SEMIGROUPS OF COMPOSITION OPERATORS AND INTEGRAL OPERATORS IN SPACES OF ANALYTIC FUNCTIONS

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1 SEMIGROUPS OF COMPOSITION OPERATORS AND INTEGRAL OPERATORS IN SPACES OF ANALYTIC FUNCTIONS O. BLASCO, M.D. CONTRERAS, S. DÍAZ-MADRIGAL, J. MARTÍNEZ, M. PAPADIMITRAKIS, AND A.G. SISKAKIS ABSTRACT. We study the maximal spaces of strong continuity on BMOA and the Bloch space B for semigroups of composition operators. Characterizations are given for the cases when these maximal spaces are V MOA or the little Bloch B. These characterizations are in terms of the weak compactness of the resolvent function or in terms of a specially chosen symbol g of an integral operator T g. For the second characterization we prove and use an independent result, namely that the operators T g are weakly compact on the above mentioned spaces if and only if they are compact.. 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 X is 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 : t Φ(t) = ϕ t from the additive semigroup of nonnegative real numbers into the composition semigroup of all analytic functions which map D into D. In other words, Φ = (ϕ t ) consists of analytic functions on D with ϕ t (D) D and for which the following three conditions hold: () ϕ is the identity in D, (2) ϕ t+s = ϕ t ϕ s, for all t, s, (3) ϕ t ϕ, as t, uniformly on compact subsets of D. Date: December 22, 2. 2 Mathematics Subject Classification. Primary 3H5, 32A37, 47B33, 47D6; Secondary 46E5. Key words and phrases. Composition semigroups, BMOA, Bloch spaces, Integration operators, weak compactness. The first and fourth authors were partially supported by the Ministerio de Ciencia e Innovación MTM /MTM. The second and third author were partially supported by the Ministerio de Ciencia e Innovación and the European Union (FEDER), project MTM C2-2, 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-33). The last author has received partial technical support by a grant from Empirikion foundation.

2 2 BLASCO, CONTRERAS, DÍAZ-MADRIGAL, MARTÍNEZ, PAPADIMITRAKIS, AND SISKAKIS It is well known that condition (3) above can be replaced by (3 ) For each z D, ϕ t (z) z, as t. Each such semigroup gives rise to a semigroup (C t ) consisting of composition operators on H(D), C t (f) := f ϕ t, f H(D). We are going to be interested in the restriction of (C t ) to certain linear subspaces H(D). Given a Banach space of analytic functions and a semigroup (ϕ t ), we say that (ϕ t ) generates a semigroup of operators on X if (C t ) is a well-defined strongly continuous semigroup of bounded operators in X. This exactly means that for every f X, we have C t (f) X for all t and lim t + C t(f) f X =. Thus the crucial step to show that (ϕ t ) generates a semigroup of operators in X is to pass from the pointwise convergence lim t + f ϕ t (z) = f(z) on D to 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 (C t ) in terms of the theory of functions. The paper [3] can be considered as the starting point in this direction. Classical choices of X treated in the literature are the Hardy spaces H p, the disk algebra A(D), the Bergman spaces A p, the Dirichlet space D and the chain of spaces Q p and Q p, which have been introduced recently and which include the spaces BMOA, Bloch as well as their little oh analogues. See [24] and [26] for definitions and basic facts of the spaces and [2], [2] and [23] 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 H p ( p < ), the Bergman spaces A p ( 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 BM OA 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 [ϕ t, X] for the composition semigroup induced by (ϕ t ) on an abstract Banach space X of analytic functions satisfying certain conditions. In section 4 we consider the maximal subspaces [ϕ t, BMOA] and [ϕ t, 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 B thus [ϕ t, B] is strictly contained in B. We also show

3 SEMIGROUPS AND INTEGRAL OPERATORS 3 that that the equality [ϕ t, BMOA] = V MOA or [ϕ t, B] = B is equivalent to the weak compactness of the resolvent operator R(λ, Γ) for the composition semigroup acting on V MOA (respectively on B ). In section 5 we study an integral operator T g on BMOA and B and we show in particular that its compactness and weak compactness are equivalent (Theorem 6). In section 6 we apply these results for T g for a special choice of the symbol g = γ (defined later in Definition 4) to obtain a characterization of the cases of equality [ϕ t, BMOA] = V MOA and [ϕ t, B] = B in 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 (ϕ t ) is a semigroup, then each map ϕ t is univalent. The infinitesimal generator of (ϕ t ) is the function ϕ t (z) z G(z) = lim, z D. t + t This convergence holds uniformly on compact subsets of D so G H(D). Moreover G satisfies () G(ϕ t (z)) = ϕ t(z) = G(z) ϕ t(z), z D, t. t z Further G has a representation (2) G(z) = (bz )(z b)p (z), z D where b D and P H(D) with Re P (z) for all z D. If G is not identically null, the couple (b, P ) is uniquely determined from (ϕ t ) and the point b is 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 [2], [7]). Recall also the notion of Koenigs function associated with a semigroup. For every non-trivial semigroup (ϕ t ) with generator G, there exists a unique univalent function h : D C, called the Koenigs function of (ϕ t ), such that. If the Denjoy-Wolff point b of (ϕ t ) is in D then h(b) =, h (b) = and (3) h(ϕ t (z)) = e G (b)t h(z) for all z D and t. Moreover, (4) h (z)g(z) = G (b)h(z), z D. 2. If the Denjoy-Wolff point b of (ϕ t ) is on D then h() = and (5) h(ϕ t (z)) = h(z) + t for all z D and t. Moreover, (6) h (z)g(z) =, z D.

4 4 BLASCO, CONTRERAS, DÍAZ-MADRIGAL, MARTÍNEZ, PAPADIMITRAKIS, AND 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 B. BMOA is the Banach space of all analytic functions in the Hardy space H 2 whose 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 [26, 2]). Namely, a function f H 2 belongs to BMOA if and only if there exists a constant C > such that f (z) 2 ( z 2 )da(z) C I, R(I) for any arc I D, where R(I) is the Carleson rectangle determined by I, that is, { R(I) := re iθ D : I } 2π < r < and eiθ I. As usual, I denotes the length of I and da(z) the normalized Lebesgue measure on D. The corresponding BMOA norm is ( /2 f BMOA := f() + sup f (z) 2 ( z )da(z)) 2. I D I 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 I R(I) R(I) f (z) 2 ( z 2 )da(z) =. 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 f L 2 (D, da) with norm f D = f() 2 + f 2 L 2 (D, da). In fact, for every f D, I R(I) f (z) 2 ( z 2 )da(z) 2 R(I) f (z) 2 da(z), as I, so D is contained in V MOA. This last inequality also implies that there is an absolute constant C such that (7) f BMOA C f D for each f D. A holomorphic function f H(D) is said to belong to the Bloch space B whenever sup z D ( z 2) f (z) <. It is well-known that B is a Banach space when it is endowed with the norm f B = f() + sup z D ( z 2) f (z).

