|f0(z)|2dA(z)→0, as |I| → 0, so Dis contained in V MOA. This last inequality also implies that there is an absolute constant Csuch that (7) kfkBMOA ≤CkfkD for each f∈ D. A holomorphic function f∈ H(D)is said to belong to the Bloch space Bwhenever supz∈D¡1−|z|2¢|f0(z)|<∞. It is well-known that Bis a Banach space when it is endowed with the norm kfkB=|f(0)|+ sup z∈D¡1−|z|2¢|f0(z)|. The closure of the polynomials in Bis called the little Bloch space and it is denoted by B0. It is also well-known that f∈ B0if and only if lim |z|→1¡1−|z|2¢|f0(z)|= 0. For more information on these Banach spaces, we refer the reader to the excellent monographs [12] or [28]. Semigroups of composition operators and integral operators in spaces of analytic functions 71 Finally at this point we would like to mention some information about univalent functions that will be needed later. Let h:D→Cbe univalent. Then h∈BMOA if and only if h∈ B, and h∈V MOA if and only if h∈ B0. Further h∈ B if and only if the discs that can be inscribed in the range of hhave bounded radii and h∈ B0if and only if the radii of these discs tend to 0as their center moves to ∞. In addition if h:D→Cis univalent and non-vanishing then log(h(z)) ∈BMOA, while if h(b) = 0 for some b∈Dthen log h(z) z−b∈BMOA. Additional information can be found in [17]. 3. The space [ϕt, X] Given a semigroup of analytic functions (ϕt)and a Banach space Xof analytic functions on the unit disk, we are interested on the maximal closed subspace of X, denoted by [ϕt, X], on which (ϕt)generates a strongly continuous semigroup (Ct) of composition operators. The existence of such a maximal subspace, as well as analytical descriptions of it, will be discussed in this section. The next result for a Banach space Xof analytic functions is contained, for the special case X=BMOA, in [4]. The proof follows the same lines as the one in [4], and for this reason we omit the details. Proposition 1. Let (ϕt)be a semigroup of analytic functions and Xa Banach space of analytic functions such that Ct:X→Xare bounded for all t≥0and supt∈[0,1] kCtkX=M < ∞. Then there exists a closed subspace Yof Xsuch that (ϕt)generates a semigroup of operators on Yand such that any other subspace of Xwith this property is contained in Y. Definition 1. We denote by [ϕt, X]the maximal subspace consisting of functions f∈Xsuch that limt→0+kf◦ϕt−fkX= 0. It is easy to see that if Zis any closed subspace of [ϕt, X]which is invariant under (Ct)(i.e. Ct(Z)⊂Zfor every t≥0) then (ϕt)generates a semigroup of operators on Z. Definition 2. Given a semigroup (ϕt)with generator Gand a Banach space of analytic functions Xwe define D(ϕt, X) := {f∈X:Gf0∈X}. Clearly D(ϕt, X)is a linear subspace of X. Theorem 1. Let (ϕt)be a semigroup with generator Gand Xa Banach space of analytic functions which contains the constant functions and such that M= supt∈[0,1] kCtkX<∞. Then, [ϕt, X] = D(ϕt, X). Proof. Let us show first [ϕt, X]⊆ D(ϕt, X).We may assume that (ϕt)is not trivial. Denote by Γthe infinitesimal generator of the operator semigroup (Ct)acting on the Banach space [ϕt, X], and by D(Γ) its domain. We will show that if f∈D(Γ) then Gf0∈X. Indeed if f∈D(Γ) then Γ(f)∈[ϕt, X]⊆Xand lim t→0+° ° ° ° 1 t(Ct(f)−f)−Γ(f)° ° ° °X = 0. 72 O. Blasco, M. D. Contreras, S. Díaz-Madrigal, J. Martínez, M. Papadimitrakis and A. G. Siskakis Since convergence in the norm of Ximplies uniform convergence on compact subsets of Dand therefore in particular pointwise convergence, for each z∈Dwe have Γ(f)(z) = lim t→0+ f(ϕt(z)) −f(z) t= lim t→0+ f(ϕt(z)) −f(ϕ0(z)) t =∂f ◦ϕt(z) ∂t ¯¯¯¯t=0 =f0(ϕ0(z)) ∂ϕt(z) ∂t ¯¯¯¯t=0 =f0(z)G(z), that is, Gf0= Γ(f)∈X, and thus D(Γ) ⊂ {f∈X:Gf0∈X}. Taking closures and bearing in mind the fact from the general theory of operator semigroups that D(Γ) is dense in [ϕt, X](see for example [10, Lemma 8, p. 620]) we get the desired inclusion. For the converse inclusion let f∈ D(ϕt, X)and write m(z) = G(z)f0(z)∈X. The argument in the proof of the converse of [4, Theorem 2.2] can be repeated here to show (f◦ϕt)0(z)−f0(z) = ˆt 0 (m◦ϕt)0(z)ds, t ≥0, z ∈D, from which we obtain (f◦ϕt)(z)−f(z) = f(ϕt(0)) −f(0) + ˆz 0ˆt 0 (m◦ϕs)0(ζ)ds dζ =f(ϕt(0)) −f(0) + ˆt 0ˆz 0 (m◦ϕs)0(ζ)dζ ds =f(ϕt(0)) −f(0) + ˆt 0 [(m◦ϕs)(z)−m(ϕs(0))] ds. It follows that for t < 1we have kf◦ϕt−fkX≤C|f(ϕt(0)) −f(0)|+ˆt 0km◦ϕs−m(ϕs(0))kXds ≤C|f(ϕt(0)) −f(0)|+ˆt 0km◦ϕskXds +Cˆt 0|m(ϕs(0))|ds ≤C|f(ϕt(0)) −f(0)|+ÃMkmkX+Csup |z|≤ρ|m(z)|!t, where C=k1kXand ρ= sups∈[0,1] |ϕs(0)|<1. Taking t→0we obtain kf◦ϕt− fkX→0, therefore D(ϕt, X)⊂[ϕt, X]. Taking closures we get the desired inclusion and this finishes the proof. ¤ Theorem 2. Let (ϕt)be a semigroup with generator Gand Xa Banach space of analytic functions such that (Ct)is strongly continuous on X. Then the infinitesimal generator Γof (Ct)is given by Γ(f)(z) = G(z)f0(z)with domain D(Γ) = D(ϕt, X). Proof. An argument similar to the one in the first part of the proof of Theorem 1 shows that if f∈D(Γ) then Γ(f)(z) = G(z)f0(z)∈Xso that D(Γ) ⊂ D(ϕt, X). On the other hand for λin the resolvent set ρ(Γ) of Γwe have D(ϕt, X) = {f∈X:Gf0∈X}={f∈X:Gf0−λf ∈X} ={f∈X:there is m∈Xsuch that m=Gf0−λf} ={f∈X:there is m∈Xsuch that f=R(λ, Γ)(m)} =R(λ, Γ)(X). Semigroups of composition operators and integral operators in spaces of analytic functions 73 Since R(λ, Γ)(X)⊆D(Γ) this gives the conclusion. ¤ 4. The maximal subspace for BMOA and B For any given semigroup (ϕt), the induced operator semigroup (Ct)is known to be strongly continuous on the little Bloch space B0and on V MOA. A proof for both spaces is contained in [25, Theorem 4.1] and an alternative proof for V MOA can be found in [4, Theorem 2.4]. For the sake of completeness we recount a short proof of the strong continuity on these two spaces. Since the polynomials are dense in B0and in V MOA and since sup0≤t≤1kCtkX→X<∞for X=B0and X=V MOA, by a use of the triangle inequality, the strong continuity requirement limt→0kf◦ϕt−fkX= 0 for f∈Xreduces to the same requirement for fa polynomial. Now if fis a polynomial then f◦ϕt−fis a function in the Dirichlet space Dand we have from (7) and [17, p. 592], kf◦ϕt−fkB≤C1kf◦ϕt−fkBMOA ≤C2kf◦ϕt−fkD, with C1and C2absolute constants. But composition semigroups are strongly continuous on D, [22, Theorem 1], so limt→0kf◦ϕt−fkX= 0 for fpolynomial and the argument is complete. Thus we have [ϕt,B0] = B0and [ϕt, V MOA] = V MOA so that for every semigroup (ϕt), B0⊆[ϕt,B]⊆ B, and V MOA ⊆[ϕt, BMOA]⊆BMOA. The question arises whether there are cases of semigroups for which equality holds at one or the other end of these inclusions. In the case of BMOA it was proved by Sarason [19] that for each of the semigroups ϕt(z) = eitzor ϕt(z) = e−tzwe have V MOA = [ϕt, BMOA]. In [4] a whole class of semigroups was identified for which this equality holds. It is easy to see that for the above semigroups of Sarason we also have B0= [ϕt,B]. For the right hand side equalities it is unknown if there are semigroups such that BMOA = [ϕt, BMOA]. For the Bloch space however we show below that there are no non-trivial semigroups such that [ϕt,B] = B, answering a relevant question from [23, page 237]. We state the result for the more general class of Bloch spaces Bα,α > 0, defined by Bα=½f∈ H(D): sup z∈D¡1−|z|2¢α|f0(z)|<∞¾, endowed with the norm kfkBα=|f(0)|+ supz∈D¡1−|z|2¢α|f0(z)|. The basic properties of these spaces