HYPERCYCLIC CONVOLUTION OPERATORS ON FRÉCHET SPACES OF ANALYTIC FUNCTIONS

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1 HYPERCYCLIC CONVOLUTION OPERATORS ON FRÉCHET SPACES OF ANALYTIC FUNCTIONS DANIEL CARANDO, VERÓNICA DIMANT, AND SANTIAGO MURO Abstract. A resut of Godefroy and Shapiro states that the convoution operators on the space of entire functions on C n, which are not mutipes of identity, are hypercycic. Anaogues of this resut have appeared for some spaces of hoomorphic functions on a Banach space. In this wor, we define the space hoomorphic functions associated to a sequence of spaces of poynomias and determine conditions on this sequence that assure hypercycicity of convoution operators. Some nown resuts come out as particuar cases of this setting. We aso consider hoomorphic functions associated to minima ideas of poynomias and to poynomias of the Schatten-von Neumann cass. Introduction This note deas with convoution operators on Fréchet spaces of hoomorphic functions associated to certain casses of poynomias. In particuar, we are interested in determining when such operators are hypercycic. Reca that convoution operators are those that commute with transations. Aso, an operator T : X X is hypercycic if there exists x X such that {T n x : n } is dense in X. In a semina wor, Godefroy and Shapiro [8] show that every convoution operator in H(C n ) which is not a scaar mutipe of identity is hypercycic. In this way, they generaize cassica resuts of Birhoff [5] and MacLane [23] on the hypercycicity of the transation and differentiation operators on H(C). Anaogues of Godefroy and Shapiro s resut for some particuar spaces of hoomorphic functions on Banach spaces are proved in [, 26, 27]. Let E be a Banach space. Given a coherent sequence A(E) = {A (E)} of Banach spaces of -homogeneous poynomias (see the definitions beow), we define a Fréchet space H ba (E) of hoomorphic function of bounded type associated to A(E), much in the spirit of hoomorphy types introduced by Nachbin [25] (see aso [3]). Under fairy genera conditions, we characterize the convoution operators on this spaces of hoomorphic functions and show that they are hypercycic whenever they are not scaar mutipes of the identity. We obtain some resuts of [, 26, 27] as particuar cases. Aso, we see that spaces of hoomorphic functions generated by poynomia minima ideas are covered by our settings, if the dua space E has the approximation property. We finay consider poynomias of the Schatten-von Neumann cass in the sense of [] and the associated space of hoomorphic functions. In the first section, we give the notions of coherence that wi be used throughout and show some genera properties. We define the Fréchet space H ba (E) of A-hoomorphic functions of bounded type and present exampes reated to some common casses of homogeneous poynomias. The second section deas with duaity questions for these spaces. Based on duaity properties for each space A (E), we characterize the dua of H ba (E) as the space of hoomorphic functions of exponentia type Exp B (E ) associated to some sequence B of Banach spaces of poynomias. Minima ideas of poynomias are particuary considered. Under certain hypotheses, Exp B (E ) Key words and phrases. Spaces of hoomorphic functions, hypercycic operators, convoution operators. The first author was partiay supported by CONICET Resoución N584-04, UBACyT Grant X08 and ANPCyT PICT The third author was partiay supported by CONICET.

2 2 DANIEL CARANDO, VERÓNICA DIMANT, AND SANTIAGO MURO is an agebra and its product can be identified with the convoution product in H ba (E) (Section 3). In the fourth section, we characterize the convoution operators on H ba (E) and show that they are hypercycic uness they are scaar mutipes of the identity. We appy these resuts to the spaces of hoomorphic functions of compact bounded type and of nuceary entire functions of bounded type to obtain some resuts of [, 27]. We aso consider hoomorphic functions generated by coherent minima ideas of poynomias. The ast section deas with Schatten-von Neumann functions of bounded type, that is, those associated to the Schatten-von Neumann homogeneous poynomias defined by Cobos, Kühn and Peetre []. We use interpoation theory to show that these spaces satisfy our hypotheses. In particuar, we obtain the hypercycicity of convoution operators on the space of hoomorphic function of Hibert-Schmidt type [26]. We refer to [4, 24] for notation and resuts regarding poynomias and hoomorphic functions in genera, and to [5, 6] for poynomia ideas. For generaities on hypercycic operators, we refer to the survey [9] and the references therein.. Definitions and properties. Throughout, E wi denote a Banach space, E its dua and P (E) the space of continuous -homogeneous poynomias on E, with the usua norm. Aso, for each, A (E) and B (E) wi be Banach spaces of -homogeneous poynomias on E containing the finite type poynomias and continuousy contained in P (E). In the seque, a poynomias wi be assumed to be continuous. For P P (E), ˇP wi denote the symmetric -inear form associated to P. Aso, if a E and r N, P a r is the ( r)-homogeneous poynomia on E defined by P a r(x) = ˇP (a,..., a, x,..., x). }{{} r In [9] we introduced the notion of coherent sequence of poynomia ideas. Here, we are interested in poynomias and anaytic functions defined on a fixed Banach space. Therefore, we adapt the definition of coherence to our setting. Definition.. We say that the sequence A(E) = {A (E)} is coherent if there exist positive constants C and D such that the foowing conditions hod for a : (i) For each P A + (E) and a E, P a beongs to A (E) and P a A (E) C P A+ (E) a, (ii) For each P A (E) and γ E, γp beongs to A + (E) and γp A+ (E) D γ P A (E). Ceary, if some A (E) is non-empty then, by condition (i), A 0 (E) is the -dimensiona space of constant functions on E. Note that the fact that A (E) contains the finite type poynomias can be deduced from condition (ii). Aso, condition (i) assures that A (E) is continuousy embedded in P (E) for every. Condition (ii) states that the product of a poynomia in A(E) by a inear functiona remains in A(E). This is not necessariy the case if we mutipy two poynomias in A(E). Thus, we introduce the foowing: Definition.2. A coherent sequence A(E) = {A (E)} is mutipicative if there exists M > 0 such that for each P A (E) and Q A (E), we have P Q A + (E) and P Q A+ (E) M + P A (E) Q A (E).