5 SEMIGROUPS AND INTEGRAL OPERATORS 5 The closure of the polynomials in B is called the little Bloch space and it is denoted by B. It is also well-known that f B if and only if ( z 2) f (z) =. lim z For more information on these Banach spaces, we refer the reader to the excellent monographs [2] or [26]. Finally at this point we would like to mention some information about univalent functions that will be needed later. Let h : D C be univalent. Then h BMOA if and only if h B, and h V MOA if and only if h B. Further h B if and only if the discs that can be inscribed in the range of h have bounded radii and h B if and only if the radii of these discs tend to as their center moves to. In addition if h : D C is univalent and non-vanishing then log(h(z)) BMOA, while if h(b) = for some b D then log h(z) z b BMOA. Additional information can be found in [6]. 3. THE SPACE [ϕ t, X]. Given a semigroup of analytic functions (ϕ t ) and a Banach space X of analytic functions on the unit disk, we are interested on the maximal closed subspace of X, denoted by [ϕ t, X], on which (ϕ t ) generates a strongly continuous semigroup (C t ) 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 X of analytic functions is contained, for the special case X = BMOA, in [4]. The proof here follows the same lines as the one in [4]. We include it here for the sake of completeness. PROPOSITION. Let (ϕ t ) be a semigroup of analytic functions and X a Banach space of analytic functions such that C t : X X are bounded for all t and sup t [,] C t X = M <. Then there exists a closed subspace Y of X such that (ϕ t ) generates a semigroup of operators on Y and such that any other subspace of X with this property is contained in Y. Proof. Consider the linear subspace of X defined by { } Y := f X : lim f ϕ t f X =. t + The hypothesis that sup t [,] C t X < together with the triangle inequality for norms shows that Y is a closed subspace of X. Thus in order to prove that (ϕ t ) generates a semigroup of operators in Y, it remains to check that if f Y then C s (f) Y for all s. To see this let s, t, then C s (f) ϕ t C s (f) X = f ϕ s ϕ t C s (f) X = f ϕ t ϕ s C s (f) X = C s (f ϕ t ) C s (f) X C s X f ϕ t f X as t +.

6 6 BLASCO, CONTRERAS, DÍAZ-MADRIGAL, MARTÍNEZ, PAPADIMITRAKIS, AND SISKAKIS Finally, if W is a subspace of X such that (ϕ t ) generates a semigroup of operators on W, then for any f W we have in particular lim f ϕ t f X =, t + thus f Y and we conclude W Y. DEFINITION. We denote by [ϕ t, X] the maximal subspace consisting of functions f X such that lim t + f ϕ t f X =. It is easy to see that if Z is any closed subspace of [ϕ t, X] which is invariant under (C t ) (i.e. C t (Z) Z for every t ) then (ϕ t ) generates a semigroup of operators on Z. DEFINITION 2. Given a semigroup (ϕ t ) with generator G and a Banach space of analytic functions X we define D(ϕ t, X) := {f X : Gf X}. Clearly D(ϕ t, X) is a linear subspace of X. THEOREM. Let (ϕ t ) be a semigroup with generator G and X a Banach space of analytic functions which contains the constant functions and such that M = sup t [,] C t X <. Then, [ϕ t, X] = D(ϕ t, X). Proof. Let us show first [ϕ t, X] D(ϕ t, X). We may assume that (ϕ t ) is not trivial. Denote by Γ the infinitesimal generator of the operator semigroup (C t ) acting on the Banach space [ϕ t, X], and by D(Γ) its domain. We will show that if f D(Γ) then Gf X. Indeed if f D(Γ) then Γ(f) [ϕ t, X] X and lim t + t (C t(f) f) Γ(f) =. X Since convergence in the norm of X implies uniform convergence on compact subsets of D and therefore in particular pointwise convergence, for each z D we have f(ϕ t (z)) f(z) f(ϕ t (z)) f(ϕ (z)) Γ(f)(z) = lim = lim t + t t + t = f ϕ t(z) t = f (ϕ (z)) ϕ t(z) t= t = f (z)g(z), t= that is, Gf = Γ(f) X, and thus D(Γ) {f X : Gf X}. Taking closures and bearing in mind the fact from the general theory of operator semigroups that D(Γ) is dense in [ϕ t, X] we get the desired inclusion. For the converse inclusion let f D(ϕ t, X) and write m(z) = G(z)f (z) X. The argument in the proof of the converse of [4, Theorem 2.2] can be repeated here to show (f ϕ t ) (z) f (z) = t (m ϕ t ) (z) ds, t, z D,

7 from which we obtain SEMIGROUPS AND INTEGRAL OPERATORS 7 (f ϕ t )(z) f(z) = f(ϕ t ()) f() + = f(ϕ t ()) f() + = f(ϕ t ()) f() + It follows that for t < we have f ϕ t f X C f(ϕ t ()) f() + t t t t t (m ϕ s ) (ζ)dsdζ (m ϕ s ) (ζ)dζds [(m ϕ s )(z) m(ϕ s ())]ds. m ϕ s m(ϕ s ()) X ds C f(ϕ t ()) f() + m ϕ s X ds + C ( C f(ϕ t ()) f() + t M m X + C sup m(z) z ρ m(ϕ s ()) ds ) where C = X and ρ = sup s [,] ϕ s () <. Taking t we obtain f ϕ t f X, therefore D(ϕ t, X) [ϕ t, X]. Taking closures we get the desired inclusion and this finishes the proof. THEOREM 2. Let (ϕ t ) be a semigroup with generator G and X a Banach space of analytic functions such that (C t ) is strongly continuous on X. Then the infinitesimal generator Γ of (C t ) is given by Γ(f)(z) = G(z)f (z) with domain D(Γ) = D(ϕ t, X). Proof. An argument similar to the one in the first part of the proof of Theorem shows that if f D(Γ) then Γ(f)(z) = G(z)f (z) X so that D(Γ) D(ϕ t, X). On the other hand for λ in the resolvent set ρ(γ) of Γ we have D(ϕ t, X) = { f X : Gf X } = { f X : Gf λf X } = { f X : there is m X such that m = Gf λf } = {f X : there is m X such that f = R(λ, Γ)(m)} = R(λ, Γ)(X). t, Since R(λ, Γ)(X) D(Γ) this gives the conclusion. Remark. Let (ϕ t ) be a non-trivial semigroup with generator G and X a Banach space of analytic functions satisfying the hypothesis of Theorem. Of particular interest are the functions in the set E λ = {f X : Gf = λf}