can be found in [29]. Theorem 3. Suppose α > 0and (ϕt)is a non-trivial semigroup of analytic functions. Then [ϕt,Bα] Bα. Proof. The result will be proved in two steps. First we show that any strongly continuous operator semigroup in Bαis uniformly continuous, therefore its infinitesimal generator is a bounded operator. In the second step we show that for composition semigroups (Ct), this implies that the infinitesimal generator is the null function hence the semigroup must be the trivial one. 74 O. Blasco, M. D. Contreras, S. Díaz-Madrigal, J. Martínez, M. Papadimitrakis and A. G. Siskakis Step 1. Each strongly continuous operator semigroup in Bαis uniformly continuous. We are going to use a theorem of Lotz [14, Theorem 3] which says that if Xis a Grothendieck space with the Dunford-Pettis property, then each strongly continuous semigroup of operators on Xis in fact uniformly continuous (for the terminology see the above cited work). On the other hand it is well-known that Bαis isomorphic to the space Hα=½f∈ H(D): sup z∈D¡1−|z|2¢α|f(z)|<∞¾, and this last space is isomorphic to the space l∞of all bounded sequences of complex numbers (see [15]). Therefore Bαis isomorphic to l∞for all positive α, and it is well known that l∞is a Grothendieck space with the Dunford–Pettis property (see [9, Chapter VII, Exercises 1 and 12]). It follows that Bαis a Grothendieck space with the Dunford–Pettis property and the theorem of Lotz applies. Step 2. If (ϕt)is a semigroup with generator Gand the induced semigroup of composition operators (Ct)is strongly continuous on Bα, then G≡0. To prove this let Γ: Bα→ Bαdenote the infinitesimal generator of (Ct). From Step 1, Γ is a bounded operator. Now for each f∈ Bαwe have Γ(f) = limt→0f◦ϕt−f t, the convergence being in the norm of Bα. But it is easy to see that convergence in Bαimplies uniform convergence on compact subsets of the disc and in particular it implies pointwise convergence for each z∈D. Thus for z∈D, Γ(f)(z) = lim t→0 f(ϕt(z)) −f(z) t=∂f(ϕt(z)) ∂t |t=0 =G(z)f0(z), therefore Γ(f) = Gf0for each f∈ Bα. Suppose that G6= 0 and recall that G(z) = (bz −1)(z−b)P(z)where P∈ H(D) with Re P(z)≥0. Since P(z)has boundary values almost everywhere on the unit circle (this for example follows from the fact that such a function Pbelongs to the Hardy spaces Hpfor all p < 1) we can find a point ξ∈∂Dsuch that G∗(ξ) = limr→1G(rξ)exists and is finite and different from zero. Now we distinguish three cases depending on α. If 0< α < 1,take f(z) = (1 −ξz)1−α, a function in Bα. Thus Γ(f)(z) = G(z)f0(z) = G(z)(α−1)ξ(1 −ξz)−α is a function in Bαand therefore bounded on Dbecause for these values of α,Bαis contained in the disc algebra [29, Proposition 9]. On the other hand taking z=rξ, 0< r < 1, we have lim r→1G(rξ)f0(rξ) = (α−1)ξlim r→1G(rξ)(1 −r)−α=∞, a contradiction. If α= 1 then take f(z) = log( 1 1−ξz ), a function in the Bloch space B=B1. Thus the function Γ(f)(z) = G(z)ξ 1−ξz belongs to B, and so it must satisfy the growth estimate [28, p. 82] for Bloch functions, that is for some constant C, |G(z)|1 |1−ξz|≤ |G(0)|+Clog 1 1−|z|, z ∈D. Semigroups of composition operators and integral operators in spaces of analytic functions 75 In particular for z=rξ we obtain |G(rξ)| ≤ (1 −r)|G(0)|+C(1 −r) log 1 1−|r| and taking r→1this implies G∗(ξ) = 0, a contradiction. Finally if α > 1let f(z) = ¡1−ξz¢1−α, a function in Bα. Thus the function Γ(f)(z) = G(z)(α−1)ξ(1 −ξz)−α belongs to Bαand hence it satisfies the estimate |Γ(f)(z)| ≤ |Γ(f)(0)|+C(1 −|z|)1−α, z ∈D, for some constant C, see [29, Theorem 18 and p. 1162]. We then obtain for z=rξ (α−1)|G(rξ)| ≤ (α−1)|G(0)|(1 −r)α+C(1 −r), and letting r→1this implies