COMPOSITION SEMIGROUPS ON BMOA AND H AUSTIN ANDERSON, MIRJANA JOVOVIC, AND WAYNE SMITH

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1 COMPOSITION SEMIGROUPS ON BMOA AND H AUSTIN ANDERSON, MIRJANA JOVOVIC, AND WAYNE SMITH Abstract. We study [ϕ t, X], the maximal space of strong continuity for a semigroup of composition operators induced by a semigroup {ϕ t} t 0 of analytic self-maps of the unit disk, when X is BMOA, H or the disk algebra. In particular, we show that [ϕ t, BMOA] BMOA for all nontrivial semigroups. We also prove, for every semigroup {ϕ t} t 0, that lim t 0 + ϕ t(z) = z not just pointwise, but in H norm. This provides a unified proof of known results about [ϕ t, X] when X {H p, A p, B 0, VMOA}. 1. Introduction Let D denote the unit disk {z : z < 1} and H(D) the set of analytic functions on D. A one-parameter semigroup {ϕ t } t 0 of analytic functions on D is a family of analytic functions ϕ t : D D that satisfies the following three conditions: (SG1) ϕ 0 is the identity, i.e. ϕ 0 (z) = z, z D; (SG2) ϕ s+t = ϕ s ϕ t, for all t, s 0; (SG3) the mapping (t, z) ϕ t (z) is continuous on [0, ) D. In the trivial case, ϕ t (z) = z for all t 0. Otherwise, we say that {ϕ t } is nontrivial. For {ϕ t }, consider the set of linear operators {C t }, where C t (f) = f ϕ t for f H(D). Then C 0 is the identity operator and C s+t = C s C t, and so {C t } is called a composition semigroup. If C t is a bounded operator on some Banach space X H(D) for all t 0, we say that the semigroup {ϕ t } acts on X. If in addition the strong continuity condition lim f ϕ t f X = 0 t 0 + holds for all f X, then it is said that {ϕ t } generates {C t }, and {C t } is strongly continuous on X. The study of composition semigroups on spaces of analytic functions began in [3], where Berkson and Porta investigated the basic structure of semigroups {ϕ t } and properties of {C t } on the Hardy spaces of the disk. A survey of further developments through 1998 of semigroups of composition operators acting on classical Banach spaces can be found in [13]. More recent papers about composition semigroups include [4], [5] and [6]. Banach spaces considered include the Hardy spaces H p, the Bergman spaces A p, the disk algebra A, the Dirichlet space D, the Bloch space B, BMOA, and their little-oh subspaces B 0 and VMOA. For a semigroup {ϕ t } that acts on a Banach space X H(D), denote by [ϕ t, X] the maximal closed subspace of X on which {C t } is strongly continuous. Thus Date: September 30, Mathematics Subject Classification. Primary: 47B33; Secondary: 47D05, 30A76. Key words and phrases. Composition semigroups, BMOA, H. 1

2 2 A. ANDERSON, M. JOVOVIC, AND W. SMITH [ϕ t, X] = X means that {ϕ t } generates {C t }, and {C t } is strongly continuous on X. Here is a quick summary of what is known: (i) If X {H p (1 p < ), A p (1 p < ), D, B 0, VMOA} and {ϕ t } is any semigroup, then [ϕ t, X] = X; (ii) For every nontrivial semigroup {ϕ t }, [ϕ t, H ] H and [ϕ t, B] B; (iii) For every semigroup {ϕ t }, VMOA [ϕ t, BMOA] and B 0 [ϕ t, B]. Statement (ii) can be proved using a functional analytic argument based on H and the Bloch space having the Dunford-Pettis property; see [5] and [9]. The space BMOA does not have the Dunford-Pettis property, so this proof is not available and the corresponding statement had remained open. Our first result resolves this problem by extending statement (ii) to include BMOA. The proof is function theoretic, and also provides a new proof of the result for H and the Bloch space; see Section 3. Theorem 1.1. Suppose H X B. Then [ϕ t, X] X. In particular, [ϕ t, BMOA] BMOA. From (SG1) and (SG3) we have the pointwise convergence ϕ t (z) z as t 0 +, which can easily be extended to uniform convergence on compact subsets of D. It was recently observed by P. Gumenyuk that this extends to uniform convergence on all of D for every semigroup {ϕ t }. Theorem 1.2. [8, Proposition 3.2] For every semigroup {ϕ t }, lim ϕ t(z) z H = 0. t 0 + We thank the referee for giving us this reference. Our proof, given in Section 4, and that in [8] are based on the fact that univalent functions have no Koebe arcs. An easy consequence of this theorem is the following corollary: Corollary 1.3. Let X be a Banach space that contains H, and let X P be the closure of the polynomials in X. For all semigroups {ϕ t }, X P [ϕ t, X]. We note that Corollary 1.3 provides a unified proof of some of the known results mentioned above. In particular, the corollary gives an alternate proof of the cases that X {H p, A p, B 0, VMOA} in statement (i) above, since polynomials are dense in these spaces, and also includes all of statement (iii). It also establishes that the natural extension of statement (iii) to H is valid: Corollary 1.4. For every semigroup {ϕ t }, the disk algebra A satisfies A [ϕ t, H ]. It is well known that a composition operator C ϕ is bounded on the disk algebra A if and only if ϕ A. Hence a semigroup {ϕ t } acts on A if and only if {ϕ t } A. Another consequence of Theorem 1.2 is the following: Corollary 1.5. A semigroup {ϕ t } satisfies {ϕ t } A if and only if [ϕ t, A] = A. Corollary 1.5 should be compared to [6, Theorem 1.1], where additional conditions are included with the statement that {ϕ t } A. Theorem 1.2 allows removal of these additional conditions. The rest of the paper is organized as follows: Section 2 contains background material. Theorem 1.1 is proved in Section 3, via another theorem involving estimates

3 COMPOSITION SEMIGROUPS ON BMOA AND H 3 of the Bloch norm. Theorem 1.2 and our results about [ϕ t, H ] are in Section 4. In addition to the theorems above, we show that for some semigroups we have A = [ϕ t, H ], while A [ϕ t, H ] for others. Notation for constants. For X and Y nonnegative quantities, the notation X Y or Y X means X CY, where the exact value of the constant C is not important. Similarly, X Y means that both X Y and Y X hold. 2. Background We begin with some basic definitions. H is the space of bounded functions in H(D), with norm f = sup{ f(z) : z D}. Let A denote the disk algebra, i.e., the subspace of H consisting of functions that extend to be continuous on D. 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; we will use the one in terms of Carleson measures; see [7], [14]. A function f H 2 belongs to BMOA if and only if there exists a constant C > 0 such that f (z) 2 (1 z 2 ) da(z) C I, R(I) for every arc I D, where R(I) is the Carleson rectangle determined by I. That is, R(I) = {re iθ D : 1 I 2π < r < 1, eiθ I}, where I denotes the length of I and da(z) is the normalized Lebesgue measure on D. The corresponding BMOA norm is f = f(0) + sup I D ( 1 I R(I) f (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 VMOA. Alternatively, 1 VMOA = {f BMOA : lim f (z) 2 (1 z 2 ) da(z) = 0}. I 0 I The Bloch space is R(I) B = {f H(D) : f B = f(0) + sup f (z) (1 z 2 ) < }. z D The closure of all polynomials in B is denoted by B 0, the little Bloch space. It is usually defined as B 0 = {f H(D) : lim z 1 f (z) (1 z 2 ) = 0}. It is well known that H BMOA B, and f f f B for all f H ; see [14]. Also if h : D C is univalent, then h BMOA if and only if h B, and h VMOA if and only if h B 0 ; see, e.g., [7, p. 283].

