Hypercyclic algebras for D-multiples of convolution operators

HYPERCYCLIC ALGEBRAS FOR D-MULTIPLES OF CONVOLUTION OPERATORS LUIS BERNAL-GONZ´ ALEZ AND MAR´ IA DEL CARMEN CALDER´ ON-MORENO Abstract. It is shown in this short note the existence, for each nonzero member of the ideal of D-multiples of convolution operators acting on the space of entire functions, of a scalar multiple of it supporting a hypercyclic algebra. 1. Introduction Let us consider the vector space L(X) of all operators on X, that is, the family of all continuous linear self-mappings T:X→X, where Xis a Hausdorff topological vector space. An operator T∈L(X) is said to be hypercyclic provided that there is some vector x0∈X(called hypercyclic for T) such that the orbit {Tnx0:n∈N}of x0under Tis dense in X(N:= {1,2, . . . }). The set of hypercyclic vectors for Twill be denoted by HC(T). In this short paper, we are concerned with the algebraic size of HC(T) when Truns over an important class of operators on the space of entire functions. For background on hypercyclic operators we refer the reader to the excellent books [5,18]. An account of concepts and results about algebraic structures inside nonlinear sets can be found in [2]. It is well known that if Xis an F-space (i.e., complete and metrizable, and hence Baire) and Tis a hypercyclic operator then the set HC(T) is residual, that is, it contains a dense Gδsubset of X; we can say that HC(T) is topologically large. Furthermore, for any topological vector space and any hypercyclic T∈L(X), the family HC(T) is algebraically large; specifically, it contains, except for 0, a dense (even T-invariant) vector subspace of X(see [8,14,19,26]). A number of criteria have been established for an operator T∈L(X) to support a closed infinite dimensional vector subspace all of whose nonzero vectors are T-hypercyclic on an F-space X; however, not every hypercyclic operators supports such a subspace (see [5, Chapter 8] and [18, Chapter 10]). If Gis a nonempty open subset of C, the symbol H(G) will stand for the class of all holomorphic functions in G. We are mainly interested in operators defined on the space H(C) of entire functions C→C. It is well known that H(C) becomes an F-space when endowed with the topology of uniform convergence on 2010 Mathematics Subject Classification. 30E10, 30H50, 30K20, 46E10, 47A16. Key words and phrases. hypercyclic operator, convolution operator, derivative operator, convexity, algebrability. 1 2 BERNAL AND CALDER ´ ON compacta. An operator T∈L(H(C)) is said to be a convolution operator provided that it commutes with translations, that is, T◦τa=τa◦Tfor all a∈C, where τaf:= f(·+a). Then T∈L(H(C)) happens to be a convolution operator if and only if Tcommutes with the derivative operator D:f7→ f0, and if and only if it is an infinite order linear differential operator with constant coefficients T= Φ(D), where Φ is an entire function with finite exponential type, that is, there exist constants A, B ∈(0,+∞) such that |Φ(z)| ≤ AeB|z|for all z∈C. For such a function Φ(z) = Pn≥0anznand f∈H(C), we have Φ(D)f=P∞ n=0 anf(n). Godefroy and Shapiro [16] proved that every non-scalar convolution operator is hypercyclic, so covering the classical Birkhoff [13] and MacLane [21] results on hypercyclicity of the translation operator τa(take