8 8 BLASCO, CONTRERAS, DÍAZ-MADRIGAL, MARTÍNEZ, PAPADIMITRAKIS, AND SISKAKIS where λ is a complex number. It is clear that E λ D(ϕ t, X). λ C If λ = then this set contains only the constant functions. Assume λ. Let b be the Denjoy-Wolff point and h the Koenigs map of (ϕ t ). If b D then G(z)h (z) = G (b)h(z), so that the equation Gf = λf can be written in terms of h as (8) h(z)f (z) = λ G (b) h (z)f(z). Suppose f, not identically zero, is an analytic function which is a solution of this equation on D. Then f (z) f(z) = λ h (z) G (b) h(z). Choose r ( b, ) such that f has no zeros on { z = r} and integrate on this circle, f (z) 2πi z =r f(z) dz = λ h (z) G (b) 2πi z =r h(z) dz. λ By the argument principle the right hand side is equal to G (b) because h has a single zero inside z < r. The left hand side is the number of zeros of f inside { z < r} and is an integer k (it is clear from the equation (8) that such a solution f satisfies f(b) = ). Thus λ = kg (b), and we conclude that the equation (8) has nonzero analytic solutions only for those values of λ. The equation then becomes h(z)f (z) = kh (z)f(z), and it is easy to see that the only analytic solutions of this are f(z) = ch(z) k with c C. Thus E λ = {} if λ kg (b) while { } E λ = ch(z) k : c C X, if λ = kg (b). Suppose now the Denjoy-Wolff point b is on D. Then G(z)h (z) = and the equation Gf = λf can be written f = λfh. The analytic solutions of this equation are f(z) = c exp(λh(z)) and { } E λ = ce λh(z) : c C X in this case.

9 SEMIGROUPS AND INTEGRAL OPERATORS 9 4. THE MAXIMAL SUBSPACE FOR BMOA AND B For any given semigroup (ϕ t ), the induced operator semigroup (C t ) is known to be strongly continuous on the little Bloch space B and on V MOA. A proof for both spaces is contained in [23, Theorem 4.] 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 B and in V MOA and since sup t C t X X < for X = B and X = V MOA, by a use of the triangle inequality, the strong continuity requirement lim t f ϕ t f X = for f X reduces to the same requirement for f a polynomial. Now if f is a polynomial then f ϕ t f is a function in the Dirichlet space D and we have from (7) and [6, page 592], f ϕ t f B C f ϕ t f BMOA C 2 f ϕ t f D, with C and C 2 absolute constants. But composition semigroups are strongly continuous on D, [2, Theorem ], so lim t f ϕ t f X = for f polynomial and the argument is complete. Thus we have [ϕ t, B ] = B and [ϕ t, V MOA] = V MOA so that for every semigroup (ϕ t ), B [ϕ t, B] B, and V MOA [ϕ t, 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 D. Sarason [8] that for each of the semigroups ϕ t (z) = e it z or ϕ t (z) = e t z we have V MOA = [ϕ t, 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 B = [ϕ t, B]. For the right hand side equalities it is unknown if there are semigroups such that BMOA = [ϕ t, BMOA]. For the Bloch space however we show below that there are no non-trivial semigroups such that [ϕ t, B] = B, answering a relevant question from [2, p. 237]. We state the result for the more general class of Bloch spaces B α, α >, defined by { B α = f H(D) : sup z D ( z 2) α f (z) < }, endowed with the norm f Bα = f() + sup z D ( z 2) α f (z). The basic properties of these spaces can be found in [27]. THEOREM 3. Suppose α > and (ϕ t ) is a non-trivial semigroup of analytic functions. Then [ϕ t, B α ] B α. Proof. The result will be proved in two steps. First we show that any strongly continuous operator semigroup in B α is uniformly continuous, therefore its

10 BLASCO, CONTRERAS, DÍAZ-MADRIGAL, MARTÍNEZ, PAPADIMITRAKIS, AND SISKAKIS infinitesimal generator is a bounded operator. In the second step we show that for composition semigroups (C t ), this implies that the infinitesimal generator is the null function hence the semigroup must be the trivial one. Step. Each strongly continuous operator semigroup in B α is uniformly continuous. We are going to use a theorem of H. Lotz [3, Theorem 3] which says that if X is a Grothendieck space with the Dunford-Pettis property, then each strongly continuous semigroup of operators on X is in fact uniformly continuous (for the terminology see the above cited work). On the other hand it is well-known that B α is isomorphic to the space { H α = f H(D) : sup z D ( z 2) α f(z) < }, and this last space is isomorphic to the space l of all bounded sequences of complex numbers (see [4]). Therefore B α is 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 and 2]). It follows that B α is a Grothendieck space with the Dunford-Pettis property and the theorem of Lotz applies. Step 2. If (ϕ t ) is a semigroup with generator G and the induced semigroup of composition operators (C t ) is strongly continuous on B α then G. To prove this let Γ : B α B α denote the infinitesimal generator of (C t ). From Step, Γ is a bounded operator. Now for each f B α we have Γ(f) =, the convergence being in the norm of B α. But it is easy to see that convergence in B α implies uniform convergence on compact subsets of the disc and in particular it implies pointwise convergence for each z D. Thus for z D, f ϕ lim t f t t Γ(f)(z) = lim t f(ϕ t (z)) f(z) t = f(ϕ t(z)) t= = G(z)f (z), t therefore Γ(f) = Gf for each f B α. Suppose that G and recall that G(z) = (bz )(z b)p (z) where P H(D) with Re P (z). Since P (z) has boundary values almost everywhere on the unit circle (this for example follows from the fact that such a function P belongs to the Hardy spaces H p for all p < ) we can find a point ξ D such that G (ξ) = lim r G (rξ) exists and is finite and different from zero. Now we distinguish three cases depending on α. If < α <, take f(z) = ( ξz) α, a function in B α. Thus Γ(f)(z) = G(z)f (z) = G(z)(α )ξ( ξz) α is a function in B α and therefore bounded on D because for these values of α, B α is contained in the disc algebra [27, Proposition 9]. On the other hand taking z = rξ, < r <, we have lim r G (rξ) f (rξ) = (α )ξ lim r G(rξ)( r) α =,