4 4 A. ANDERSON, M. JOVOVIC, AND W. SMITH Next, we review some fundamental results about the structure of composition semigroups. A reference for these is Section 3 of [3]; see also Section 3 of [13]. Every nontrivial semigroup of analytic functions {ϕ t } t 0 has a unique common fixed point b with ϕ t(b) 1 for all t 0, called the Denjoy-Wolff point of {ϕ t } t 0. Under a normalization, the Denjoy-Wolff point b may be assumed to be 0 or 1. If b = 0, then (2.1) ϕ t (z) = h 1 (e ct h(z)), where h is a univalent function from D onto a spirallike domain Ω, h(0) = 0, Re c 0, and we ct Ω for each w Ω, t 0. If b = 1, then (2.2) ϕ t (z) = h 1 (h(z) + ct), where h is a univalent function from D onto a close-to-convex domain Ω, h(0) = 0, where Re c 0, and w + ct Ω for each w Ω, t 0. If {ϕ t } t 0 is a semigroup, then each map ϕ t is univalent. The infinitesimal generator of {ϕ t } t 0 is the function ϕ t (z) z G(z) = lim, z D. t 0 + t This convergence holds uniformly on compact subsets on D so G H(D). G has a representation G(z) = (bz 1)(z b)p (z), z D, where b is the Denjoy-Wolff point of {ϕ t } t 0, P H(D) with Re P (z) 0 for all z D. We will use the following result from [5]: Theorem 2.1. [5, Theorem 1] Let {ϕ t } t 0 be a semigroup with generator G and X a Banach space of analytic functions which contains the constant functions and such that sup 0 t 1 C t <. Then [ϕ t, X] = {f X : Gf X}. We will now review some basic facts about prime ends introduced by Carathéodory in order to describe the boundary behavior of a univalent function h from D onto a simply connected domain Ω C { }; see Section 2.4 in [10]. A crosscut C of Ω is an open Jordan arc in Ω such that C\C consists of one or two points on Ω. Here C denotes the closure of C in the Riemann sphere. If C is a crosscut of Ω, then Ω\C has exactly two components. The diameter of a set E C { } in the spherical metric is denoted diam # E. A null-chain (C n ) n 0 of Ω is defined as a sequence of crosscuts of Ω such that (a) C n C n+1 = for all n; (b) C n separates C 0 and C n+1 for all n; (c) diam # C n 0 as n. Let V n be the component of Ω\C n not containing C 0, and define V n similarly for (C n). The null-chains (C n ) and (C n) are called equivalent if, for every sufficiently large m, there exists n such that V n V m and V n V m. This is an equivalence relation on the set of all null-chains of Ω. The equivalence classes are called the prime ends of Ω. A point a C { } is called a principal point of the prime end P if there exists a null-chain (C n ) representing P such that C n {a} in the spherical metric as n. The set I(P ) = n V n is non-empty, compact and connected in

5 COMPOSITION SEMIGROUPS ON BMOA AND H 5 C { }. We call I(P ) the impression of P. If I(P ) is a single point we call the prime end degenerate. We call a prime end P accessible if there exists a Jordan arc that lies, except for one endpoint on Ω, in Ω and intersects all but finitely many crosscuts of every null-chain (C n ) that represents P. We will also need the following result from univalent function theory which states that univalent functions have no Koebe arcs. For our purposes, it may be stated as follows: Lemma 2.2. [11, Lemma 9.3 and Corollary 9.1] Suppose that h : D C is univalent, {η n } is a sequence of Jordan arcs in D, and h(η n ) converges to a point w 0 C { }, i.e., h(z) w 0, z η n, n. Then the Euclidean diameter of η n satisfies diam η n 0, as n. 3. Proof of Theorem 1.1 Theorem 3.1. Given any nontrivial semigroup {ϕ t }, there exists f H such that lim inf f ϕ t f B 1. t 0 Proof. Let {ϕ t } be a nontrivial semigroup, and let b be the corresponding Denjoy- Wolff point. After normalization, we may assume that b is either 0 or 1. First we deal with the case that b = 0, so that each ϕ t is given by (2.1). When Re c = 0 in (2.1), the {ϕ t } are rotations of the disk. An infinite interpolating Blaschke product f with all its zeros on the radius [0, 1) satisfies lim sup f (r) (1 r) δ for some δ > 0; see [2, Lemma 2.8]. However, as f extends to be analytic in a neighborhood of every point e iθ D except 1, we have lim sup f (re iθ ) (1 r) = 0 (e iθ 1). If ϕ t (z) = ze iat for real a 0, then, for all t between 0 and 2π/ a, f ϕ t f B sup [ f (ϕ t (r))ϕ t(r) f (r) (1 r)] 0<r<1 δ. Replacing f with f/δ gives the result. Next consider the case that Re c > 0 in (2.1), so that {ϕ t } does not consist of automorphisms. Since Ω is spirallike about 0, we can choose w 0 Ω such that w 0 = inf{ w : w Ω}. Then [0, w 0 ) Ω. For all sufficiently large values of n, let C n be the connected component of {w Ω : w w 0 = 1/n} that intersects [0, w 0 ). Then (C n ) is a null-chain that represents an accessible prime end P with principal point w 0. By [11, Corollary 9.3] (see also [10]), lim h(rγ 0 ) exists (and is equal to w 0 ), where γ 0 D corresponds to P. Thus, lim ϕ t (rγ 0 ) = h 1 (e ct w 0 ) D, t > 0.