3 HYPERCYCLIC CONVOLUTION OPERATORS ON FRÉCHET SPACES OF ANALYTIC FUNCTIONS 3 It is not difficut to see that not every coherent sequence is mutipicative (see [0]). The space of hoomorphic functions of bounded type H b (E) is, in some sense, associated to the sequence {P (E)} of a homogeneous poynomias. Anaogousy, we can define the space of hoomorphic functions of bounded type associated to any coherent poynomia sequence: Definition.3. Let A(E) = {A (E)} be a coherent sequence. We define the space of A- hoomorphic functions of bounded type on E by { H ba (E) = f H(E) : d f(0) A (E) for a and d f(0)!! A (E) } 0. Athough the definition of H ba (E) invoves the derivatives at 0, the same condition hods for the derivatives at any point a E, as the foowing emma shows. Lemma.4. Let A(E) = {A (E)} be a coherent sequence and f H ba (E). Then, for a a E, im d f(a)! = 0. A (E) Therefore, the function τ a (f) = f(a + ) beongs to H ba (E). Proof. Let f = Thus d f(a)! = =0 P H ba (E), with P = d f(0)! f(x + a) = = =0 =0 ( ) ˇP (a, x ) = A (E). Then =0 = ( ) ˇP (a, x ). ( ) (P ) a. The partia sums of this series beong to A (E) since A(E) is a coherent sequence. For 0 < ε <, et N N such that P A (E) =max(,n) ( ) (P ) a A (E) =max(,n) ( ε ) C a ε C a ( ) C P A (E) a =max(,n) ( ) ε <. if N. We have Therefore, the series converges in A (E) and d f(a) A (E).! Aso, for N, d f(a)! A (E) ε C a ( = Since ε > 0 is arbitrary, we concude that d f(a)! ( ) ε ) 0. A (E) ε C a ( ε). Lemma.5. Let A(E) = {A (E)} be a coherent mutipicative sequence. If f, g H ba (E) then fg H ba (E). Therefore, H ba (E) is an agebra.

4 4 DANIEL CARANDO, VERÓNICA DIMANT, AND SANTIAGO MURO Proof. Tae f, g H ba (E) with Tayor expasions f = A (E). Then d (fg)(0)! = ε > 0, tae A > 0 such that P and g = =0 Q, where P, Q P Q beongs to A (E), since A(E) is mutipicative. For each =0 max{ P A (E), Q A (E)} Aε, =0 for every 0. Then, d (fg)(0)! A (E) ( M P A (E) Q A (E) (( + )A 2 ) Mε Mε. Therefore d (fg)(0)! 0 and thus fg H ba(e). A (E) =0 ) We define in H ba (E) a sequence of seminorms {p n } n, p n (f) = d f(0)! A (E) n, =0 for f H ba (E). ) It is easy to see that (H ba (E), {p n } n is a Fréchet space. Moreover, for each f H ba (E), the partia sums of the Tayor series expansion of f about the origin converges to f in H ba (E). Next, we present exampes of coherent mutipicative sequences. The coherence is shown in [9]. It is immediate that the sequence of Exampes.6 and.8 are mutipicative. For the other exampes see [4, Exercise 2.63] and [0]. Moreover, Boyd and Lassae showed that the product of two integra poynomias with vaues in a Banach agebra is aso integra [4]. Exampe.6. Let A(E) be the sequence of homogeneous poynomias, A (E) = P (E),. Then, A(E) is coherent and mutipicative and H ba (E) = H b (E). Exampe.7. Let A(E) be the sequence of nucear poynomias, A (E) = PN (E),. Then, A(E) is coherent and mutipicative and H ba (E) is the space of nuceary entire functions of bounded type H Nb (E) defined by Gupta and Nachbin (see [20, 4]). Exampe.8. Suppose A(E) is the sequence of extendibe poynomias, that is, A (E) = P e (E),. Then, A(E) is coherent and mutipicative. Moreover, an appication of [8, Proposition 4] gives that H ba (E) is the space of a f H(E) such that, for any Banach space G E, there is an extension f H b (G) of f. Exampe.9. Let A(E) be the sequence of integra poynomias, A (E) = P I (E),. Then, A(E) is coherent and mutipicative and H ba (E) is the space of integra hoomorphic functions of bounded type H bi (E) defined in [2].