11 SEMIGROUPS AND INTEGRAL OPERATORS a contradiction. If α = then take f(z) = log( ), a function in the Bloch space B = B ξz. Thus the function ξ Γ(f)(z) = G(z) ξz belongs to B, and so it must satisfy the growth estimate [26, p. 82] for Bloch functions, that is for some constant C, G(z) ξz G() + C log z, z D. In particular for z = rξ we obtain G(rξ) ( r) G() + C( r) log r and taking r this implies G (ξ) =, a contradiction. Finally if α > let f(z) = ( ξz ) α, a function in Bα. Thus the function Γ(f)(z) = G(z)(α )ξ( ξz) α belongs to B α and hence it satisfies the estimate Γ(f)(z) Γ(f)() + C( z ) α, z D, for some constant C, see [27, Theorem 8 and p. 62]. We then obtain for z = rξ (α ) G(rξ) (α ) G() ( r) α + C( r), and letting r this implies G (ξ) = lim r G(rξ) =, a contradiction. This completes the proof. Suppose now that X is either V MOA or the little Bloch space B so that the second dual X is BMOA or B respectively. Let (ϕ t ) be a semigroup on D and let (C t ) be the induced semigroup of composition operators on X. Since each ϕ t is univalent each C t is maps X into itself, and the restriction S t = C t X, is a strongly continuous semigroup on X. LEMMA. Using the preceding notation for the semigroups (S t ) and (C t ) on X and X respectively we have St = C t for each t. Proof. By the definition of the adjoint operator, S t X = S t = C t X. But X is weak dense in X and the conclusion follows. THEOREM 4. Let (ϕ t ) be a semigroup and X be one of the spaces V MOA or B. Denote by Γ the generator of the induced composition semigroup (S t ) on X and let λ ρ(γ). Then the following are equivalent. () [ϕ t, X ] = X;

12 2 BLASCO, CONTRERAS, DÍAZ-MADRIGAL, MARTÍNEZ, PAPADIMITRAKIS, AND SISKAKIS (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, [, Theorem VI.4.2]. We proceed to show that () and (3) are equivalent. () (3). Take λ ρ(γ) a big real number and f X. Writing the resolvent as a Laplace transform [, Theorem, page 622], we have S t R(λ, Γ)(f) = From this we obtain R(λ, Γ)(f) = S t R(λ, Γ)(f) R(λ, Γ)(f) = (e λt ) thus e λu S u (f) du, e λu S t+u (f) du = e λt e λu S u (f) du. S t R(λ, Γ)(f) R(λ, Γ)(f) ( e λt e λu S u du + t t e λu S u (f) du t t t e λu S u (f) du, ) e λu S u du f, see [, Corollary 5, page 69] for justification of the finiteness of the integrals. Consequently and so, recalling that S t lim S t R(λ, Γ) R(λ, Γ) = t = C t and that S t commutes with R(λ, Γ) we have lim C t R(λ, Γ) R(λ, Γ) =. t Thus if f X, for the function F = R(λ, Γ) (f) we have lim C t(f ) F =, t which says that R(λ, Γ) (f) = F [ϕ t, X ] = X, i.e. R(λ, Γ) (X ) X. This gives the result for big real values of λ and the resolvent equation gives the inclusion for any other λ ρ(γ). (3) (). To show this put Y = [ϕ t, X ] then X Y X. The restriction of (C t ) on Y is a strongly continuous semigroup with generator (f) = Gf, D( ) = {f Y : Gf Y }. It is clear that D(Γ) D( ) so that is an extension of Γ. Let λ be a big real number such that λ ρ(γ) ρ( ). An argument similar to the one in the proof of Lemma shows that R(λ, Γ) X = R(λ, Γ), R(λ, Γ) Y = R(λ, )

13 We have SEMIGROUPS AND INTEGRAL OPERATORS 3 D( ) = R(λ, )(Y ) = R(λ, Γ) Y (Y ) R(λ, Γ) (X ) X. Recalling that D( ) is dense in Y we have [ϕ t, X ] = Y = D( ) X, and this finishes the proof. COROLLARY. Let (ϕ t ) be a semigroup of functions in D, let Γ be the generator of (C t ) on the space V MOA or on the space B and let λ ρ(γ). Then () [ϕ t, BMOA] = V MOA if and only if R(λ, Γ) is weakly compact on V MOA. (2) [ϕ t, B] = B if and only if R(λ, Γ) is weakly compact on B. 5. THE INTEGRAL OPERATOR T g. We are going to use a certain integral operator T g defined on analytic functions by T g (f)(z) = f(ξ)g (ξ)dξ, f H(D), where its symbol g is an analytic function on D. This operator (also called in the literature the Volterra operator or the generalized Cesàro operator) was first considered by Ch. Pommerenke [6] and has been widely studied in several recent papers ([2], [], [22], [25]). We will use properties of this operator for a particular choice of the symbol g in order to better describe information for the maximal space of strong continuity in various cases. Before doing so we present some facts for T g acting on the spaces of our concern. To describe the symbols g for which the integral operator T g is bounded or compact on spaces like BMOA or B we need to consider the logarithmically weighted versions of those spaces. They are defined as follows. DEFINITION 3. Let f : D C an analytic function. () We say that f belongs to BMOA log if ( ) 2 log 2 f 2 I := sup f (z) 2 ( z 2 )dm(z) I D I R(I) <. The subspace V MOA log of BMOA log contains by definition all the functions f such that ( ) 2 log 2 I lim f (z) 2 ( z 2 )dm(z) I I R(I) =. (2) We say that f belongs to the weighted Bloch space B log if ( ) sup( z 2 ) log z D z 2 f (z) <.

14 4 BLASCO, CONTRERAS, DÍAZ-MADRIGAL, MARTÍNEZ, PAPADIMITRAKIS, AND SISKAKIS The subspace B log, contains all functions f that satisfy ( ) lim ( z z 2 ) log z 2 f (z) =. It is clear that BMOA log V MOA and B log B. The following theorem characterizes boundedness of the operators T g on BMOA and Bloch spaces. THEOREM 5. Let X = V MOA or respectively, X = B. Then the following are equivalent () T g is bounded on X; (2) T g is bounded on X ; (3) g BMOA log, respectively g B log. Proof. For the space V MOA, this result is contained in [22, Corollary 3.3]. For the space B, the proof is almost contained in [25, Theorem 2.]. The only thing we have to check is that if g B log, then T g is bounded on B. To show this recall that if f B then, [26, page 2], Write Then lim z f(z) ( ) =. log z 2 ( ) M = sup( z 2 ) log z D z 2 g (z). ( z 2 ) T g (f) (z) = ( z 2 ) f(z) g (z) ( ) = ( z 2 ) log z 2 g f(z) (z) ( ) log z 2 That is T g (f) is also in B. M f(z) ( ) as z. log z 2 The next theorem is about the compactness and weak compactness of T g on the spaces of our interest. THEOREM 6. Let X = V MOA or respectively X = B. Suppose that T g is bounded on X, that is, g BMOA log, respectively g B log. Then the following are equivalent () T g is weakly compact on X; (2) T g is compact on X; (3) T g is weakly compact on X ; (4) T g is compact on X ; (5) T g (X ) X; (6) g V MOA log, respectively g B log,.