6 6 A. ANDERSON, M. JOVOVIC, AND W. SMITH Since ϕ t is univalent and bounded, ϕ t is in the Dirichlet space, and ϕ t B 0. Hence lim ϕ t(rγ 0 ) (1 r) = 0. Letting f be an infinite interpolating Blaschke product with zeros confined to the radius {rγ 0 : 0 < r < 1}, we have lim sup f (rγ 0 ) (1 r) δ for some δ > 0. However, f is continuous on D, so for fixed t > 0 Thus, for all t > 0, (3.1) lim f (ϕ t (rγ 0 )) = f (h 1 (e ct w 0 )) <. f ϕ t f B lim sup f (ϕ t (rγ 0 ))ϕ t(rγ 0 ) f (rγ 0 ) (1 r) δ, and replacing f by f/δ gives f ϕ t f B 1. It remains to consider the case that the Denjoy-Wolff point b = 1, so that the ϕ t are given by ϕ t (z) = h 1 (h(z) + ct) from (2.2). If the ϕ t are automorphisms, then the map w w + ct is an automorphism of Ω. It follows that Ω is a half-plane or strip, and Ω in C consists of impressions of degenerate prime ends which are not fixed under w w + ct, t > 0. Let w 0 C be one such impression, and let γ 0 be the corresponding point in D. Then ϕ t (γ 0 ) D but ϕ t (γ 0 ) γ 0 for all t > 0. Let f be a bounded function (such as an interpolating Blaschke product with zero sequence along {rγ 0 : 0 < r < 1}) such that for some δ > 0, but lim sup f (rγ 0 ) (1 r) δ, lim f (rγ) (1 r) = 0 for all γ D, γ γ 0. We require that f is well-behaved away from γ 0, i.e., f extends continuously to D {γ 0 }. Now fix some t > 0. Since γ 0 is not a fixed point of ϕ t, composition with ϕ t moves the radius [0, γ 0 ) away to where f is well-behaved. For γ t = ϕ t (γ 0 ), our requirement ensures that f extends to be continuous at γ t, so lim f (ϕ t (rγ 0 )) = f (γ t ). Since ϕ t is an automorphism, ϕ t is bounded on D. For fixed t > 0, f ϕ t f B lim sup f (ϕ t (rγ 0 ))ϕ t(rγ 0 ) f (rγ 0 ) (1 r) δ. As before, replacing f by f/δ gives the result. In the non-automorphism case, for t > 0 the map given for w Ω by w w + ct is not onto. Let t > 0 and w Ω \ (Ω + ct). Then there is t 0 (0, t] such that w 0 = w ct 0 Ω, but (w 0, w] Ω. As in the case b = 0, w 0 is the principal point of an accessible prime end, and the same argument terminating with (3.1) completes the proof.

7 COMPOSITION SEMIGROUPS ON BMOA AND H 7 Proof of Theorem 1.1: Each test function f in Theorem 3.1 is in H, and hence in X from the hypothesis that H X. Since X B, the Closed Graph Theorem shows that B X and bounding the Bloch norm away from 0 bounds the X norm as well. Thus it follows from Theorem 3.1 that f / [ϕ t, X], and so [ϕ t, X] X. 4. Proof of Theorem 1.2 and Corollaries Proof of Theorem 1.2: In the case where the Denjoy-Wolff point b of {ϕ t } is on the boundary of D, we may assume b = 1 and ϕ t (z) = h 1 (h(z) + ct), z D. As before, h is a univalent function from D to a close-to-convex domain Ω which has the property that w + ct Ω for each w Ω and all t > 0, for some c C with Re c 0. If c = 0, the result is trivial. So assume c 0. Suppose ϕ t (z) does not converge uniformly to z in D. Then there exist some δ > 0 and infinite sequences {t n }, t n 0 + as n, and {z n } D such that Letting w n = h(z n ) Ω, δ ϕ tn (z n ) z n, n 1. ϕ tn (z n ) z n = h 1 (h(z n ) + ct n ) h 1 (h(z n )) = h 1 (w n + ct n ) h 1 (w n ). The points w n + ct n and w n are endpoints of a line segment in Ω which pulls back to the Jordan arc η n = {h 1 (w n + ct) : 0 t t n } D. Since t n 0 and Ω is compact in the Riemann sphere, we may pass to a subsequence of {w n } and assume the line segment [w n, w n + t n ] = h(η n ) w 0 Ω { }. However, diam η n δ, contradicting Lemma 2.2. Therefore, ϕ t (z) z 0 uniformly in D as t 0 +. In the case where {ϕ t } has Denjoy-Wolff point inside D, minor modifications of the same argument yield uniform convergence. We may assume ϕ t (z) = h 1 (h(z)e ct ), z D, t 0, where Re c 0 and Ω = h(d) is a spirallike domain. convergence fails, we have δ ϕ tn (z n ) z n = h 1 (w n e ctn ) h 1 (w n ). Supposing that uniform Let η n = {h 1 (w n e ct ) : 0 t t n }. Passing to a subsequence as before, h(η n ) w 0 Ω { } while diam η n δ, a contradiction of Lemma 2.2. Thus, ϕ t (z) z uniformly in D. We can now prove Corollaries 1.3, 1.4, and 1.5. Proof of Corollary 1.3: The closed graph theorem and Theorem 1.2 show that lim ϕ t z X lim ϕ t z H = 0, t 0 t 0 i.e., the function f(z) = z is in [ϕ t, X]. To extend this to any function in X P is straightforward. The inequality ϕ t (z) n z n n ϕ t (z) z

8 8 A. ANDERSON, M. JOVOVIC, AND W. SMITH and linearity show any polynomial is in [ϕ t, X], and since [ϕ t, X] is closed, X P [ϕ t, X]. Corollary 1.4 follows immediately from Corollary 1.3 since A is the closure of the polynomials in H. Proof of Corollary 1.5: If A = [ϕ t, A], then the semigroup {ϕ t } acts on A and hence {ϕ t } A. Next, for the converse, assume that {ϕ t } A. Since the norm on A is the same as the H norm, Theorem 1.2 shows that the function f(z) = z is in [ϕ t, A]. As in the previous proof, this extends to any function in A, and so A [ϕ t, A]. Since the reverse inclusion is trivial, A = [ϕ t, A] and the proof of the corollary is complete. In Proposition 4.3 below, we will show that there are semigroups {ϕ t } such that A [ϕ t, H ]. In the other direction, it is known that when {ϕ t } consists of rotations or dilations of D, then VMOA = [ϕ t, BMOA]; see [4]. In the setting of H we have the following analog: Proposition 4.1. If {ϕ t } consists of rotations or dilations of D, then A = [ϕ t, H ]. Proof. In light of Corollary 1.4 we need only show [ϕ t, H ] A for such {ϕ t }. If the semigroup consists of rotations, then ϕ t (z) = e iat z for some real a 0. If f [ϕ t, H ], then (4.1) lim t 0 + f(eiat z) f(z) = 0. It follows from (4.1) that f A; see the proof of Theorem 3.5 in [2]. Hence [ϕ t H ] A as required. If {ϕ t } consists of dilations, they have the form ϕ t (z) = e ct z for Re c > 0, and any f H is uniformly continuous on ϕ t (D) for t > 0. I.e., f ϕ t A. Thus, f [ϕ t, H ] means f is the uniform limit of the functions f ϕ t, implying f A. Next, we turn to the demonstration that there is a semigroup {ϕ t } with A [ϕ t, H ]. First we need a proposition. Proposition 4.2. There exists an infinite Blaschke product B such that B (z)(1 z) H. Proof. Let B be the interpolating Blaschke product whose zeroes are a k = 1 2 k for k = 1, 2,... We write B = 1 σ k, where Let f(θ) = σ k (z) = a k z 1 a k z k=1 (z D). 1 a k (1 a k ) 2 + θ 2 ( π θ < π). As in the proof of [1, Lemma 1], e iθ a k 2 = (1 a k ) 2 + 4a k sin 2 (θ/2). Hence, for 0 < θ < π/4,