Φ(z) = eaz,a6= 0) and of D (take Φ(z) = z), respectively. For every non-scalar convolution operator T, the set HC(T) contains, except for 0, a closed infinite dimensional vector subspace of H(C): see [23–25] and also [2, Section 4.5] and [18, Section 10.1]. Therefore, many linear combinations of hypercyclic functions with respect to a convolution operator are still hypercyclic. But things seem to be more disappointing with respect to multiplication of hypercyclic functions. Let us give an overview on the short history of this topic. Firstly, we have that for every a6= 0, every f∈HC(τa) and every n∈Nwith n≥2, the n-power fn6∈ HC(τa) (see [3, Cor. 2.4]). As a positive result, it was proved in [3, Th. 2.3] (see also [4]) the existence of a function f∈H(C) –in fact, of a residual subset of them– such that fn∈HC(D) for all n∈N. Going into this area in more depth, it has been recently shown in [7] the existence (for any prescribed nonconstant Φ∈H(C) of subexponential type, meaning that for given ε > 0 there is a constant A=A(ε)∈(0,+∞) such that |Φ(z)| ≤ A eε|z|for all z∈C) of an infinitely generated multiplicative group consisting, except for the constant function 1, of non-vanishing Φ(D)-hypercyclic entire functions. The mentioned result in [3, Th. 2.3] was extended by Bayart and Matheron who proved that there is even a residual set of functions in H(C) generating a hypercyclic algebra for the derivative operator, that is, every non-null function in one of these algebras is D-hypercyclic [5, Th. 8.26]. Recall that the algebra generated by a function fis nothing but the set {Q◦f:Qpolynomial, Q(0) = 0}. The existence of (one-generated) algebras of hypercyclic functions for Dwas also independently proved by Shkarin in [25] and extended –also with residuality of the set of generating functions– by B`es, Conejero and Papathanasiou in [9] to operators of the form P(D), where Pis a nonconstant polynomial with P(0) = 0. Recently, important advances have been carried out by the last three mentioned authors in [10,12], but let us first recall the notion of algebrability. If Xis a (linear) algebra then a subset A⊂Xis called algebrable whenever A∪ {0}contains a subalgebra that is not finitely generated, and provided that Xis a Fr´echet algebra, Ais called dense algebrable if such infinitely generated subalgebra can be taken HYPERCYCLIC ALGEBRAS FOR D-MULTIPLES OF CONVOLUTION OPERATORS 3 dense in X. For any A⊂Cthe symbol conv(A) denotes its convex hull. If S⊂C is an arc, then Sis said to be strictly convex if for each pair of distinct z, w ∈S one has S∩conv({z, w}) = {z, w}. As usual, the open unit disc {z:|z|<1} will be denoted by D, and the space H(C)Nis assumed to be endowed with the product topology. Theorem 1.1. Let Φ∈H(C)be of finite exponential type. (a) ([10]) Assume that the level set {z:|Φ(z)|= 1}contains a nontrivial, strictly convex compact arc Γsuch that conv(Γ∪{0})\(Γ∪{0})⊂Φ−1(D). Then the set of entire functions that generate a hypercyclic algebra for Φ(D)is residual in H(C). (b) ([12]) Assume that |Φ(0)|<1and that the level set {z:|Φ(z)|= 1} contains a nontrivial, strictly convex compact arc Γsuch that conv(Γ ∪ {0})\Γ⊂Φ−1(D). Then the set of sequences f= (fj)∞ j=1 that generate a dense hypercyclic algebra for Φ(D)that is not contained in a finitely generated