Convolution operators supporting hypercyclic algebras

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Convolution operators supporting hypercyclic algebras ✩ Juan Bès a , J. Alberto Conejero b,∗ , Dimitris Papathanasiou a a

Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403, USA b Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, València, Spain

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 24 November 2015 Available online xxxx Submitted by D. Ryabogin

We show that any convolution operator induced by a non-constant polynomial that vanishes at zero supports a hypercyclic algebra. This partially solves a question raised by R. Aron. © 2016 Elsevier Inc. All rights reserved.

Dedicated to Professor Richard Aron on the occasion of his 70th birthday Keywords: Algebrability Hypercyclic algebras Convolution operators Hypercyclic subspaces MacLane operator

1. Introduction In 2001 Aron, García and Maestre [4] called the attention to the wide range of examples supporting the following “principle”: In many different settings one encounters a problem which, at first glance, appears to have no solution at all. And, in fact, it frequently happens that there is a large linear subspace of solutions to the problem. This originated growing interest in the study of large algebraic structures within nonlinear settings, giving rise to the notions of lineability, spaceability and algebrability, to name a few [1]. This has been the case, for instance, in the study of the set of hypercyclic vectors. It is well known that for any operator T on a topological vector space X, the set HC (T ) = {f ∈ X : {f, T f, T 2 f, . . . } is dense in X} ✩

This work is supported in part by MICINN and FEDER, Project MTM2013-47093-P, and by GVA, Project ACOMP/2015/005.

* Corresponding author. E-mail addresses: [email protected] (J. Bès), [email protected] (J.A. Conejero), [email protected] (D. Papathanasiou). http://dx.doi.org/10.1016/j.jmaa.2016.01.029 0022-247X/© 2016 Elsevier Inc. All rights reserved.

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of hypercyclic vectors for T is either empty or contains a dense infinite-dimensional linear subspace (but the origin), see [18]. In fact, when HC (T ) is non-empty it sometimes contains (but zero) a closed and infinite dimensional linear subspace, while other times the only closed subspaces it contains (but zero) are of finite dimension [8,14]; see also [7, Ch. 8] and [15, Ch. 10]. When X is a topological algebra it is natural to ask whether HC (T ) can contain, or must always contain, a subalgebra (but the origin) whenever it is non-empty. Both questions have been answered by considering convolution operators on the space X = H(C) of entire functions on the complex plane C, endowed with the compact-open topology; that convolution operators (other than scalar multiples of the identity) are hypercyclic was established by Godefroy and Shapiro [13], see also [10,16,5]. Aron et al. [2,3] showed that no translation operator τa τa (f )(z) = f (z + a) f ∈ H(C), z ∈ C can support a hypercyclic algebra, in a very strong way: Indeed, for any positive integer p and any f ∈ H(C), the non-constant elements of the orbit of f p under τa are those entire functions for which the multiplicities of their zeros are integer multiples of p. In stark contrast with this operator they also showed that the collection of entire functions f for which every power f n (n = 1, 2, . . . ) is hypercyclic for the operator D of complex differentiation is residual in H(C). Later Shkarin [17, Th. 4.1] showed that HC (D) contained both a hypercyclic subspace and a hypercyclic algebra, and with a different approach Bayart and Matheron [7, Th. 8.26] also showed that the set of f ∈ H(C) that generate an algebra consisting entirely (but the origin) of hypercyclic vectors for D is residual in H(C). The abovementioned solutions by Aron et al., Bayart and Matheron, and Shkarin bear the question of which convolution operators on H(C) support a hypercyclic algebra. In this note we consider the following question: Question 1 (Aron). Let Φ be a non-constant polynomial. Does Φ(D) support a hypercyclic algebra? The purpose of this note is to show that the techniques by Bayart and Matheron provide an affirmative answer for the case when Φ(0) = 0: Theorem 1. Let Ω be a simply connected planar domain and H(Ω) the space of holomorphic functions on Ω endowed with the compact open topology. Let Φ be a non-constant polynomial with Φ(0) = 0. Then the set of functions f ∈ H(Ω) that generate a hypercyclic algebra for Φ(D) is residual in H(Ω). 2. Proof of Theorem 1 The proof of Theorem 1 follows that of [7, Th. 8.26]. We postpone the proof of Proposition 2 for later. Proposition 2. Let Φ be a polynomial with Φ(0) = 0. Then for each pair (U, V ) of non-empty open subsets of H(Ω) and each m ∈ N there exist P ∈ U and q ∈ N so that  Φ(D)q (P j ) = 0

for 0 ≤ j < m,

Φ(D) (P ) ∈ V. q

m

(2.1)

Proof of Theorem 1. For any g ∈ H(Ω) and α ∈ Cm , we let gα := α1 g +· · ·+αm g m . Let (Vk )k be a countable local basis of open sets of H(Ω). For each (k, s, m) ∈ N3 we let A(k, s, m) denote the set of f ∈ H(Ω) that satisfy the following property ∀α ∈ Cm with αm = 1 and α∞ ≤ s, ∃q ∈ N : Φ(D)q (fα ) ∈ Vk .