15 SEMIGROUPS AND INTEGRAL OPERATORS 5 Proof. For the little Bloch space, this result follows from general theory of weakly compact operators, the fact that B is isomorphic to c and [25, Theorem 2.3]. We have to address the case X = V MOA. The equivalence between statements (2), (4), and (6) is included in [22, Theorem 3.6 and page 3]. From general theory of weakly compact operators we know that (), (3), and (5) are equivalent and (2) implies (). Therefore we only have to prove that (5) implies (6). To do this suppose that T g (BMOA) V MOA and g / V MOA log. This implies that there is some δ > and some sequence I n of intervals in D such that I n and log 2 4π I n I n R(I n ) g (z) 2 dm(z) δ for all n. Now let z In denote the point in the middle of the internal side of R(I n ) and take f BMOA. Clearly, for every z D, T g (f) (z) f(z In )g (z) = (f(z) f(z In ))g (z). Then, integrating and following literally the argument in [5, page 58, lines 7-4 from above], we deduce that, for some constant c, T g (f) (z) f(z In )g (z) 2 c dm(z) I n R(I n ) log 2 4π f 2 BMOA g 2. I n These estimates imply that T g (f) (z) 2 dm(z) I n R(I n) 2 f(z I n ) 2 g (z) 2 c dm(z) I n R(I n ) log 2 4π f 2 BMOA g 2 I n f(z In ) 2 c 2 log 2 4π δ I n log 2 4π f 2 BMOA g 2 I n Now, we notice that if we are able to construct some f BMOA such that this would imply that lim sup n lim sup n f(z In ) log 4π I n >, T g (f) (z) 2 dm(z) > I n R(I n ) and, hence, T g (f) / V MOA obtaining in this way a contradiction. The rest of the proof is devoted to the construction of such function f. Taking a subsequence, we may assume that the intervals I n accumulate to some point of D and, without loss of generality, we may take this point

16 6 BLASCO, CONTRERAS, DÍAZ-MADRIGAL, MARTÍNEZ, PAPADIMITRAKIS, AND SISKAKIS to be the point. Consider the function p(z) = log ( z) 2 ( + z) 2 + ( z) 2 which is the conformal mapping of the unit disc D onto the open set E = {w : Im(w) < π} \ {w : Re(w), Im(w) = }. It is easy to prove the following properties of p: p(z) c z + 2 whenever z +, z D; 2 p(z) c log z whenever z, z D; Im(p(ζ)) = ±π whenever < arg ζ < π 2, ζ D; Im(p(ζ)) = whenever π 2 < arg ζ < π, ζ D. We write z n = z In and δ n = z n I n, z n = z n e iθn, θ n < π. By our assumption, we have that δ n, θ n. There are exactly two cases. Case. There is some positive constant c and a subsequence of (z n ) such that δ n cθ 2 n. In this case, and restricting to this subsequence, we have z n c(δ n + θ n ) c δ n and, hence, 2 p(z n ) c log z n c log c log 4π δ n I n. Since p has bounded imaginary part, it belongs to BMOA and we are done just taking f = p. Case 2. The case δ n θ 2. n We now consider u n = ( δ n ) z n z n, δ n = z n, λ n = + u n + u n and construct the functions p n (z) = p ( ) z u n λ n. u n z Observe that arg u n = arg z n = θ n and that u n = δ n = z n. z u Also, the function λ n n u n z maps u n to and to. Furthermore, there is an interval J n of D of length J n u n = δ n which is centered at a point e iφn such that φ n θ n cδ n with the property Im(p n (ζ)) = ±π, ζ J n, Im(p n (ζ)) =, ζ T \ J n.

17 SEMIGROUPS AND INTEGRAL OPERATORS 7 Since J n θ n, by taking a subsequence, we may assume that the intervals J n are disjoint, that J n+ is closer to than J n and that δ k < +. Note that θ 2 k= k (9) λ n z u n ū n z + = + z + u n u n 2 ū n z. This shows that δn z u n λ n ū n z + C z δn and hence p n (z) C and we can define the analytic function in the ( z ) 2 unit disc given by f(z) = p n (z). n= Notice that we have that the boundary values of the harmonic function Imf(z) are absolutely bounded by π and, hence, that f belongs to BMOA. A calculation shows that Therefore and, hence, and λ n z n u n u n z n = On the other hand, using + ( 2 δn δ n ). δ n δ n + u n λ z n u n n u n z n c δ n p n (z n ) c log δ n c log 4π J n. u k z n z n u k, z n θ n, u k θ k, one easily gets from (9) that { λ z n u k k + u k z n c u k u k z n and, hence, { c δ k p k (z n ) θ 2, k < n, k, k > n. c δ k θ 2 n c δk θ k, k < n, c δk θ n, k > n,

18 8 BLASCO, CONTRERAS, DÍAZ-MADRIGAL, MARTÍNEZ, PAPADIMITRAKIS, AND SISKAKIS From these estimates we get f(z n ) p n (z n ) p k (z n ) p k (z n ) k<n k>n c log 4π I n c δ k θ 2 c δ k k<n k θ 2 k>n n c log 4π I n c δ k θ 2 k n k c log 4π I n and this finishes the proof. The above result and in particular the implication (5) (6) answers in the negative a question from [22] where it was asked if there are functions g such that T g is weakly compact but not compact in V MOA. 6. APPLICATIONS We are going now to apply these properties of T g for a special choice of the symbol g. DEFINITION 4. Given a semigroup (ϕ t ) with generator G and Denjoy-Wollf point b, we define the function γ(z) : D C as follows () If b D let (2) If b D let γ(z) = γ(z) = b ξ b G(ξ) dξ, G(ξ) dξ. Notice that γ(z) is analytic on D and that when b D then γ(z) = h(z), the Koenigs function for (ϕ t ). We call γ(z) the associated g-symbol of (ϕ t ). The following proposition shows the connection between [ϕ t, X] and integral operators. PROPOSITION 2. Let (ϕ t ) be a semigroup with associated g-symbol γ(z). Let also X be a Banach space of analytic functions with the properties: (i) X contains the constant functions, (ii) For each b D, f X f(z) f(b) z b X, (iii) If (C t ) is the induced semigroup on X then sup t [,] C t X <. Then [ϕ t, X] = X (T γ (X) C). Proof. Let G be the generator of (ϕ t ). First observe that {f X : Gf X} = {f X : f γ X}.