hypercyclic algebra for Φ(D)is residual in H(C)N. In particular, the set HC(Φ(D)) is dense algebrable. As applications of the last theorem, it is proved in [10] or [12] that, provided a6= 0,|b| ≤ 1,0< c ≤1,|d|<1 and k, n ∈N, the operators cos(aD), sin(aD), (aDk+b)n,D eD,eD−cI support hypercyclic algebras, while sin(aD), cos(D) and aDk+dsupport dense, infinitely generated hypercyclic algebras. With the aim of contributing to shedding light on this research line, and taking advantage of Theorem 1.1, we prove in this short note that every D-multiple of a nonzero convolution operator possesses some scalar multiple supporting a hypercyclic algebra. 2. D-multiples of convolution operators and hypercyclic algebras Geometrical properties of analytic functions play a central role in the results of [10,12]. We recall in the next lemma a convexity property due to R. Nevanlinna, which can be found, for instance, in [17, Theorem 2.33]. Lemma 2.1. If f∈H(D)is one-to-one and satisfies f0(0) = 1 and f(0) = 0 then the set f(rD)is a convex domain for every r∈(0,2−√3). Let us denote by Lcvl(H(C)) the ring of all convolution operators on H(C), and by DLcvl(H(C)) the ideal generated by the differentiation operator D, that is, DLcvl(H(C)) := {DS :S∈Lcvl(H(C))}. Next, we state and prove our result. We denote by Tthe unit circle {z∈C: |z|= 1}. Theorem 2.2. Assume that T∈DLcvl(H(C)) \{0}. Then there is ρ > 0such that, for every λ∈Cwith |λ|> ρ, the set HC(λ T)is dense algebrable. 4 BERNAL AND CALDER ´ ON Proof. We are going to apply Theorem 1.1. By assumption, there is a convolution operator Ssuch that T=DS. Therefore, there exists a nonzero entire function Ψ(z) with finite exponential type such that S= Ψ(D). If Φ(z) := zΨ(z), then we get T= Φ(D), Φ has finite exponential type, Φ 6= 0 and Φ(0) = 0. Let N∈N be the order of the zero of Φ at the origin or, that is the same, N= min{n∈N: Φ(n)(0) 6= 0}. According to the local representation theorem (see, e.g., [1, Section 3.3]), there exist α > 0 as well as a one-to-one function ϕ:αD→Csuch that ϕN= Φ on αD. Note that ϕ(0) = 0. Consider the inverse mapping ψ:ϕ(αD)→αDof ϕ. Since the image ϕ(αD) is an open subset of Ccontaining 0, there exists β > 0 such that βD⊂ϕ(αD). Then ψ∈H(βD), ψ(0) = 0, ϕ◦ψ(z) = z(z∈βD) and ψis one-to-one on βD. In particular, ψ0(0) 6= 0. Define the function γ:D→Cas γ(z) := ψ(βz) β ψ0(0). Note that γ∈H(D), γ(0) = 0, γ0(0) = 1 and γis one-to-one on D. According to Lemma 2.1, the set γ(rD) is a convex domain for each r∈(0,2−√3). But this is equivalent to the fact that κ(P)>0, κdenoting the signed curvature at the points Pof the curve Γr:= γ(rT), where Tis oriented counterclockwise (see [17, Remark on p. 166]). If P= (x(t), y(t)) (t∈R) is a (2π-periodic) smooth parametrization of Γr(obtained, for instance, via composition of γwith (rcos t, r sin t)), then κ(P) = x0(t)y00(t)−y0(t)x00(t) ((x0(t))2+ (y0(t))2)3/2, which is a continuous function of P. Therefore, a segment joining two points P, Q in Γrcan intersect this curve only at P, Q (otherwise κwould be ≤0 at some point of Γr). In other words, each curve Γris strictly convex. But neither dilations nor rotations alter the strict convexity of an arc as well as the convexity of a domain, so that all curves ψ(β rT) are strictly convex and all domains ψ(β rD) (0 < r < 2−√3) are