5 HYPERCYCLIC CONVOLUTION OPERATORS ON FRÉCHET SPACES OF ANALYTIC FUNCTIONS 5 2. Coherent sequences and duaity. Let A(E) = {A (E)} be a coherent sequence. The Bore transform β : H ba (E) H(E ) assigns to each eement ϕ H ba (E) the hoomorphic function β(ϕ) H(E ), given by β(ϕ)(γ) = ϕ(exp γ) = ϕ(e γ ). If ϕ A (E), we have two natura ways to identify ϕ with an eement in H ba (E) : ϕ ϕ H ba (E) C ) or H ba (E) C f ϕ f ϕ ( d f(0) ). ( d f(0)! Thus, the Bore transform induces two different poynomia Bore transforms: β : A (E) P (E ) where β (ϕ) = ( β( ) ϕ) and B : A (E) P (E ) given by B (ϕ) = β(ϕ). Note that for γ E, β (ϕ)(γ) = ϕ γ! and B (ϕ)(γ) = ϕ ( γ ). In the poynomia setting it is more common to use the mapping B than the mapping β. Moreover, it is not necessary to dea with hoomorphic functions in order to define the poynomia Bore transform B. Indeed, for a Banach space of -homogeneous poynomias A (E), we can define B : A (E) P (E ) by B (ϕ)(γ) = ϕ ( γ ), for every γ E. Aso, we can express the hoomorphic Bore transform β in terms of the B s: ( ) B ϕ A (E) β(ϕ) =.! Notation: In the seque, the expression =0 A (E) = B (E ) wi aways mean that the poynomia Bore transform B : A (E) B (E ) is an isometric isomorphism. The foowing emma states that in order to have this duaity, A (E) must be sma : Lemma 2.. If A (E) = B (E ), then finite type poynomias are dense in A (E): P f (E)A = A (E). Proof. Suppose there exists P A (E) P f (E)A. Then there is ϕ A (E) such that ϕ(p ) = and ϕ P f (E) 0. For every γ E, ϕ(γ ) = 0 and then B (ϕ)(γ) = 0. Thus B (ϕ) = 0 and therefore ϕ = 0 in A (E), which is a contradiction. Since the Tayor expansion about the origin of a function f H ba (E) converges in H ba (E), we have Coroary 2.2. Let {A (E)} and {B (E )} be coherent sequences such that A (E) = B (E ). Then finite type poynomias are dense in H ba (E). Exampes 2.3. We exhibit two simpe situations of coherent sequences where the duaity A (E) = B (E ) hods. First, if P n A (E) is the space of approximabe n-homogeneous poynomias, then Pn A (E) is (isometricay) the space of integra poynomias P n I (E ) [3]. Second, if E has the approximation property, the dua of P n N (E) coincides with Pn (E ) [20]. Note that in both cases, the sequence of dua spaces (i.e. {P n (E )} n and {P I (E )} n respectivey) is coherent and mutipicative. Remar 2.4. Now we use resuts from [5] on minima, maxima and dua (or adoint) poynomia ideas to show how to obtain other exampes in which the duaity reation A (E) = B (E ) hods.

6 6 DANIEL CARANDO, VERÓNICA DIMANT, AND SANTIAGO MURO Suppose A is a minima idea and et α be its associated -symmetric tensor norm. If E has the bounded approximation property, then A (E) identifies isometricay with s,α E and then A (E) is isometricay isomorphic to B (E ) = A dua (E ) via the Bore transform B, see [5, Coroary 4.3]. On the other hand, if we start with a maxima idea B, et A = (B dua ) min. Again, if E has the bounded approximation property, the Bore transform B is an isometric isomorphism between A (E) and B (E ). The foowing proposition states that if the duas of A (E) form a coherent sequence of spaces of poynomias, then {A (E)} inherits the coherence. Proposition 2.5. Let {B (E )} be a coherent sequence with constants C and D, and suppose A (E) = B (E ) for a. Then, {A (E)} is a coherent sequence with constants D and C. Proof.First, observe that if ξ E and a E then (ξ + ) a = ξ(a)ξ. Thus, for every ψ B (E ), B (ψ)((ξ+ ) a ) = ξ(a)b (ψ)(ξ ) = ξ(a)ψ(ξ) = (aψ)(ξ) = B + (aψ)(ξ+ ). This impies that, for every P P + f P a A (E) = (E), B (ψ)(p a) = B + (aψ)(p ) and sup B ψ B (E ) = + (aψ)(p ) D a P A + (E). By the density resut in Lemma 2., we obtain that for every P A + (E) and every a E, P a beongs to A (E) and P a A (E) D a P A+ (E). To prove the second condition of coherence, note that if γ and ξ are in E and ψ B + (E ) we have, by the poarization formua, B + (ψ)(γξ ) = B = = 2 + ( + )! 2 + ( + )! ε,...,ε + =± ε,...,ε + =± ε ε + B + (ψ)( (ε γ + (ε ε + )ξ) +) ε ε + ψ ( ε γ + (ε ε + )ξ ) = ψ γ (ξ) = B (ψ γ)(ξ ). This impies that if P is a finite type -homogeneous poynomia on E, then B (ψ γ)(p ). And thus, for every P P f (E), γp A+ (E) = sup B ψ B (E ) = (ψ γ)(p ) C γ P A (E). + (ψ)(γp ) = Therefore, again by Lemma 2., for every P A (E) the poynomia γp beongs to A + (E) and γp A+ (E) C γ P A (E). In order to study the dua of H ba (E), we need the foowing Definition 2.6. Let B(E) = {B (E)} be a coherent sequence. We define the hoomorphic functions of B-exponentia type on E, Exp B (E) = { f H(E) : d f(0) B (E) for a and im sup d f(0) B < }. A cassica resut of Gupta states that, for E with the approximation property, the Bore transform defines a duaity between the space of nuceary entire functions of bounded type over E, H Nb (E), and the space of hoomorphic mappings of exponentia type on E, Exp(E ) [20, 4]. In an anaogous way, we prove the foowing:

7 HYPERCYCLIC CONVOLUTION OPERATORS ON FRÉCHET SPACES OF ANALYTIC FUNCTIONS 7 Proposition 2.7. Let {B (E )} be a coherent sequence and et {A (E)} be such that A (E) = B (E ) for a. Then the Bore transform is a vector space isomorphism between H ba (E) and Exp B (E ). Proof. Let ϕ H ba (E). Since ϕ is continuous, there are constants c, R > 0 such that ϕ(g) cp R (g), for every g H ba (E). In particuar, for each, if g beongs to A (E), we get ϕ(g) cr g A (E). Then ϕ A (E) A (E) cr, for every. Moreover, since d β(ϕ)(0)! (γ) = ϕ A (E) ( γ! ) we have that d β(ϕ)(0) = B ( ϕ A (E)). Then d β(ϕ)(0) B (E ) = ϕ A (E) A (E) c R. Therefore, β(ϕ) Exp B (E ). The Bore transform β is inective as a consequence of Coroary 2.2. To see that it is aso surective, et ψ Exp B (E ) and A = sup d ψ(0) B (E ). For each g H ba(e), we define Since =0 ϕ(g) = =0 B ( d ψ(0) )( d g(0)! ϕ(g) d ψ(0) B (E ) d g(0)! A d g(0) A (E)! = p (g), A A (E) we have ϕ H ba (E). Finay, simpe computations show that β(ϕ) = ψ. =0 ). 3. Mutipication in H ba (E). Suppose that B(E) = {B (E)} is a coherent mutipicative sequence. Then, we can see that Exp B (E) is an agebra. Indeed, if f, g Exp B (E), with A = sup d f(0) B (E) and A 2 = sup d g(0) B (E), we have d (fg)(0) B (E) and d (fg)(0) B (E) M(A + A 2 ), where M is the mutipicative constant of the sequence B(E) (see the proof of Lemma.5 for some detais). This fact aows us to introduce a mutipication on H ba (E), ust copying the product in Exp B (E) via the Bore transform: Definition 3.. Let {B (E )} be a coherent mutipicative sequence and et {A (E)} be such that A (E) = B (E ) for a. For ϕ, ψ H ba (E) we define the product in H ba (E), by ϕ ψ = β ( β(ϕ)β(ψ) ). We woud aso ie to define the natura convoution product in H ba (E) : if ϕ, ψ H ba (E), then ϕ ψ H ba (E) is the inear functiona given by ϕ ψ(f) = ψ(x ϕ(τ x f)). We prove that this convoution product is we defined in a few steps. First we define: Definition 3.2. Let ϕ H ba (E) and et f H ba (E). We define ϕ f : E C as foows, ϕ f(x) = ϕ(τ x f) = ϕ(f(x + )). Therefore, the desired convoution product can be rewritten as ϕ ψ(f) = ψ(ϕ f). For this to be we defined, we must show that ϕ f H ba (E) and that f ϕ f is continuous. This wi be done in the foowing emma and theorem.

8 8 DANIEL CARANDO, VERÓNICA DIMANT, AND SANTIAGO MURO Lemma 3.3. Let {B (E )} be a coherent mutipicative sequence and suppose A (E) = B (E ) for every. Let P A (E) and ϕ A (E),. Then the -homogeneous poynomia x ϕ(p x ) beongs to A (E) and x ϕ(p x ) A (E) M ϕ A (E) P A (E). Proof. If P is a finite type poynomia then x ϕ(p x ) is aso a finite type poynomia and thus beongs to A (E). We can therefore define a inear operator ( ) T : Pf (E), A (E) A (E) P = [ ] N = γ x ϕ(p x ). If P = N = γ and ψ A (E) then ψ(t (P )) = Then, for every ψ A (E), N = (B (ϕ)b (ψ))(γ ) = B (B (ϕ)b (ψ))(p ). ψ(t (P )) M ϕ A (E) ψ A (E) P A (E). Therefore, T is continuous and, by Lemma 2., can be extended to A (E). By density, it easiy foows that T (P )(x) = ϕ(p x ) for every x E and every P A (E). Theorem 3.4. Let {B (E )} be a coherent mutipicative sequence and {A (E)} be such that A (E) = B (E ) for every. Let f H ba (E) and ϕ H ba (E). Then ϕ f beongs to H ba (E) and the appication T ϕ : H ba (E) H ba (E) f ϕ f is a continuous inear operator. Proof. If f = =0 P, P A (E), then ( ϕ f(x) = ϕ(τ x f) = ϕ =0 =0 ( ) ) (P ) x = =0 = ( ) ϕ ( (P ) x ). From Lemma 3.3, the poynomia x ϕ ( (P ) x ) beongs to A (E). Aso, proceeding as in the beginning of the proof of Proposition 2.7, we have that there exist constants c, R > 0 such that A cr. Let ε > 0. Since f H (E) ba (E), there exists c ε > 0 such that P A (E) ϕ A (E) c ε ε for a. Then d (ϕ f)(0)! beongs to A (E) because d (ϕ f)(0)! (x) = ( ( = ) ϕ (P ) x ) and, by Lemma 3.3, ( ) x ϕ(p x ) ( ) A(E) M ϕ A (E) A P A (E) (E) = = ( ) c M R P A (E) = ( ) c c ε (Mε) (MRε) = = c c ε (Mε) <, ( MRε) +