(2.2)

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Each such A(k, s, m) is open and dense in H(Ω), thanks to Proposition 2. By Baire’s Theorem, A = ∩k,s,m∈N A(k, s, m) is residual in H(Ω). Let f ∈ A, and let g be in the algebra generated by f . Since a vector is hypercyclic if and only if any non-zero scalar multiple of it is hypercyclic, we may assume g = fα = α1 f + α2 f 2 + · · · + αm−1 f m−1 + f m . Then g is clearly hypercyclic for Φ(D). Indeed, given any non-empty open set U of H(Ω), let k ∈ N so that Vk ⊂ U . Pick s > α∞ . Then since f ∈ A(k, s, m) we know that there exists q satisfying (2.2). Hence Φ(D)q g = Φ(D)q (α1 f + α2 f 2 + · · · + αm−1 f m−1 + f m ) ∈ Vk ⊂ U.

2

k Proof of Proposition 2. Let Φ(z) = z r j=0 aj z j , with a0 = 0 and r ∈ N, and let (A, B) ∈ U × V be polynomials. Enlarging the degree of B if necessary, we may assume that degree(B) = p ∈ rN and p > m. It suffices to show the following. Claim 1. For any large n ∈ rN there exist (c0 , . . . , cp ) = (c0 (n), . . . , cp (n)) ∈ Cp+1 so that 

R := Rn = z n q := qn =

m r n

p

j=0 cj z

+ (m −

j

and

1) pr

satisfy the following: (i) Φ(D)q ((A + R)j ) = 0 for 0 ≤ j < m, (ii) Φ(D)q ((A + R)m ) = Φ(D)q (Rm ) = B, (iii) Rn → 0 in H(Ω). n→∞

To show the claim, notice that for each 0 ≤  ≤ m − 1 and each s ∈ N we have the inequality degree(As R ) ≤ constant + (m − 1)n < (m − 1)p + mn = rq k for all large n. Hence, since Φ(D)q = ( =0 a D )q Drq , it follows that for large n we have Φ(D)q ((A + R)j ) = 0 for 0 ≤ j < m, and Φ(D)q ((A + R)m ) = Φ(D)q (Rm ). So (i) holds, as well as the first equality in (ii), regardless of the selection c = (c0 , . . . , cp ). Next, for each s ∈ N and multi-index γ = (γ0 , . . . , γs ) ∈ Ns+1 we let 0 |γ| =

s 

γj and γ =

j=0

s 

jγj =

j=0

s 

jγj .

j=1

Also, for c = (c0 , . . . , cs ) ∈ (C \ {0})s+1 and |γ| = m, we let γ

c =

s  j=0

γ cj j

  m m! . and = γ0 !γ1 ! . . . γs ! γ

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With this notation we have ⎛



Φ(D)q (Rm ) = ⎝

β∈Nk+1 :|β|=q 0



=

(α,β)∈A



=

(α,β)∈A

⎞ ⎛   q  aβ Drq+β ⎠ ⎝ β

 α∈Np+1 : |α|=m

⎞   m cα z nm+α ⎠ α

    m q  cα aβ Drq+β z nm+α α β     (nm + α )! m q   β−(m−1)p z α− cα aβ α β ( α − β − (m − 1)p)!

where

 A = (α, β) ∈ Np+1 × N0k+1 : |α| = m, |β| = q, and rq + β ≤ nm + α  0

 = (α, β) ∈ Np+1 × N0k+1 : |α| = m, |β| = q, and mp − p ≤ α  − β . 0 Thus

q

m

Φ(D) (R ) =

p 



i=0 (α,β)∈Ai

    m q (nm + α )! i α c aβ z α β i!

where for each i = 0, . . . , p Ai = {(α, β) ∈ A : α  − β = i + (m − 1)p}. In particular, B =

p

i=0 bi z

i

bi =

= Φ(D)q (Rm ) if and only if c = (c0 , . . . , cp ) is a solution of the system  (α,β)∈Ai

    m q (nm + α )! cα aβ α β i!