19 SEMIGROUPS AND INTEGRAL OPERATORS 9 This is clear in the case b D because then Gγ = Gh =. In the case b D we have G(b) = and the assumption (ii) on X gives G(z)f (z) X G(z) z b f (z) = f (z) γ (z) X. Now write m(z) = f (z) X, then where c C. Thus f(z) = γ (z) D(ϕ t, X) = {f X : Gf X} m(ξ)γ (ξ)dξ + c = T γ (m)(z) + c, = {f X : f γ = m for some m X} = {f X : f = mγ for some m X} = {f X : f(z) = = X (T γ (X) C), m(ξ)γ (ξ)dξ + c for some m X and c C} and the conclusion follows by taking closures and using Theorem. In view of the Proposition 2 it would be desirable to have the operator T γ bounded on X. The following proposition says that this is not always the case. PROPOSITION 3. Let X = BMOA or X = B, and (ϕ t ) be a semigroup with Denjoy -Wolff point on D and associated g-symbol γ. Then T γ is not bounded on X. Proof. Denote by h the Koenigs function of (ϕ t ). Then h is a univalent function on D with h() = and such that the range of h has the following geometrical property: w h(d) w + t h(d), for all t. Recall that a univalent function belongs to B if and only if it belongs to V MOA and that such a function belongs B if and only if whenever D(w, r) are discs contained in its range with the centers w moving to the boundary of the range, then the radii tend to zero. It follows from above geometric property of the range of h that h is not in B and 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 T γ is bounded on BMOA, then γ = h BMOA log which means, sup I D ( ) 2 log 2 I I R(I) h (z) 2 ( z 2 )dm(z) <.

20 2 BLASCO, CONTRERAS, DÍAZ-MADRIGAL, MARTÍNEZ, PAPADIMITRAKIS, AND SISKAKIS This implies lim I h (z) 2 ( z 2 )dm(z) =, I R(I) that is h V MOA, getting a contradiction. Therefore T γ is not bounded on BMOA. Similarly if T γ is bounded on the Bloch space, then sup( z 2 ) log z D ( z 2 ) h (z) <. This implies that h belongs to B getting again a contradiction. Therefore T γ is not bounded on B. Remark 2. If the Denjoy-Wolff point of (ϕ t ) is inside D and there is a regular boundary fixed point for some (and then for all) ϕ t, then we can show a similar result on B. Recall that a regular boundary fixed point is a point a D such that ϕ s (a) = a and ϕ s(a) for some s > in the sense of angular limits and angular derivative. It then follows that there is a β (, ) such that ϕ t (a) = a, and ϕ t(a) = e βt for all t, and that for the generator G of (ϕ t ) we have G(z) lim G(z) =, and lim z a z a z a = β where the limits are taken non-tangentially, see [8]. In particular taking G(ra) r < real we obtain lim r (r )a = β. Let γ be the associated g-symbol for (ϕ t ), then γ (z) = z b G(z) where b D is the Denjoy-Wolff point. We have, sup z D ( z 2 ) log( z 2 ) γ (z) sup r (,) therefore T γ is not bounded on B. = sup r (,) =, ( r 2 ) log( r 2 ) ra b G(ra) ra b ( + r) log( r 2 ) r G(ra) Undoubtedly, the above proposition and remark are a handicap for our applications because the boundedness of T γ is 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 = B. Suppose (ϕ t ) is a semigroup with associated g-symbol γ(z) and suppose γ BMOA log, respectively γ B log. Then [ϕ t, X ] = X if and only if γ V MOA log, respectively γ B log,.

21 SEMIGROUPS AND INTEGRAL OPERATORS 2 Proof. If γ V MOA log, respectively γ B log,, then by Theorem 6 we have T γ (X ) X. It then follows from Proposition 2 that [ϕ t, X ] = X (T γ (X ) C) X X = X and we have equality because X [ϕ t, X ]. Conversely suppose [ϕ t, X ] = X. Then from Proposition 2 we must have X (T γ (X ) C) = X and in particular () X (T γ (X ) C) X. The hypothesis on γ implies that T γ is bounded on X and X contains the constants therefore T γ (X ) C X. This in view of () implies T γ (X ) C X and then T γ (X ) X because X contains the constants. Using Theorem 6 again we conclude γ V MOA log, respectively γ B log,. COROLLARY 3. (see [4, Theorem 3.]) Let (ϕ t ) be a semigroup with generator G(z). Assume that for some < α <, Then ( z ) α G(z) = O(), z. [ϕ t, BMOA] = V MOA, and [ϕ t, B] = B. Proof. Let γ be the associated g-symbol for (ϕ t ). The assumption implies that there is a constant C such that ( z ) α γ (z) C for z D. Therefore γ has boundary values ( α)-hölder continuous, in particular γ V MOA log and γ B log,. The conclusion follows by applying Corollary 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 (ϕ t ) with generator G(z) = (bz )(z b)p (z), the associated g-symbol γ is a univalent function when b D because it coincides with the Koenigs map h. We observe that γ is also univalent when b D. The easiest way to see this is to show that γ is in this case a close-to-convex function (see [7, page 68] for the definition). Indeed the function g(z) = (/ b) log( bz) is a convex univalent function for b D and it can be checked easily that Re( γ g ) = Re( P ). We are going to consider also the function () ψ(z) = dξ, z D. P (ξ)