9 HYPERCYCLIC CONVOLUTION OPERATORS ON FRÉCHET SPACES OF ANALYTIC FUNCTIONS 9 for arbitrary sma ε > 0. Moreover, since d (ϕ f)(0)! (c c ε) Mε A (E) ( MRε) + Mε MRε, for every ε sufficienty sma, it foows that ϕ f H ba (E). Finay, the appication T ϕ is continuous because p r (ϕ f) = d (ϕ f)(0) ( )! A (E) r c r M R P A (E) = c =0 P A (E) M =0 = cp M(R+r) (f). =0 =0 = ( ) r R = c P A (E) M ( R + r ) =0 We can now define the foowing: Definition 3.5. For ϕ, ψ H ba (E), the convoution product ϕ ψ H ba (E) is defined by for f H ba (E). As a consequence of Theorem 3.4 we have ϕ ψ(f) = ψ(x ϕ(τ x f)) = ψ(ϕ f), Coroary 3.6. If ϕ H ba (E), then the appication M ϕ : H ba (E) H ba (E) ψ ψ ϕ is a continuous inear operator when we consider the strong dua topoogy on H ba (E). Proof. Just note that M ϕ is the transpose of T ϕ. Next we show that the two products defined on H ba (E) are actuay the same. Proposition 3.7. Let {B (E )} be a coherent mutipicative sequence and {A (E)} such that A (E) = B (E ) for every. Let ϕ, ψ H ba (E). Then ϕ ψ = ϕ ψ. Proof. Since finite type poynomias are dense in H ba (E), it is sufficient to verify that, for each γ E and 0, ϕ ψ(γ ) = ϕ ψ(γ ). For g Exp B (E ), β (g)(γ ) = B ( d g(0) ) (γ ) = d g(0)(γ). Then, ϕ ψ(γ ) = β (β(ϕ)β(ψ))(γ ) = d ( β(ϕ)β(ψ) ) (0)(γ) = ( ) d ( β(ϕ) ) (0)(γ)d ( β(ψ) ) (0)(γ) = =0 =0 ( ) ϕ(γ )ψ(γ ). On the other hand, since (ϕ γ )(x) = ϕ(τ x γ ) = ( =0 ) ϕ(γ )γ(x), we obtain ϕ ψ(γ ) = ψ ( ϕ γ ) ( ) = ϕ(γ )ψ(γ ) = ϕ ψ(γ ). =0

10 0 DANIEL CARANDO, VERÓNICA DIMANT, AND SANTIAGO MURO As an immediate consequence, we have: Coroary 3.8. The convoution product in H ba (E) is commutative. 4. Convoution operators and hypercycicity. As usua, by a convoution operator T, we mean a continuous operator that commutes with transations τ x, i.e., T τ x = τ x T. Next we characterize the convoution operators on H ba (E). Proposition 4.. Let {B (E )} be a coherent sequence and {A (E)} such that A (E) = B (E ) for every. Then for every convoution operator T : H ba (E) H ba (E) there exists a unique functiona ϕ H ba (E) such that T (f) = ϕ f, for every f H ba (E). If, in addition, {B (E )} is mutipicative, then for every ϕ H ba (E), T ϕ is a convoution operator on H ba (E). Proof. Let T : H ba (E) H ba (E) be a convoution operator and et ϕ = δ 0 T, i.e. ϕ(f) = T (f)(0) for f H ba (E). Then ϕ H ba (E) and T (f)(x) = [ τ x T (f) ] (0) = T (τ x f)(0) = ϕ(τ x f) = ϕ f(x), for every f H ba (E) and x E. The uniqueness of ϕ foows from the identity () T (e γ ) = ϕ e γ = [ x ϕ(τ x e γ ) ] = ϕ(e γ )e γ = β(ϕ)(γ)e γ and the fact that β is one to one. The second assertion is immediate by Theorem 3.4. Proposition 4. and the resuts in the previous sections aow us to estabish the foowing: Coroary 4.2. Let {B (E )} be a coherent mutipicative sequence and et {A (E )} be such that A (E) = B (E ) for every. Then ψ T β (ψ) is an agebra isomorphism from Exp B (E ) onto the agebra of convoution operators on H ba (E). Now we are ready to prove the announced resut about hypercycicity of convoution operators. We foow the steps of the proof of [26, Theorem 3.]. Theorem 4.3. Suppose that E is separabe. Let {B (E )} be a coherent sequence and {A (E)} be such that A (E) = B (E ) for every. Then, every convoution operator T : H ba (E) H ba (E) which is not a scaar mutipe of the identity is hypercycic. Proof. Let ϕ H ba (E) be the inear functiona given in Proposition 4., which satisfies T (f) = ϕ f. Since T is not a scaar mutipe of the identity, it foows that ϕ is not a scaar mutipe of δ 0. Since E is separabe, by Coroary 2.2, H ba (E) is separabe. Therefore, we can use the Hypercycicity Criterion [7, 22]. First, note that span{e γ : γ U} is dense in H ba (E) for any nonempty open set U E. Indeed, if ψ H ba (E) and ψ(e γ ) = 0 for every γ U, then β(ψ) 0 in U and we have β(ψ) = 0. This means that ψ is 0. Aso, the fact that ϕ is not a scaar mutipe of δ 0 impies that β(ϕ) is not a constant function. Indeed, if β(ϕ) was constant then λ = ϕ() = β(ϕ)(0) = β(ϕ)(γ) = ϕ(e γ ) for a γ E. But, on the other hand, λ = λδ 0 (e γ ) for a γ E and we woud have that ϕ = λδ 0. We wi now prove that T satisfies the Hypercycicity Criterion. Let V = {γ E : β(ϕ)(γ) < } and W = {γ E : β(ϕ)(γ) > }.