(0 ≤ i ≤ p).

(2.3)

We finish the proof of the claim using the following remark. Remark 3. For each fixed 0 ≤  ≤ p, the following occurs: (a) Each (α, β) ∈ A must satisfy α0 = · · · = α−1 = 0. Otherwise, if αs > 0 with 0 ≤ s ≤  − 1 we’d have since |α| = m that pm − (p − ) =  + (m − 1)p ≤ α  − β ≤ α  ≤ sαs + p(m − αs ) = pm − (p − s)αs ≤ pm − (p − s), a contradiction. (b) If (α, β) ∈ A satisfies that α > 0, then β = (q, 0, . . . , 0) and α = 1, αp = m −1, and αj = 0 for j = , p. p This is forced from (a) and the inequalities pm − (p − ) = α  − β ≤ α  = j= jαj ≤ α + p(m − α ) = pm − (p − )α . (c) Let A := A \ {((0, . . . , 0,  1 , 0, . . . , 0, m − 1), (q, 0, . . . , 0))}. Then from (a) and (b) each (α, β) ∈ A th

satisfies that α0 = · · · = α = 0.   satisfies |β| = q and β ∈ {0, . . . , }, then βq ≤ q  and |aβ | ≤ (max{|a0 |, . . . , |ak |}) . (d) If β ∈ Nk+1 0

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Now, thanks to Remark 3 the system (2.3) is upper triangular and thus solvable, and any solution to (2.3) satisfies (ii) for sufficiently large n. To see (iii), it suffices to show that there exists w > 1 so that for each  = 0, 1, . . . , p we have  cp− = O

wn 1 [(mn + mp)!] m

 as n → ∞.

(2.4)

Condition (2.4) ensures that Rn → 0 in H(Ω) as for each M > 0 we have M n+i |ci | → 0. Indeed, by n→∞

n→∞

(2.4) and Stirling’s formula 

m M (n+i)m wmn M n+i |ci | ≤ (mn + mp)!   (M w)mn+mp =o ( mn+mp )mn+mp e mn+mp  eM w =O → 0. n→∞ mn + mp

So we finish by proving (2.4) by induction on . Taking i = p in (2.3) we get —since in this case (α, β) ∈ Ap ⇔ α = (0, . . . , 0, m) and β = (q, 0, . . . , 0)— that    m q α β )! a c (nm + α α β (α,β)∈Ap    m q = aq0 cm p (nm + mp)!. (0, . . . , 0, m) (q, 0, . . . , 0)

p!bp =



Thus cm p =

p!bp aq0 (nm + mp)!

(2.5)

and (2.4) holds for  = 0. Inductively, suppose there exists w−1 > 1 so that  cp−j = O



n w−1

(n → ∞)

1

[(mn + mp)!] m

for each j = 0, . . . ,  − 1. We want to show that for some w > 1  cp− = O

wn 1 [(mn + mp)!] m

 (n → ∞).

(2.6)

Now, taking i = p −  in (2.3) we have by Remark 3 (b) and (c) that (p − )! bp− =

 (α,β)∈Ap−

   m q β α )! a c (nm + α α β

= mcp− cm−1 (nm + mp − )! aq0 + Kn , p where Kn =



(α,β)∈Ap−

m q  α

β

aβ cα (nm + α )!. Also, thanks to (2.5) we have that

(2.7)

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1

cm−1 (nm + mp − )! = p

1

(p!bp )1− m [(nm + mp)!] m −1 1− 1 (a0 m )q j=0 (nm + mp − j) 1

∼

[(nm + mp)!] m 1 1− m

(a0

)q n

(n → ∞).

(2.8)

 ∈ {(mp − j, j) : j = 0, . . . , }, and thus by Remark 3 (d) Now, let (α, β) ∈ Ap− be fixed. Notice that ( α, β) q  that β ≤ q  and |aβ | ≤ a∞ . Moreover, thanks to Remark 3 (c) and our inductive hypothesis we also have  |c | = α

αp−+1 |cp−+1

p · · · cα p |

αp−+1 +···+αp 

n w−1

=O

1

[(nm + mp)!] m   nm w−1 . =O (nm + mp)!