G∗(ξ) = limr→1G(rξ)=0, a contradiction. This completes the proof. ¤ Suppose now that Xis either V MOA or the little Bloch space B0so that the second dual X∗∗ is BMOA or Brespectively. Let (ϕt)be a semigroup on Dand let (Ct)be the induced semigroup of composition operators on X∗∗. Since each ϕtis univalent each Ctis maps Xinto itself, and the restriction St=Ct|X, is a strongly continuous semigroup on X. Lemma 1. Using the preceding notation for the semigroups (St)and (Ct)on X and X∗∗ respectively we have S∗∗ t=Ct for each t≥0. Proof. By the definition of the adjoint operator, S∗∗ t|X=St=Ct|X. But Xis weak∗dense in X∗∗ and the conclusion follows. ¤ Theorem 4. Let (ϕt)be a semigroup and Xbe one of the spaces V MOA or B0. Denote by Γthe generator of the induced composition semigroup (St)on Xand let λ∈ρ(Γ). Then the following are equivalent. (1) [ϕt, X∗∗] = X; (2) R(λ, Γ) is weakly compact on X; (3) R(λ, Γ)∗∗(X∗∗)⊂X. Proof. Conditions (2) and (3) are well known to be equivalent for every bounded operator on every Banach space X, [10, Theorem VI.4.2]. We proceed to show that (1) and (3) are equivalent. (1) ⇒(3). Take λ∈ρ(Γ) a big real number and f∈X. Writing the resolvent as a Laplace transform [10, Theorem 11, p. 622], R(λ, Γ)(f) = ˆ∞ 0 e−λuSu(f)du, we have St◦R(λ, Γ)(f) = ˆ∞ 0 e−λuSt+u(f)du =eλt ˆ∞ t e−λuSu(f)du. 82 O. Blasco, M. D. Contreras, S. Díaz-Madrigal, J. Martínez, M. Papadimitrakis and A. G. Siskakis Then [ϕt, X] = X∩(Tγ(X)⊕C). Proof. Let Gbe the generator of (ϕt). First observe that {f∈X:Gf0∈X}={f∈X:f0 γ0∈X}. This is clear in the case b∈∂Dbecause then Gγ0=Gh0= 1. In the case b∈Dwe have G(b) = 0 and the assumption (ii) on Xgives G(z)f0(z)∈X⇐⇒ G(z) z−bf0(z) = f0(z) γ0(z)∈X. Now write m(z) = f0(z) γ0(z)∈X, then f(z) = ˆz 0 m(ξ)γ0(ξ)dξ +c=Tγ(m)(z) + c, where c∈C. Thus D(ϕt, X) = {f∈X:Gf0∈X}={f∈X:f0 γ0=mfor some m∈X} ={f∈X:f0=mγ0for some m∈X} ={f∈X:f(z) = ˆz 0 m(ξ)γ0(ξ)dξ +cfor some m∈Xand c∈C} =X∩(Tγ(X)⊕C), and the conclusion follows by taking closures and using Theorem 1. ¤ In view of the Proposition 2 it would be desirable to have the operator Tγbounded on X. The following proposition says that this is not always the case. Proposition 3. Let X=BMOA or X=B, and (ϕt)be a semigroup with Denjoy–Wolff point on ∂Dand associated g-symbol γ. Then Tγis not bounded on X. Proof. Denote by hthe Koenigs function of (ϕt). Then his a univalent function on Dwith h(0) = 0 and such that the range of hhas the following geometrical property: w∈h(D) =⇒w+t∈h(D),for all t≥0. Recall that a univalent function belongs to B0if and only if it belongs to V MOA and that such a function belongs B0if and only if whenever D(w, r)are discs contained in its range with the centers wmoving to the boundary of the range, then the radii tend to zero. It follows from above geometric property of the range of hthat his not in B0and therefore also it is not in V MOA. On the other hand the associated g-symbol of the semigroup is γ(z) = h(z). Assume Tγis bounded on BMOA, then γ=h∈BMOAlog ⊂V MOA and this is a contradiction. Similarly if we assume Tγis bounded on Bthen γ=h∈ Blog ⊂ B0 and we obtain again a contradiction. ¤ Remark 1. If the Denjoy–Wolff point of (ϕt)is inside Dand there is a regular boundary fixed point for some (and then for all) ϕt, then we can show a similar result on B. Recall that a regular boundary fixed point is a point a∈∂Dsuch that Semigroups of composition operators and integral operators in spaces of analytic functions 83 ϕs(a) = aand ϕ0 s(a)6=∞for some s > 0in the sense of angular limits and angular derivative. It then follows that there is a β∈(0,∞)such that ϕt(a) = aand ϕ0 t(a) = eβt for all t, and that for the generator Gof (ϕt)we have lim z→aG(z) = 0 and lim z→a G(z) z−a=β, where the limits are taken non-tangentially, see [8]. In particular taking r < 1real we obtain limr→1G(ra) (r−1)a=β. Let γbe the associated g-symbol for (ϕt), then γ0(z) = z−b G(z)where b∈Dis the Denjoy–Wolff point. We have, sup z∈D (1 −|z|2) log( 1 1−|z|2)|γ0(z)| ≥ sup r∈(0,1) (1 −|r|2) log( 1 1−|r|2)¯¯¯¯ ra −b G(ra)¯¯¯¯ = sup r∈(0,1) |ra −b|(1 + r) log( 1 1−|r|2)¯¯¯¯ 1−r G(ra)¯¯¯¯ =∞, therefore Tγis not bounded on B. Undoubtedly, the above proposition and remark are a handicap for our applications because the boundedness of Tγis desired. However, in many families of examples our techniques work to give a description of the maximal space of strong continuity. Corollary 2. Let X=V MOA, or respectively X=B0. Suppose (ϕt)is a semigroup with associated g-symbol γ(z)and suppose γ∈BMOAlog, respectively γ∈ Blog. Then [ϕt, X∗∗] = Xif and only if γ∈V MOAlog, respectively γ∈ Blog,0. Proof. If γ∈V MOAlog, respectively γ∈ Blog,0, then by Theorem 6 we have Tγ(X∗∗)⊂X. It then follows from Proposition 2 that [ϕt, X∗∗] = X∗∗ ∩(Tγ(X∗∗)⊕C)⊂X∗∗ ∩X=X and we have equality because X⊂[ϕt, X∗∗]. Conversely suppose [ϕt, X∗∗] = X. Then from Proposition 2 we must have X∗∗ ∩(Tγ(X∗∗)⊕C) = Xand in particular (9) X∗∗ ∩(Tγ(X∗∗)⊕C)⊂X. The hypothesis on γimplies that Tγis bounded on X∗∗ and X∗∗ contains the constants therefore Tγ(X∗∗)⊕C⊂X∗∗. This in view of (9) implies Tγ(X∗∗)⊕C⊂X and then Tγ(X∗∗)⊂Xbecause Xcontains the constants. Using Theorem 6 again we conclude γ∈V MOAlog, respectively γ∈ Blog,0.¤ Corollary 3. (see [4, Theorem 3.1]) Let (ϕt)be a semigroup with generator G(z). Assume that for some 0< α < 1, (1 −|z|)α G(z)=O(1),|z| → 1. Then [ϕt, BMOA] = V MOA and [ϕt,B] = B0. 84 O. Blasco, M. D. Contreras, S. Díaz-Madrigal, J. Martínez, M. Papadimitrakis and A. G. Siskakis Proof. Let γbe the associated g-symbol for (ϕt). The assumption implies that there is a constant Csuch that (1 − |z|)α|γ0(z)| ≤ Cfor z∈D. Therefore γ has boundary values (1 −α)-Hölder continuous, in particular γ∈V MOAlog and γ∈ Blog,0. The conclusion follows by applying Corollary 2. ¤ 7. Final remarks and open questions In this section we obtain some additional information, mostly on the Koenigs function and the maximal space of strong continuity for semigroups acting on BMOA or on B. We also present some questions which we could not answer in this article. Recall that for any semigroup (ϕt)with generator G(z) = (bz −1)(z−b)P(z), the associated g-symbol γis a univalent function when b∈∂Dbecause it coincides with the Koenigs map h. We observe that γis also univalent when b∈D. The easiest way to see this is to show that γis in this case a close-to-convex function (see [18, p. 68] for the definition). Indeed the function g(z) = (1/¯ b) log(1−¯ bz)is a convex univalent function for b∈Dand it can be checked easily that Re(γ0 g0) = Re(1 P)≥0. We are going to consider also the function (10) ψ(z) = ˆz 0 1 P(ξ)dξ, z ∈D. Since Re(1/P)≥0this is a univalent function [11, Theorem 2.16]. The growth estimate |1/P(z)| ≤ C1+|z| 1−|z|for z∈Dfor the function 1/P of non-negative real part says that (1 − |z|)|ψ0(z)| ≤ 2Cso ψ∈ B and since ψis univalent it follows that ψ∈BMOA. Proposition 4. Let X=BMOA or X=Band let (ϕt)be a semigroup with generator G(z) = (bz −1)(z−b)P(z), Koenigs function hand associated g-symbol γ and let ψ(z)be given by (10). Then the following hold: (1) ψ(z), γ(z)∈ D(ϕt, X), (2) If h∈X, then h∈ D(ϕt, X). (3) (i) If b∈D, then (z−b) log h(z) z−b∈[ϕt, X]. Moreover, log h(z) z−b∈ D(ϕt, X)if and only if G(z)∈X, (ii) If b∈∂D, then zlog h(z) z∈[ϕt, X]. Moreover, log h(z) z∈ D(ϕt, X)if and only if G(z)∈X, (4) If C\h(D)has nonempty