11 HYPERCYCLIC CONVOLUTION OPERATORS ON FRÉCHET SPACES OF ANALYTIC FUNCTIONS Then V, W E are open sets, and they are nonempty. Indeed, if W = (V = ) then β(ϕ) ) woud be a nonconstant bounded entire function. Let ( β(ϕ) H V (E) = span{e γ, γ V } and H W (E) = span{e γ, γ W }. As we have observed, H V (E) and H W (E) are dense in H ba (E). For γ V, T (e γ ) = β(ϕ)(γ)e γ (see ()). Then T (H V (E)) H V (E). Aso, T n (e γ ) = β(ϕ)(γ) n e γ, and since β(ϕ)(γ) < for γ V, we obtain that T n (f) 0, for every f n H V (E). e γ For γ W, et S(e γ ) = β(ϕ)(γ). Since {eγ, γ W } is ineary independent (see the proof of [, Lemma 2.3]), we can ineary extend S to H W (E). Then S(H W (E)) H W (E) and e γ S n (e γ ) = β(ϕ)(γ) n. Thus Sn (f) 0, for every f H W (E), since β(ϕ)(γ) > for γ W. n ( Finay, T S(e γ e γ ) ) = T = e γ and therefore T Sf = f for a f H W (E). β(ϕ)(γ) By the Hypercycicity Criterion, T is hypercycic. By Theorem 3.4, if B(E ) is a coherent mutipicative sequence, each ϕ H ba (E) defines a convoution operator f ϕ f. As mentioned in the previous proof, this operator is not a scaar mutipe of the identity uness ϕ is a scaar mutipe of δ 0. Therefore, we have: Coroary 4.4. Suppose that E is separabe. Let {B (E )} be a coherent mutipicative sequence and {A (E)} such that A (E) = B (E ) for every. For every ϕ H ba (E) which is not a scaar mutipe of δ 0, the operator is hypercycic. T ϕ : H ba (E) H ba (E) f T ϕ (f) = ϕ f Now we appy the previous resuts to different spaces of hoomorphic functions. Exampe 4.5. In [] the authors study differentiation operators in H bc (E), the space of hoomorphic functions of compact bounded type on E (that is: f = P n H bc (E) whenever each P n is an approximabe n-homogeneous poynomia and P n n 0, where denotes the usua norm). They show that if the differentiation operator is constructed from an entire function of exponentia type on C, then it is hypercycic. By Exampes 2.3, this resut is a particuar case of Coroary 4.4. Indeed, every such differentiation operator in H ba (E) is a convoution operator: if Φ(z) = c n z n is an exponentia type function and a E, we define h(γ) = c n γ(a) n. Then, h Exp B (E ) and β (h)(f) = c n d n f(0)(a) for a f H ba (E). Therefore, β (h) f(x) = h(τ x f) = c n (d n τ x f)(0)(a) = c n d n f(x)(a). Exampe 4.6. Consider the space H Nb (E) of nuceary entire functions of bounded type. If E is separabe with the approximation property, Exampes 2.3 and Proposition 4. assert that the convoution operators on H Nb (E) are precisey the operators T ϕ, ϕ H Nb (E). Such an operator is hypercycic whenever it is not a scaar mutipe of the identity by Theorem 4.3. This ast statement answers a question of Aron and Marose in [2]. For E a dua Banach space and a sighty different definition of nucear poynomias, Petersson obtained a stronger version of this resut [27]. Exampe 4.7. Let {A } be a sequence of minima ideas. If E has the bounded approximation property, A (E) can, by Remar 2.4, be identified with B (E ) = A dua (E ). Therefore, if E is aso separabe and {B (E )} is coherent and mutipicative, the convoution operators on