22 22 BLASCO, CONTRERAS, DÍAZ-MADRIGAL, MARTÍNEZ, PAPADIMITRAKIS, AND SISKAKIS Since Re(/P ) this is a univalent function [, Theorem 2.6]. The growth estimate /P (z) C + z z for z D for the function /P of nonnegative real part says that ( z ) ψ (z) 2C so ψ B and since ψ is univalent it follows that ψ BMOA. PROPOSITION 4. Let X = BMOA or X = B and let (ϕ t ) be a semigroup with generator G(z) = (bz )(z b)p (z), Koenigs function h and associated g-symbol γ and let ψ(z) be given by (). Then the following hold () ψ(z), γ(z) D(ϕ t, X), (2) If h X then h D(ϕ t, X). (3) (i) If b D then (z b) log h(z) z b [ϕ t, X]. Moreover log h(z) z b D(ϕ t, X) if and only if G(z) X, (ii) If b D then z log h(z) z [ϕ t, X]. Moreover log h(z) z D(ϕ t, X) if and only if G(z) X, (4) If C \ h(d) has nonempty interior then log(h(z) c) belongs to D(ϕ t, X) for each c in the interior of C \ h(d). Proof. (). We have G(z)ψ (z) = (bz )(z b) X therefore ψ D(ϕ t, X). For γ(z), since G(z)γ (z) = if b D while G(z)γ (z) = z b if b D the conclusion follows. (2). Since G(z)h (z) = if b D and G(z)h (z) = G (b)h(z) if b D the assertion is clear. (3)(i). We observe that G (b)γ(z) = = = b b b G (b)(ξ b) dξ G(ξ) (ξ b)h (ξ) dξ h(ξ) [ + (ξ b) ( log h(ξ) ξ b = z b + (z b) log h(z) z b ) ] dξ b log h(ξ) ξ b dξ. Now log h(z) z b belongs to BMOA (and to B) so the function h(ξ) b log ξ b dξ is continuous on D, hence it belongs to V MOA (and to B ). Since (z b) log h(z) z b = G (b)γ(z) (z b) + log h(ξ) b ξ b dξ, with γ D(ϕ t, X) [ϕ t, X] we conclude that (z b) log h(z) z b [ϕ t, X].

23 SEMIGROUPS AND INTEGRAL OPERATORS 23 Finally let q(z) = log h(z) z b and observe that ( h G(z)q (z) (z) = G(z) h(z) ) z b = G (b) G(z) z b = G (b) ( bz )P (z). This says that q(z) D(ϕ t, X) if and only if ( bz )P (z) X and this is equivalent to G(z) = (z b)( bz )P (z) X. (3)(ii). Let m(z) = z h(z). This is a bounded function on D hence it belongs to BMOA and to B. Observe that ξh (ξ) T γ (m)(z) = h(ξ) dξ. A calculation similar to the one in (3)(i) gives T γ (m)(z) = z log h(z) z + z log h(ξ) ξ dξ, and the three terms in the right hand side are elements of X. Thus T γ (m)(z) X with m(z) X, and Proposition 2 implies that T γ (m)(z) [ϕ t, X]. By the argument of (3)(i), z conclude z log h(z) z [ϕ t, X]. h(ξ) log ξ dξ is in V MOA (and in B ) and we To show the last assertion let q(z) = log h(z) z (. Then h zg(z)q (z) (z) = zg(z) h(z) ) z = zg(z)h (z) G(z) h(z) = z h(z) G(z). Now G(z)q (z) X is equivalent to zg(z)q (z) X. Thus q D(ϕ t, X) if z and only if h(z) G(z) X and since z h(z) is a function in X we 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 > such that h(z) c > δ for z D, so for the case b D, G(z)q h (z) (z) = G(z) h(z) c = h(z) c, is a bounded function so it is in BMOA and the conclusion follows in this case. Finally if b D then G(z)q h (z) (z) = G(z) h(z) c = h(z) G (b) h(z) c, which is also bounded on D so the result follows.

24 24 BLASCO, CONTRERAS, DÍAZ-MADRIGAL, MARTÍNEZ, PAPADIMITRAKIS, AND SISKAKIS COROLLARY 4. (see [4, Theorem 3.3]) If (ϕ t ) is a semigroup with generator G(z) and Denjoy-Wolff b D and either [ϕ t, BMOA] = V MOA or [ϕ t, B] = B, then z (2) lim z G(z) =. 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 B. This implies log h(z) z b V MOA B therefore, ( = lim ( z ) log h(z) ) ( h (z) = lim ( z ) z z b z h(z) ) z = lim z ( z )h (z) h(z) = lim z and since G (b) the conclusion follows. G (b)( z ), G(z) If F (z) is analytic with ReF (z) on D, the Herglotz theorem says that there is a nonnegative Borel measure µ on D such that F (z) = 2π e iθ + z e iθ dµ(θ) + iimf (). z The following proposition says that the presence of point masses for the measure µ associated with /P in the case b = implies automatically that [ϕ t, BMOA] is strictly larger than V MOA. COROLLARY 5. Let X = BMOA or X = B and let (ϕ t ) be a semigroup with generator G(z) = (z b)( bz )P (z) such that [ϕ t, BMOA] = V MOA or [ϕ t, B] = B. Then the Herglotz measure for /P (z) has no point masses on D. Proof. From Proposition 4 and the hypothesis we have that the function ψ(z) = P (ξ) dξ belongs to V MOA in the first case or to B in the second case. Since this function is univalent it belongs to V MOA in both cases. But then the operator T ψ (f)(z) = f(ξ)ψ (ξ)dξ = f(ξ) P (ξ) dξ is a compact operator on the Hardy space H 2, see []. It follows then by [9, Theorem 4] that µ has no point masses on D. COROLLARY 6. Let (ϕ t ) be a semigroup with Denjoy-Wolff point b D and Koenigs function h. If either [ϕ t, BMOA] = V MOA or [ϕ t, B] = B then h ( p< H p ) \ BMOA. Proof. First observe that the Koenigs function cannot belong to BM OA. Otherwise from Proposition 4-(2), we would have h D(ϕ t, BMOA) [ϕ t, BMOA]

25 SEMIGROUPS AND INTEGRAL OPERATORS 25 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 z log h(z) V MOA so in particular log h(z) z h(z) log V MOA. This implies e z z H p for all p <, [6, page 596], so h H p for all p <. The argument for the Bloch case is similar and is omitted. Remark 3. One could expect, for the case of Denjoy-Wolff point on the boundary, to obtain a result similar to (2) for the generator G(z), under the assumption [ϕ t, BMOA] = V MOA or [ϕ t, B] = B. In this case, using Proposition 4 one obtains log h(z) z B which means and in terms of G(z), Note that lim ( (z) z z )h h(z) =, lim z z G(z) =. G(ξ) dξ G(ξ) dξ = h(z) is never bounded, and finer analysis is required. We can however obtain a substitute of (2) if we consider integral averages of /G(z). COROLLARY 7. Let (ϕ t ) be a semigroup with generator G(z) and Denjoy-Wolff point b D. If either [ϕ t, BMOA] = V MOA or [ϕ t, B] = B then for every < p < ( 2π lim( r) r ) /p G(re it ) p dt =. Proof. As observed in the previous Remark, we have lim z ( z ) h (z) h(z) =. Since from Corollary 6 the function h belongs to all Hardy spaces we have 2π ( r) p 2π G(re it ) p dt = sup z =r and the right hand side goes to as r. ( r) p h (re it ) p h(re it ) p h(re it ) p dt ( ( z ) p h (z) p ) h p, h(z) p We now present some open questions. On the basis of Theorem 3 which says that no non-trivial semigroup (ϕ t ) induces a strongly continuous composition operator semigroup (C t ) 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.