interior, then log(h(z)−c)belongs to D(ϕt, X)for each cin the interior of C\h(D). Proof. (1) We have G(z)ψ0(z) = (bz −1)(z−b)∈X, therefore ψ∈ D(ϕt, X). For γ(z), since G(z)γ0(z)=1if b∈∂Dwhile G(z)γ0(z) = z−bif b∈D, the conclusion follows. (2) Since G(z)h0(z) = 1 if b∈∂Dand G(z)h0(z) = G0(b)h(z)if b∈D, the assertion is clear. (3)(i) We observe that G0(b)γ(z) = ˆz b G0(b)(ξ−b) G(ξ)dξ =ˆz b (ξ−b)h0(ξ) h(ξ)dξ Semigroups of composition operators and integral operators in spaces of analytic functions 85 =ˆz b·1+(ξ−b)µlog h(ξ) ξ−b¶0¸dξ =z−b+ (z−b) log h(z) z−b−ˆz b log h(ξ) ξ−bdξ. Now log h(z) z−bbelongs to BMOA (and to B) so the function ´z blog h(ξ) ξ−bdξ is continuous on D, hence it belongs to V MOA (and to B0). Since (z−b) log h(z) z−b=G0(b)γ(z)−(z−b) + ˆz b log h(ξ) ξ−bdξ, with γ∈ D(ϕt, X)⊂[ϕt, X]we conclude that (z−b) log h(z) z−b∈[ϕt, X]. Finally, let q(z) = log h(z) z−band observe that G(z)q0(z) = G(z)µh0(z) h(z)−1 z−b¶=G0(b)−G(z) z−b=G0(b)−(¯ bz −1)P(z). This says that q(z)∈ D(ϕt, X)if and only if (¯ bz −1)P(z)∈Xand this is equivalent to G(z) = (z−b)(¯ bz −1)P(z)∈X. (3)(ii) Let m(z) = z h(z). This is a bounded function on Dhence it belongs to BMOA and to B. Observe that Tγ(m)(z) = ˆz 0 ξh0(ξ) h(ξ)dξ. A calculation similar to the one in (3)(i) gives Tγ(m)(z) = zlog h(z) z+z−ˆz 0 log h(ξ) ξdξ, and the three terms in the right hand side are elements of X. Thus Tγ(m)(z)∈X with m(z)∈X, and Proposition 2 implies that Tγ(m)(z)∈[ϕt, X]. By the argument of (3)(i), z−´z 0log h(ξ) ξdξ is in V MOA (and in B0) and we conclude zlog h(z) z∈[ϕt, X]. To show the last assertion let q(z) = log h(z) z. Then zG(z)q0(z) = zG(z)µh0(z) h(z)−1 z¶=zG(z)h0(z) h(z)−G(z) = z h(z)−G(z). Now G(z)q0(z)∈Xis equivalent to zG(z)q0(z)∈X. Thus q∈ D(ϕt, X)if and only if z h(z)−G(z)∈Xand since z h(z)is a function in Xwe obtain the desired conclusion. (4) Observe that q(z) = log(h(z)−c)belongs to BMOA (and to B). By the hypothesis there is a δ > 0such that |h(z)−c|> δ for z∈D, so for the case b∈∂D, G(z)q0(z) = G(z)h0(z) h(z)−c=1 h(z)−c, is a bounded function so it is in BMOA and the conclusion follows in this case. Finally if b∈Dthen G(z)q0(z) = G(z)h0(z) h(z)−c=G0(b)h(z) h(z)−c, which is also bounded on Dso the result follows. ¤ 86 O. Blasco, M. D. Contreras, S. Díaz-Madrigal, J. Martínez, M. Papadimitrakis and A. G. Siskakis Corollary 4. (see [4, Theorem 3.3]) If (ϕt)is a semigroup with generator G(z) and Denjoy–Wolff b∈Dand either [ϕt, BMOA] = V MOA or [ϕt,B] = B0, then (11) lim |z|→1 1−|z| G(z)= 0. Proof. By Proposition 4 and our hypothesis, we have (z−b) log h(z) z−b∈V MOA or respectively (z−b) log h(z) z−b∈ B0. This implies log h(z) z−b∈V MOA ⊂ B0, therefore 0 = lim |z|→1(1 −|z|)µlog h(z) z−b¶0= lim |z|→1(1 −|z|)µh0(z) h(z)−1 z¶ = lim |z|→1(1 −|z|)h0(z) h(z)= lim |z|→1 G0(b)(1 −|z|) G(z), and since G0(b)6= 0 the conclusion follows. ¤ If F(z)is analytic with ReF(z)≥0on D, the Herglotz theorem says that there is a nonnegative Borel measure µon ∂Dsuch that F(z) = ˆ2π 0 eiθ +z eiθ −zdµ(θ) + iImF(0). The following proposition says that the presence of point masses for the measure µ associated with 1/P implies automatically that [ϕt, BMOA]is strictly larger than V MOA. Corollary 5. Let X=BMOA or X=Band let (ϕt)be a semigroup with generator G(z) = (z−b)(¯ bz−1)P(z)such that [ϕt, BMOA] = V MOA or [ϕt,B] = B0. Then the Herglotz measure for 1/P(z)has no point masses on ∂D. Proof. From Proposition 4 and the hypothesis we have that the function ψ(z) = ´z 0 1 P(ξ)dξ belongs to V MOA in the first case or to B0in the second case. Since this function is univalent it belongs to V MOA in both cases. But then the operator Tψ(f)(z) = ˆz 0 f(ξ)ψ0(ξ)dξ =ˆz 0 f(ξ)1 P(ξ)dξ is a compact operator on the Hardy space H2, see [1]. It follows then by [21, Theorem 4] that µhas no point masses on ∂D.