12 2 DANIEL CARANDO, VERÓNICA DIMANT, AND SANTIAGO MURO H ba (E) are those of the form T ϕ, ϕ H ba (E). They are hypercycic if they are not scaar mutipes of the identity. For exampe, we can tae A to be the minima idea associated to the tensor norm η [2, 7]. In this case, {B } is the coherent mutipicative sequence of extendibe poynomias. In the next section, we present other exampes in which the hypotheses of Theorem 4.3 and Coroary 4.4 hod. Namey, we consider the hoomorphic functions of bounded type associated to the Schatten-von Neumann poynomias in the sense of Cobos, Kühn and Peetre []. 5. Hoomorphic Schatten-von Neumann functions of bounded type. Suppose H is a separabe Hibert space. Let us first reca the definition of Hibert-Schmidt n-homogeneous poynomias on H, which wi be denoted S2 n (H). For finite type poynomias it is possibe to define an inner product in the foowing way: if y, z H,, y n,, z n = z, y n. Then S2 n(h) is the competion of the space of finite type poynomias Pn f (H) with this inner product. Note that if {e i } i is an orthonorma basis of H and P, Q S2 n (H), then P, Q = ˇP (e i,..., e in ) ˇQ(e i,..., e in ). i,...,i n= Aso note that the Bore transform is an isometric isomorphism between (S n 2 (H)) and S n 2 (H ). Cobos, Kühn and Peetre in [] define Schatten-von Neumann casses of mutiinear functionas on H. We adapt their definition to homogeneous poynomias on H. To this end, throughout this section we wi denote by S n (H) and Sn (H) the spaces of n-homogeneous nucear and approximabe poynomias on H, respectivey. Since H has the approximation property and the Radon-Niodym property, the Bore transform is an isometric isomorphism between (S n (H)) and P n (H ), and aso between (S n (H)) and S n (H ). We use the compex interpoation method [6, 4] to define Schatten poynomias. Foowing [], we define: Definition 5.. The Schatten-von Neumann p-cass of n-homogeneous poynomias on H is defined as S n p (H) = [S n (H), S n (H)] θ, with p = θ and 0 < θ <. Here, [Sn (H), Sn (H)] θ denotes the space obtained by compex interpoation from the pair (S n (H), Sn (H)), with parameter θ. The foowing resut, which is the poynomia version of [, Theorem 3.] and can be proved anaogousy, shows that this definition is consistent with the definition of S n 2 (H). Proposition 5.2. We have the foowing isometric isomorphisms [S n (H), P n (H)] /2 = [S n (H), S n (H)] /2 = S n 2 (H). For the rea method of interpoation the previous proposition hods with equivaent norms (see []). In the proof of [, Theorem 4.5], the refexivity of S n p (H) is proven. In fact, this can be seen as a consequence of the foowing resut: Proposition 5.3. If < p, q < are such that p + q isometric isomorphism between (Sp n (H)) and Sq n (H ) =, then the Bore transform is an Proof. We now that the statement hods when p = 2. Next, assume < p < 2. By the Reiteration Theorem [4, 4.6.] for the compex method, S n p (H) = [S n (H), S n (H)] θ = [S n (H), S n 2 (H)] η,

13 HYPERCYCLIC CONVOLUTION OPERATORS ON FRÉCHET SPACES OF ANALYTIC FUNCTIONS 3 where θ = η 2. Then Sn p (H) = [S n (H), Sn 2 (H)] 2θ. In the foowing two cases, the Bore transform is an isomorphism B n : (S n (H)) P n (H ) and, B n : (S n 2 (H)) S n 2 (H ). Then B n : [(S n(h)), (S2 n(h)) ] 2θ [P n (H ), S2 n(h )] 2θ is an isomorphism. Since S2 n (H) is refexive, and by a duaity theorem [4, Coroary ], we have [(S n (H)), (S2 n (H)) ] 2θ = [S n (H), S2 n (H)] 2θ = (Sn p (H)). On the other hand, [P n (H ), S n 2 (H )] 2θ = [S n (H ), S n 2 (H )] 2θ = [S n (H ), S n (H )] ν, with ν = 22θ + ( 2θ) = θ (the first equaity foows from [4, Theorem ] and the ast one from the Reiteration Theorem). Therefore, B n : (Sp n (H)) [S n(h ), S (H n )] ν = Sq n (H ) is an isomorphism, with q = ν = θ, that is, q = p. For the case 2 < p <, we have S n p (H) = [S n (H), S n (H)] θ = [S n 2 (H), S n (H)] η, where η = 2θ. We proceed anaogousy to obtain the desired resut. Coroary 5.4. For < p <, the Schatten-von Neumann casses S n p (H) are refexive. In [0] it is shown that interpoation of coherent sequences is coherent. We wi now show that interpoation of mutipicative coherent sequences is mutipicative. Proposition 5.5. Let E be Banach space and et {A 0 (E)}, {A (E)} be coherent mutipicative sequences (with constants M 0 and M, respectivey). Then, the sequence {A θ (E)} is mutipicative, where A θ (E) = [A0 (E), A (E)] θ, for every 0 < θ < (with constant M θ 0 M θ ). Proof. Since {A (E)} are mutipicative, for = 0,, we can define a continuous biinear mapping Φ, : A (E) A (E) A + (E) (P, Q) P Q. It foows that Φ, M +. Then, by the Mutiinear Interpoation Theorem [4, Theorem 4.4..], (P, Q) P Q defines a mapping Φ θ, : Aθ (E) Aθ (E) Aθ + (E), which is continuous and has norm ess than or equa to ( M0 θ M θ ) +. That is, if P A θ (E), Q A θ (E), then P Q Aθ + (E) and From Propositions 5.3 and 5.5 we have P Q A θ + (E) ( M θ 0 M θ ) + P A θ (E) Q A θ (E). Coroary 5.6. For every p, the sequence of Schatten-von Neumann p-casses of homogeneous poynomias {S p (H)} is mutipicative. Let H bsp (H) be the hoomorphic functions of bounded type of the Schatten-von Neumann p-cass on H, then by Theorems 3.4 and 4.3, we have Coroary 5.7. Let H be separabe Hibert space. For p, every convoution operator on H bsp (H) which is not a scaar mutipe of the identity is hypercycic. Moreover, each ϕ H bsp (H) defines a convoution operator T ϕ.