Hence       (α,β)∈A

p−

        m q β α mn )! = O n w−1 (n → ∞). a c (nm + α α β 

(2.9)

So by (2.7), (2.8) and (2.9) we have  cp− = O



mn n2 w−1 1

q

[(nm + mp)!] m a0m − r1

m Thus any w > w−1 a0

(n → ∞).

satisfies (2.6), and Claim 1 holds. The proof of Proposition 2 is now complete. 2

We conclude this note with the following natural questions: Question 2. Can we eliminate the assumption Φ(0) = 0 in Theorem 1? It is also natural to ask for examples of algebras generated by bigger sets. In fact, the simplest case of the existence of algebras generated by two functions is still open. Question 3 (Seoane-Sepúlveda). Does there exist a pair of algebraically independent hypercyclic functions f, g ∈ H(C) that altogether generate an algebra consisting entirely (but the origin) of hypercyclic vectors for D on H(C)? Finally, we may also ask about the existence of algebras of frequently hypercyclic vectors (respectively, of upper frequently hypercyclic vectors) for convolution operators on H(C). We note that every convolution operator Φ(D) that is not a scalar multiple of the identity is frequently hypercyclic [11]. Indeed, Φ(D) has a frequently hypercyclic subspace whenever Φ ∈ H(C) is transcendental [12], but when Φ is a non-constant polynomial the operator Φ(D) has an upper frequently hypercyclic subspace but has no frequently hypercyclic subspace [6,9]. Question 4. Which convolution operators support an algebra of frequently hypercyclic (respectively, of upper frequently hypercyclic) vectors?

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References [1] R.M. Aron, L. Bernal-González, D. Pellegrino, J.B. Seoane-Sepúlveda, Lineability: The Search for Linearity in Mathematics, Monographs and Research Notes in Mathematics, Chapman and Hall/CRC, 2015. [2] R.M. Aron, J.A. Conejero, A. Peris, J.B. Seoane-Sepúlveda, Powers of hypercyclic functions for some classical hypercyclic operators, Integral Equations Operator Theory 58 (4) (2007) 591–596. [3] R.M. Aron, J.A. Conejero, A. Peris, J.B. Seoane-Sepúlveda, Sums and products of bad functions, in: Function Spaces, in: Contemporary Mathematics, vol. 435, American Mathematical Society, Providence, RI, 2007, pp. 47–52. [4] R. Aron, D. García, M. Maestre, Linearity in non-linear problems, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 95 (1) (2001) 7–12. [5] R. Aron, D. Markose, On universal functions, in: Satellite Conference on Infinite Dimensional Function Theory, J. Korean Math. Soc. 41 (1) (2004) 65–76. [6] F. Bayart, R. Ernst, Q. Menet, Non-existence of frequently hypercyclic subspaces for P (D), Israel J. Math. (2016), in press. [7] F. Bayart, É. Matheron, Dynamics of Linear Operators, Cambridge Tracts in Mathematics, vol. 179, Cambridge University Press, Cambridge, 2009. [8] L. Bernal-González, A. Montes-Rodríguez, Non-finite-dimensional closed vector spaces of universal functions for composition operators, J. Approx. Theory 82 (3) (1995) 375–391. [9] J. Bès, Q. Menet, Existence of common and upper frequently hypercyclic subspaces, J. Math. Anal. Appl. 432 (1) (2015) 10–37. [10] G.D. Birkhoff, Démonstration d’un théorème élémentaire sur les fonctions entières, C. R. Math. Acad. Sci. Paris 189 (2) (1929) 473–475. [11] A. Bonilla, K.-G. Grosse-Erdmann, On a theorem of Godefroy and Shapiro, Integral Equations Operator Theory 56 (2) (2006) 151–162. [12] A. Bonilla, K.-G. Grosse-Erdmann, Frequently hypercyclic subspaces, Monatsh. Math. 168 (3–4) (2012) 305–320. [13] G. Godefroy, J.H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (2) (1991) 229–269. [14] M. González, F. León-Saavedra, A. Montes-Rodríguez, Semi-Fredholm theory: hypercyclic and supercyclic subspaces, Proc. Lond. Math. Soc. (3) 81 (1) (2000) 169–189. [15] K.-G. Grosse-Erdmann, A. Peris, Linear Chaos, Universitext, Springer, London, 2011. [16] G.R. MacLane, Sequences of derivatives and normal families, J. Anal. Math. 2 (1952) 72–87. [17] S. Shkarin, On the set of hypercyclic vectors for the differentiation operator, Israel J. Math. 180 (2010) 271–283. [18] J. Wengenroth, Hypercyclic operators on non-locally convex spaces, Proc. Amer. Math. Soc. 131 (6) (2003) 1759–1761 (electronic).