¤ Corollary 6. Let (ϕt)be a semigroup with Denjoy-Wolff point b∈∂Dand Koenigs function h. If either [ϕt, BMOA] = V MOA or [ϕt,B] = B0then h∈ (∩p<∞Hp)\BMOA. Proof. First observe that the Koenigs function cannot belong to BMOA. Otherwise from Proposition 4(2), we would have h∈ D(ϕt, BMOA)⊂[ϕt, BMOA] so h∈V MOA by our assumption, which is impossible in view of the geometric property of the range of h. On the other hand again from Proposition 4(3)(ii) and our assumption we have zlog h(z) z∈V MOA so in particular log h(z) z∈V MOA. This implies elog h(z) z∈Hpfor all p < ∞, [17, p. 596], so h∈Hpfor all p < ∞. The argument for the Bloch case is similar and is omitted. ¤ Semigroups of composition operators and integral operators in spaces of analytic functions 87 Remark 2. One could expect, for the case of Denjoy–Wolff point on the boundary, to obtain a result similar to (11) for the generator G(z), under the assumption [ϕt, BMOA] = V MOA or [ϕt,B] = B0. In this case, using Proposition 4 one obtains log h(z) z∈ B0which means lim |z|→1(1 −|z|)h0(z) h(z)= 0, and in terms of G(z), lim |z|→1 1−|z| G(z) 1 ´z 0 1 G(ξ)dξ = 0. Note that ´z 0 1 G(ξ)dξ =h(z)is never bounded, and finer analysis is required. We can however obtain a substitute of (11) if we consider integral averages of 1/G(z). Corollary 7. Let (ϕt)be a semigroup with generator G(z)and Denjoy–Wolff point b∈∂D. If either [ϕt, BMOA] = V MOA or [ϕt,B] = B0then for every 0< p < ∞, lim r→1(1 −r)µˆ2π 0 1 |G(reit)|pdt¶1/p = 0. Proof. As observed in the previous Remark, we have lim|z|→1(1 −|z|)h0(z) h(z)= 0. Since from Corollary 6 the function hbelongs to all Hardy spaces we have ˆ2π 0 (1 −r)p |G(reit)|pdt =ˆ2π 0 (1 −r)p|h0(reit)|p |h(reit)|p|h(reit)|pdt ≤sup |z|=rµ(1 −|z|)p|h0(z)|p |h(z)|p¶khkp, and the right hand side goes to 0as r→1.¤ We now present some open questions. On the basis of Theorem 3 which says that no non-trivial semigroup (ϕt)induces a strongly continuous composition operator semigroup (Ct)on the Bloch space, it is natural to expect that the same is true for BMOA. The space BMOA however does not posses the Dunford–Pettis property (see [5]) and the method of proof for the Bloch space does not work. Question 1. Is it true that for every non-trivial semigroup (ϕt)the space [ϕt, BMOA]is strictly smaller than BMOA? Suppose now (ϕt)has its Denjoy–Wolff point on the boundary so its generator can be written G(z) = ¯ b(b−z)2P(z)with Re(P(z)) ≥0. This generator cannot satisfy the condition (1 −|z|)α G(z)=O(1),as |z| → 1, for 0< α < 1, of Corollary 3 which implies [ϕt, BMOA] = V MOA. Indeed for z=rb we have (1 −|z|)α G(z)=(1 −|r|)α b(1 −r)2P(rb)=¯ b (1 −r)2−αP(rb), and the growth estimate |P(z)| ≤ C1+|z| 1−|z|for the function Pof non-negative real part implies (1 −r)2−αP(rb)→0as r→1for 0< α < 1. 88 O. Blasco, M. D. Contreras, S. Díaz-Madrigal, J. Martínez, M. Papadimitrakis and A. G. Siskakis In addition in all examples we can work it turns out that for this case the space [ϕt, BMOA]is strictly larger than V MOA. This leads to Question 2. Suppose (ϕt)is a semigroup with Denjoy–Wolff point on the boundary. Is it true in this case that [ϕt, BMOA]is strictly larger than V MOA and [ϕt,B] is strictly larger than B0? Acknowledgement. We would like to thank the referee for suggesting improvements in the presentation of the article. References [1] Aleman, A., and A. G. Siskakis: Integration operators on Hp. - Complex Variables Theory Appl. 28, 1995, 149–158. [2] Aleman, A., and A. G. Siskakis: Integration operators on Bergman spaces. - Indiana