14 4 DANIEL CARANDO, VERÓNICA DIMANT, AND SANTIAGO MURO The case p = 2 of this resut (Hibert-Schmidt hoomorphic functions of bounded type) was proved by Petersson in [26]. Acnowedgements 5.8. We woud ie to than Pabo Sevia-Peris for teaching us interpoation theory during his stay in Buenos Aires in We aso want to express our gratitude to the referee for his/her usefu comments and for suggesting us the statement of Coroary 4.2. References [] Aron, Richard; Bès, Juan. Hypercycic differentiation operators. Function spaces (Edwardsvie, IL, 998), 39 46, Contemp. Math., 232, Amer. Math. Soc., Providence, RI, 999. [2] Aron, Richard; Marose, Dinesh. On universa functions. Sateite Conference on Infinite Dimensiona Function Theory. J. Korean Math. Soc. 4 (2004), no., [3] Bergh, Jöran; Löfström, Jörgen. Interpoation spaces. An introduction. Grundehren der Mathematischen Wissenschaften, No Springer-Verag, Berin-New Yor, 976. [4] Boyd, Christopher; Lasae, Sivia. Persona communication. [5] Birhoff, George D. Démonstration d un théorème éémentaire sur es fonctions entières. C. R. Acad. Sci. Paris 89 (929), [6] Caderón, Aberto P. Intermediate spaces and interpoation, the compex method. Stud. Math. 24 (964), [7] Carando, Danie. Extendibe poynomias on Banach spaces. J. Math. Ana. App. 233 (999), no., [8] Carando, Danie. Extendibiity of poynomias and anaytic functions on p. Studia Math. 45 (200), no., [9] Carando, Danie; Dimant, Verónica; Muro, Santiago. Coherent sequences of poynomia ideas on Banach spaces. Preprint. [0] Carando, Danie; Dimant, Verónica; Muro, Santiago. Hoomorphic Functions and Coherent poynomia ideas on Banach spaces (wor in progress). [] Cobos, Fernando; Kühn, Thomas; Peetre, Jaa. Schatten-von Neumann casses of mutiinear forms. Due Math. J. 65 (992), no., [2] Dimant, Verónica; Gaindo, Pabo; Maestre, Manue; Zaduendo, Ignacio. Integra hoomorphic functions. Studia Math. 60 (2004), no., [3] Dineen, Seán. Hoomorphy types on a Banach space. Studia Math. 39 (97), [4] Dineen, Seán. Compex anaysis on infinite-dimensiona spaces. Springer Monographs in Mathematics. Springer-Verag London, Ltd., London, 999. [5] Foret, Kaus. Minima ideas of n-homogeneous poynomias on Banach spaces. Resut. Math. 39 (200), no. 3-4, [6] Foret, Kaus. On ideas of n-homogeneous poynomias on Banach spaces. Proceedings of the Fest-Cooquium in honour of Professor A. Maios, University of Athens (2002), [7] Gethner, Robert M.; Shapiro, Joe H. Universa vectors for operators on spaces of hoomorphic functions. Proc. Amer. Math. Soc. 00 (987), no. 2, [8] Godefroy, Gies; Shapiro, Joe H. Operators with dense, invariant, cycic vector manifods. J. Funct. Ana. 98 (99), no. 2, [9] Grosse-Erdmann, Kar-Goswin. Universa famiies and hypercycic operators. Bu. Amer. Math. Soc. (N.S.) 36 (999), no. 3, [20] Gupta, Chaitan P. On the Magrange theorem for nuceary entire functions of bounded type on a Banach space. Indag. Math. 32 (970), [2] Kirwan, Pádraig; Ryan, Raymond A. Extendibiity of homogeneous poynomias on Banach spaces. Proc. Amer. Math. Soc. 26 (998), no. 4, [22] Kitai, Caro. Invariant cosed sets for inear operators, Dissertation, Univ. Toronto, 982. [23] MacLane, Gerad R. Sequences of derivatives and norma famiies. J. Anayse Math. 2 (952), [24] Muica, Jorge. Compex Anaysis in Banach Spaces, Math. Studies 20, North-Hoand, Amsterdam (986). [25] Nachbin, Leopodo. Topoogy on spaces of hoomorphic mappings. Springer-Verag, New Yor, 969. [26] Petersson, Henri. Hypercycic convoution operators on entire functions of Hibert-Schmidt hoomorphy type. Ann. Math. Baise Pasca 8 (200), no. 2, [27] Petersson, Henri. Hypercycic subspaces for Fréchet space operators. J. Math. Ana. App. 39 (2006), no. 2,

15 HYPERCYCLIC CONVOLUTION OPERATORS ON FRÉCHET SPACES OF ANALYTIC FUNCTIONS 5 Departamento de Matemática - Pab I, Facutad de Cs. Buenos Aires, (428) Buenos Aires, Argentina. E-mai address: dcarando@dm.uba.ar Exactas y Naturaes, Universidad de Departamento de Matemática, Universidad de San Andrés, Vito Dumas 284, (B644BID) Victoria, Buenos Aires, Argentina. E-mai address: vero@udesa.edu.ar Departamento de Matemática - Pab I, Facutad de Cs. Buenos Aires, (428) Buenos Aires, Argentina. E-mai address: smuro@dm.uba.ar Exactas y Naturaes, Universidad de