convex. Equivalently, all curves Ψr:= ψ(rT) are strictly convex and all domains ψ(rD) (0 < r < (2 −√3)β) are convex. Define ρ:= ((2 −√3)β)−N. Fix λ∈Cwith |λ|> ρ and set Φλ:= λΦ. Note that λ T = Φλ(D) and |Φλ(0)|= 0 <1. Consider the level set L:= {z∈C:|Φλ(z)|= 1} and the curve Γ := Ψ|λ|−1/N . This curve is a strictly convex compact arc, and if z∈Γ then |ϕ(z)|=|λ|−1/N , hence |Φλ(z)|=|λ|·|ϕ(z)|N=|λ|·|λ|−1= 1. Therefore z∈ L, so Γ ⊂ L. Denote, as usual, by ∂A and Athe boundary and HYPERCYCLIC ALGEBRAS FOR D-MULTIPLES OF CONVOLUTION OPERATORS 5 the closure, respectively, of a set A⊂C. From the maximum modulus principle, we get Γ = ∂ψ(|λ|−1/N D) and ψ(|λ|−1/N D) = Γ ∪ψ(|λ|−1/N D)⊂ {z∈C:|ϕ(z)|≤|λ|−1/N }. Finally, take z0∈conv(Γ ∪{0})\Γ. Since ψ(|λ|−1/N D) is a convex set containing Γ∪{0}, we get |ϕ(z0)|≤|λ|−1/N . But z06∈ Γ, so |ϕ(z0)|<|λ|−1/N and |Φλ(z0)|= |λ| · |ϕ(z0)|N<|λ| · |λ|−1= 1. Consequently, we obtain conv(Γ ∪ {0})\Γ⊂ (Φλ)−1(D), and an application of Theorem 1.1(b) yields the desired conclusion.  Remarks 2.3. 1. Recall that an operator Ton a Hausdorff topological space X is said to be supercyclic whenever there is a vector x0∈X–called supercyclic for T– whose projective orbit {λTnx0:n∈N}is dense in X. The set of supercyclic vectors for Tis denoted by SC(T). Observe that Theorem 2.2 also shows, in particular, that SC(T) is dense algebrable for every T∈DLcvl(H(C)) \{0}. 2. Part(b) of Theorem 1.1 tells us that, in fact, there are –in the topological sense– many algebraically independent sequences of functions generating algebras contained, except for 0, in the sets HC(λ T) of the conclusion of our theorem. 3. Note that the number ρin Theorem 2.2 is obtained constructively in the proof. Specifically, we can take ρ= ((2 −√3)β)−N, where βis the radius of a disc with center at 0 where the inverse of a local holomorphic branch of the Nth-root of Φ is defined, and Nis the valence of Φ at the origin. This allows us to exhibit explicit examples. For instance, the operator T:f∈H(C)7−→ µf0+f0(·+ 1) ∈H(C), where |µ| ≥ 3e, supports hypercyclic algebras. In order to prove this, we first notice that T= Φ(D) with Φ(z) = µz +zez. Then Φ(0) = 0. Also, Φ0(0) = µ+ 1 6= 0, so that N= 1. If z1, z2are distinct points of Dand Lis the segment joint z1to z2, then by assuming Φ(z1) = Φ(z2) we get µ|z1−z2|>2e|z1−z2|=ZL 2e|dz| ≥ ZL|ez+zez||dz|=ZL|Φ0(z)−µ||dz| ≥ZL (Φ0(z)−µ)dz =|(Φ(z2)−µz2)−(Φ(z1)−µz1)|=µ|z1−z2|, which is absurd. Hence Φ(z1)6= Φ(z2) and Φ is one-to-one on αD, where α= 1. Given w∈Cwith |w|<2.10 + √3, we have for all z∈Tthat |(Φ(z)−w)−(µz −w)|=|Φ(z)−µz|=|zez|< e|z|=e <3e−(2.10 + √3) <|µ|−|w|≤|µz −w|. According to Rouch´e’s theorem (see, e.g., [1]), and taking into account that the value wis reached exactly once by µz in the geometric interior Dof the Jordan curve T(namely, at z=w/µ), we deduce that wis reached exactly once by Φ in 6 BERNAL AND CALDER ´ ON D, so that we can take as β, ρ the respective quantities β= 2.10 + √3, ρ = ((2 −√3)(2.10 + √3))−1<((2 −√3)(2 + √3))−1= 1. The selection λ:= 1 in Theorem 2.2 ensures the existence of a dense infinitely generated algebra contained in HC(T). In the next and final remark, we will make use of the following assertion, known as Hadamard’s factorization theorem, that can be found in [20, pp. 68–70]. Lemma 2.4. Assume that