Univ. Math. J. 46, 1997, 337–356. [3] Berkson, E., and H. Porta: Semigroups of analytic functions and composition operators. - Michigan Math. J. 25, 1978, 101–115. [4] Blasco, O.,M. D. Contreras,S. Díaz-Madrigal,J. Martinez, and A. G. Siskakis: Semigroups of composition operators in BMOA and the extension of a theorem of Sarason. - Integral Equations Operator Theory 61, 2008, 45–62. [5] Castillo, J. M. F., and M. Gonzalez: New results on the Dunford–Pettis property. - Bull. London Math. Soc. 27 (1995), 599–605. [6] Contreras, M. D., and S. Díaz-Madrigal: Fractional iteration in the disk algebra: prime ends and composition operators. - Rev. Mat. Iberoam. 21, 2005, 911–928. [7] Contreras, M. D., and S. Díaz-Madrigal: Analytic flows in the unit disk: angular derivatives and boundary fixed points. - Pacific J. Math. 222, 2005, 253–286. [8] Contreras, M. D.,S. Díaz-Madrigal, and Ch. Pommerenke: On boundary critical points for semigroups of analytic functions. - Math. Scand. 98, 2006, 125–142. [9] Diestel, J.: Sequences and series in Banach spaces. - Springer-Verlag, New York, 1984. [10] Dunford, N., and J. T. Schwartz: Linear operators. Part I. General theory. - Wiley Classics Library, John Wiley & Sons Inc., New York, 1988. [11] Duren, P. L.: Univalent functions. - Springer-Verlag, New York, 1983. [12] Garnett, J. B.: Bounded analytic functions. - Pure and Applied Mathematics 96, Academic Press, New York, 1981. [13] Laitila, J.,S. Miihkinen, and P. Nieminen: Essential norms and weak compactness of integration operators. - Arch. Math. (Basel) 97, 2011, 39–48. [14] Lotz, H. P.: Uniform convergence of operators on L∞and similar spaces. - Math. Z. 190, 1985, 207–220. [15] Lusky, W.: On the isomorphic classification of weighted spaces of holomorphic functions. - Acta Univ. Carolin. Math. Phys. 41, 2000, 51–60. [16] Papadimitrakis, M., and J. A. Virtanen: Hankel and Toeplitz transforms on H1: Continuity, compactness and Fredholm propoerties. - Integral Equations Operator Theory 61, 2008, 573–591. [17] Pommerenke, Ch.: Schlichte Funktionen und analytische Funktionen von beschrÄankter mittlerer Oszillation. - Comment. Math. Helv. 52, 1977, 591–602. [18] Pommerenke, Ch.: Boundary behaviour of conformal maps. - Springer-Verlag, Berlin, 1992. [19] Sarason, D.: Function of vanishing mean oscillation. - Trans. Amer. Math. Soc. 207, 1975, 391–405. Semigroups of composition operators and integral operators in spaces of analytic functions 89 [20] Shoikhet, D.: Semigroups in geometrical function theory. - Kluwer Academic Publishers, Dordrecht, 2001. [21] Siskakis, A. G.: The Koebe semigroup and a class of averaging operators on Hp(D). - Trans. Amer. Math. Soc. 339, 1993, 337–350. [22] Siskakis, A. G.: Semigroups of composition operators on the Dirichlet space. - Results Math. 30, 1996, 165–173. [23] Siskakis, A. G.: Semigroups of composition operators on spaces of analytic functions, a review. - Contemp. Math. 213, 1998, 229–252. [24] Siskakis, A. G., and R. Zhao: A Volterra type operator on spaces of analyitc functions. - Contemp. Math. 232, 1990, 299–311. [25] Wirths, K. J., and J. Xiao: Recognizing Qp,0functions as per Dirichlet space structure. - Bull. Belg. Math. Soc. Simon Stevin 8, 2001, 47–59. [26] Xiao, J.: Holomorphic Qclasses. - Lecture Notes in Math. 1767, Springer-Verlag, 2001. [27] Yoneda, R.: Integration operators on weighted Bloch spaces. - Nihonkai Math. J. 12, 2001, 123–133. [28] Zhu, K.: Operator theory in function spaces. - Marcel Dekker, Inc., New York and Basel, 1990. [29] Zhu, K.: Bloch type spaces of analytic functions. - Rocky Mountain J. Math. 23, 1993, 1143– 1177. Received 28 October 2011 •Accepted 27 August 2012