26 26 BLASCO, CONTRERAS, DÍAZ-MADRIGAL, MARTÍNEZ, PAPADIMITRAKIS, AND SISKAKIS Question. Is it true that for every non-trivial semigroup (ϕ t ) the space [ϕ t, BMOA] is strictly smaller than BMOA? Suppose now (ϕ t ) has its Denjoy-Wolff point on the boundary so its generator can be written G(z) = b(b z) 2 P (z) with Re(P (z)). This generator cannot satisfy the condition ( z ) α G(z) = O(), as z, for < α <, of Corollary 3 which implies [ϕ t, BMOA] = V MOA. Indeed for z = rb we have ( z ) α G(z) = ( r )α b( r) 2 P (rb) = b ( r) 2 α P (rb), and the growth estimate P (z) C + z z for the function P of non-negative real part implies ( r) 2 α P (rb) as r for < α <. In addition in all examples we can work it turns out that for this case the space [ϕ t, BMOA] is strictly larger than V MOA. This leads to Question 2. Suppose (ϕ t ) is a semigroup with Denjoy-Wolff point on the boundary. Is it true in this case that [ϕ t, BMOA] is strictly larger than V MOA and [ϕ t, B] is strictly larger than B? REFERENCES [] A. Aleman and A. G. Siskakis, Integration operators on H p, Complex Variables 28 (995), [2] A. Aleman and A. G. Siskakis, Integration operators on Bergman spaces, Indiana Univ. Math. J., 46 (997), [3] E. Berkson and H. Porta, Semigroups of analytic functions and composition operators, Michigan Math. J. 25 (978) 5. [4] Blasco, O.; Contreras, M. D.; Díaz-Madrigal, S.; Martinez, J.; Siskakis, A. G. Semigroups of composition operators in BMOA and the extension of a theorem of Sarason, Integral Equations Operator Theory 6 (28), [5] J. M. F. Castillo and M. Gonzalez, New results on the Dunford-Pettis property, Bull. London Math. Soc. 27 (995), [6] M.D. Contreras and S. Díaz-Madrigal, Fractional iteration in the disk algebra: prime ends and composition operators, Revista Mat. Iberoamericana 2 (25) [7] M.D. Contreras and S. Díaz-Madrigal, Analytic flows in the unit disk: angular derivatives and boundary fixed points, Pacific J. Math. 222 (25) [8] M.D. Contreras, S. Díaz-Madrigal, and Ch. Pommerenke, On boundary critical points for semigroups of analytic functions, Math. Scand. 98 (26) [9] J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag, New York, 984. [] N. Dunford and J. T. Schwartz, Linear operators, Part I. Wiley Classics Library, John Wiley & Sons Inc., New York, 988. [] P. L. Duren, Univalent functions. Springer-Verlag, New York, 983. [2] J.B. Garnett, Bounded Analytic Functions, Pure and Applied Mathematics, Vol. 96 Academic Press, New York, 98. [3] H. P. Lotz, Uniform convergence of operators on L and similar spaces, Math. Z. 9 (985)

27 SEMIGROUPS AND INTEGRAL OPERATORS 27 [4] W. Lusky, On the isomorphic classification of weighted spaces of holomorphic functions, Acta Univ. Carolin. Math. Phys. 4 (2) 5 6. [5] M. Papadimitrakis and J.A. Virtanen, Hankel and Toeplitz Transforms on H : Continuity, Compactness and Fredholm Propoerties, Integral Equations Operator Theory 6 (28), [6] Ch. Pommerenke, Schlichte Funktionen und analytische Funktionen von beschräankter mittlerer Oszillation, Comment. Math. Helv. 52 (977), [7] Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 992. [8] D. Sarason, Function of vanishing mean oscillation, Trans. Amer. Math. Soc. 27 (975) [9] A. G. Siskakis, The Koebe semigroup and a class of averaging operators on Hp(D), Trans. Amer. Math. Soc. 339 (993), [2] A. G. Siskakis, Semigroups of composition operators on the Dirichlet space, Results Math. 3 (996), [2] A. G. Siskakis, Semigroups of composition operators on spaces of analytic functions, a review, Contemp. Math. 23 (998) [22] A. G. Siskakis and R. Zhao, A Volterra type operator on spaces of analyitc functions, Contemp. Math. 232 (99), [23] K. J. Wirths and J. Xiao, Recognizing Q p, functions as per Dirichlet space structure, Bull. Belg. Math. Soc. 8 (2), [24] J. Xiao, Holomorphic Q Classes, Lecture Notes in Math. 767, Springer-Verlag 2. [25] R. Yoneda, Integration operators on weighted Bloch spaces, Nihonkai Math. J. 2 (2) [26] K. Zhu, Operator Theory in Function spaces, Marcel Dekker, Inc., New York and Basel, 99. [27] K. Zhu, Bloch type spaces of analytic functions, Rocky Mountain J. Math. 23 (993) DEPARTAMENTO DE ANÁLISIS MATEMÁTICO, UNIVERSIDAD DE VALENCIA, 46 BUR- JASSOT, VALENCIA, SPAIN. address: Oscar.Blasco@uv.es CAMINO DE LOS DESCUBRIMIENTOS, S/N, DEPARTAMENTO DE MATEMÁTICA APLI- CADA II, ESCUELA TÉCNICA SUPERIOR DE INGENIEROS, UNIVERSIDAD DE SEVILLA, 492, SEVILLA, SPAIN. address: contreras@us.es CAMINO DE LOS DESCUBRIMIENTOS, S/N, DEPARTAMENTO DE MATEMÁTICA APLI- CADA II, ESCUELA TÉCNICA SUPERIOR DE INGENIEROS, UNIVERSIDAD DE SEVILLA, 492, SEVILLA, SPAIN. address: madrigal@us.es DEPARTAMENTO DE ANÁLISIS MATEMÁTICO, UNIVERSIDAD DE VALENCIA, 46 BUR- JASSOT, VALENCIA, SPAIN. address: Josep.Martinez@uv.es DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CRETE, KNOSSOS AVENUE 74 9 IRAKLION CRETE, GREECE address: papadim@math.uoc.gr DEPARTMENT OF MATHEMATICS, ARISTOTLE UNIVERSITY OF THESSALONIKI, 5424 THESSALONIKI, GREECE. address: siskakis@math.auth.gr