f∈H(C)\{0}has finite exponential growth order ρ. Let Π(z)be the canonical product formed with the zeros of f. Then there exists a polynomial Qof degree not greater than ρsuch that f(z) = eQ(z)Π(z)for all z∈C. Remark 2.5. Suppose that T∈Lcvl(H(C)) is invertible. Then there is an operator S∈L(H(C)) such that ST =I=TS, where Iis the identity on H(C). Since Tis a convolution operator, one has Tτ−a=τ−aTfor all a∈C. Taking inverses, we get τaS=Sτafor all a∈C, so that Sis also a convolution operator. Hence there are Φ,Ψ∈H(C) with finite exponential type such that T= Φ(D) and S= Ψ(D). If by 1we represent the constant function 1(z) = 1, we obtain: 1(D) = I= Φ(D)Ψ(D) = (Φ ·Ψ)(D). But the (linear) mapping Φ ∈ {entire functions with finite exponential type} 7→ Φ(D)∈Lcvl(H(C)) is one-to-one because Φ(D)6= 0 as soon as Φ 6= 0: indeed, according to the Malgrange–Ehrenpreis theorem (see [6,15,22]), any nonzero convolution operator is even surjective. Therefore, Φ(z)·Ψ(z) = 1 for all z∈C and so Φ is an entire function without zeros having exponential growth order ρ≤1. It follows from Lemma 2.4 that Φ(z) = eaz+bfor some scalars a, b ∈C. Consequently, we have T= Φ(D) = eaD+b=eb·eaD =eb·τa, that is, Tis a nonzero scalar multiple of a translation operator (conversely, if T=Mτawith M6= 0 and a∈Cthen Tis invertible: indeed, it possesses inverse M−1τ−a). Now, assume that T∈Lcvl(H(C)) is invertible (so that T=Mτaas above), that f∈H(C) and that (nk)⊂Nis a strictly increasing sequence such that fk(z)→z(k→ ∞) compactly in C, where fk(z) := Mnk(f(z+nka))2. Now, we proceed similarly to [3] (in fact, the following is a special case of [10, Theorem 2.1], but we keep the proof for the sake of completeness). Since fcannot be identically 0, any fkis not identically 0 either, so that we can find a closed Jordan curve Jsurrounding 0 such that fk(z)6= 0 (z∈J, k ∈N). Now, we invoke the argument principle (see, e.g., [1, p. 152]) and the fact that the identity z7→ zvanishes only at 0 to deduce that Nk→1 (k→ ∞), where by Nkwe have denoted the number of zeros of fkinside J, counting multiplicities. But, clearly, HYPERCYCLIC ALGEBRAS FOR D-MULTIPLES OF CONVOLUTION OPERATORS 7 each Nkis an even integer, hence (Nk) cannot tend to 1, a contradiction. Hence f26∈ HC(T) for any f∈H(C). In other words, Tcannot support hypercyclic algebras. In view of these considerations, of Theorem 2.2, and of the fact that Tis invertible if and only if Tis not in any ideal (D−cI)Lcvl(H(C)) (c∈C), we want to conclude this paper by expressing what is our conjecture: A nonzero operator T∈Lcvl(H(C)) supports hypercyclic algebras if and only if Tis not invertible; if this is the case, then HC(T) is dense algebrable. In particular, our conjecture claims for an affirmative answer to Question 3(ii) in [11] whether the only hypercyclic convolution operators on H(C) not supporting hypercyclic algebras must be scalar multiples of translations. Acknowledgements. The authors are grateful to the referee for helpful comments and suggestions. 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Departamento de An´ alisis Matem´ atico, Facultad de Matem´ aticas, Instituto de Matem´ aticas Antonio de Castro Brzezicki, Universidad de Sevilla, Avenida Reina Mercedes, 41080 Sevilla, Spain. E-mail address:[email protected], [email protected]