9 COMPOSITION SEMIGROUPS ON BMOA AND H 9 (4.2) (4.3) (4.4) B (e iθ ) σ k(e iθ ) σ j (e iθ ) j k k=1 k=1 With a k = 1 2 k and θ 0, we have f(θ) = k = 2 k 2 2k + θ 2 2 2k θ 2 2 k 2 k θ 1 a 2 k 1 a k e iθ f(θ) k + θ k 2 2k + 2 k <θ 2 2k + θ 2 2 2k <θ 2 2 k 2 k θ 2 (4.5) 2 θ + 2θ θ 2 = 4 θ. The approximation from (4.3) to (4.4) invokes the inequality 1 x 1 x + y 1 2x, 0 < y x. The sums in (4.4) are geometric, which we estimate by their largest terms. Hence f(θ) 1 θ, 0 < θ π/4. Note that B extends to be continuous on D \ D ε, where D ε is a small disk centered at 1. Also 1 e iθ θ, for 0 < θ π. Hence there is M > 0 such that (1 e iθ )B (e iθ ) θ f(θ) M, 0 < θ π. Applying the same estimates to partial sums we see the finite Blaschke products B n = n 1 σ k satisfy B n(e iθ )(1 e iθ ) M for all θ, and by the maximum principle B n(z)(1 z) M for all z D. Also, B n(z) B (z) pointwise on D. Thus for z D, (1 z)b (z) = lim (1 n z)b n(z) M. Thus, B (z)(1 z) H. Proposition 4.3. There exist semigroups {ϕ t } such that A [ϕ t, H ]. Proof. The function G 1 (z) = (1 z) is the (infinitesimal) generator of a semigroup with ϕ t (z) = e t z + 1 e t and Denjoy-Wolff point on D. Also, G 2 (z) = z(1 e z) is the generator of a semigroup with ϕ t (z) = t z (e t 1)z+1 and inner Denjoy- Wolff point; see [13]. Let B be the Blaschke product from Proposition 4.2. Then Proposition 4.2 shows that G 1 B and G 2 B are in H. Whether {ϕ t } is generated by G 1 or G 2, Theorem 2.1 implies B [ϕ t, H ] for the Blaschke product B / A.

10 10 A. ANDERSON, M. JOVOVIC, AND W. SMITH Acknowledgment. The authors want to thank the referee for carefully reading the paper and for the valuable suggestions that improved it. References [1] P. Ahern and D. Clark, On inner functions with H p -derivative, Michigan Math. J. 21 (1974), [2] A. Anderson, M. Jovovic and W. Smith, Some integral operators acting on H, Integr. Equ. Oper. Theory 80 (2014), [3] E. Berkson and H. Porta, Semigroups of analytic functions and composition operators, Michigan Math. J. 25 (1978), [4] O. Blasco, M. Contreras, S. Díaz-Madrigal, J. Martínez, and A. Siskakis, Semigroups of composition operators in BMOA and the extension of a theorem of Sarason, Integr. Equ. Oper. Theory 61 (2008), [5] O. Blasco, M. Contreras, S. Díaz-Madrigal, J. Martínez, M. Papadimitrakis, and A. Siskakis, Semigroups of composition operators and integral operators in spaces of analytic functions, Ann. Acad. Sci. Fenn. Math. 38 (2013), [6] M.D. Contreras and S. Díaz-Madrigal, Fractional iteration in the disk algebra: prime ends and composition operators, Revista Mat. Iberoamericana 21 (2005) [7] J. Garnett, Bounded Analytic Functions, Academic Press [8] P. Gumenyuk, Angular and unrestricted limits of one-parameter semigroups in the unit disk, J. Math. Anal. Appl. 417 (2014) [9] H. Lotz, Uniform convergence of operators on L and similar spaces, Math Z. 190 (1985), [10] Ch. Pommerenke, Boundary Behavior of Conformal Maps, Springer-Verlag [11] Ch. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht [12] L. Rubel, A. Shields, and B. Taylor, Mergelyan Set and the Modulus of Continuity of Analytic Functions, J. Approx. Theory 15 (1975), [13] A. Siskakis, Semigroups of composition operators on spaces of analytic functions, a review, Studies on composition operators (Laramie, WY, 1996), , Contemp. Math., 213, Amer. Math. Soc., Providence, RI, [14] K. Zhu, Operator theory in function spaces, 2nd ed., Mathematical Surveys and Monographs 138. Amer. Math. Soc., Providence, Department of Mathematics, University of Hawaii, Honolulu, Hawaii address: austina@hawaii.edu Department of Mathematics, University of Hawaii, Honolulu, Hawaii address: jovovic@math.hawaii.edu Department of Mathematics, University of Hawaii, Honolulu, Hawaii address: wayne@math.hawaii.edu