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Fixed points and orbits of non-convolution operators

Abstract

A continuous linear operator T on a Fréchet space F is hypercyclic if there exists a vector fF (which is called hypercyclic for T) such that the orbit { T n f:nN} is dense in F. A subset M of a vector space F is spaceable if M{0} contains an infinite-dimensional closed vector space. In this paper note we study the orbits of the operators T λ , b f= f (λz+b) (λ,bC) defined on the space of entire functions and introduced by Aron and Markose (J. Korean Math. Soc. 41(1):65-76, 2004). We complete the results in Aron and Markose (J. Korean Math. Soc. 41(1):65-76, 2004), characterizing when T λ , b is hypercyclic on H(C). We characterize also when the set of hypercyclic vectors for T λ , b is spaceable. The fixed point of the map zλz+b (in the case λ1) plays a central role in the proofs.

1 Introduction

Let us denote by F a complex infinite dimensional Fréchet space. A continuous linear operator T defined on F is said to be hypercyclic if there exists a vector fF (called hypercyclic vector for T) such that the orbit ({ T n f:nN}) is dense in F. We refer to the books [1, 2] and the references therein for further information on hypercyclic operators. From a modern terminology, a subset M of a vector space F is said to be spaceable if M{0} contains an infinite-dimensional closed vector space. The study of spaceability of (usually pathological) subsets is a natural question which has been studied extensively (see [1] Chapter 8 or the recent survey [3] and the references therein).

In 1991, Godefroy and Shapiro [4] showed that every continuous linear operator L:H(C)H(C) which commutes with translations (these operators are called convolution operators) and which is not a multiple of the identity is hypercyclic. This result unifies two classical results by Birkhoff and MacLane (see the survey [5]).

In [5], Aron and Markose introduced new examples of hypercyclic operators on H(C) which are not convolution operators. Namely, T λ , b f= f (λz+b), λ,bC. In the first section we show that if λD and bC then T λ , b is not hypercyclic on H(C). This result together with the results in [5] and [6] shows the following characterization: T λ , b is hypercyclic on H(C) if and only if |λ|1. Thus, we complete the results of Aron and Markose [5] and Fernández and Hallack [6] characterizing when T λ , b (λ,bC) is hypercyclic. Let us denote by HC(T) the set of hypercyclic vectors for T. In Section 3 we characterize when HC( T λ , b ) is spaceable. Namely HC( T λ , b ) is spaceable if and only if |λ|=1. During the proofs, it is essential to take into account the fixed point of the map zλz+b (λ1).

2 Characterizing the hypercyclicity of T λ , b

The proof of this result follows the ideas of the proof of Proposition 14 in [5].

Theorem 2.1 For any λD and bC and for any fH(C), the sequence T λ , b n f0 uniformly on compact subsets of . Therefore T λ , b is not hypercyclic on H(C).

Proof Set φ(z)=λz+b, λD and bC. Since λ1, φ(z) has a fixed point z 0 . Indeed, z 0 = b 1 λ . We denote by φ n (z) the sequence of the iterates defined by

φ n (z)=φφ(ntimes),

an easy computation yields

φ n (z)= λ n z+ 1 λ n 1 λ b.

Let us observe that the iterates of the operator T λ , b have the form

T λ , b n f(z)= λ n ( n 1 ) 2 f ( n ) ( λ n z + ( 1 λ n ) b 1 λ ) = λ n ( n 1 ) 2 f ( n ) ( φ n ( z ) ) ,

where f ( n ) denotes the n th derivative of f. It is well known that if λD then z 0 is an attractive fixed point, that is, φ n (z) converges to the fixed point z 0 uniformly on compact subsets. Indeed, let R>0. If |z|R, then

| φ n ( z ) z 0 | =| λ n z+ ( 1 λ n ) b 1 λ b 1 λ | | λ | n R+ | λ | n | 1 λ | |b|0

as n. Thus, there exists n 0 such that if |z|R then | φ n (z) z 0 |<1/2 for all n n 0 .

If n n 0 and |z|R, we have by the Cauchy inequality

| f ( n ) ( φ n ( z ) ) | Cn! 2 n ,where C=max { | f ( w ) | : | w | 1 } .

Now, it follows from Stirling’s formula that n!e n n + 1 / 2 e n . Hence, if |z|R and n n 0 , then

| T λ , b n f ( z ) | Cn! 2 n | λ | n ( n 1 ) 2 Ce n 1 / 2 ( 2 n | λ | ( n 1 ) / 2 e ) n ,

and since 2n | λ | ( n 1 ) / 2 0 as n, we conclude that max | z | R | T λ , b n f(z)|0, as n, as desired. We point out that this is a refinement of the argument by Aron and Markose. One of the referees chased the constants and recovered the factor n 1 / 2 that was missing but that does not break the argument. □

Theorem 13 in [5] and Theorem 2.1 give the following characterization.

Theorem 2.2 For any λC and bC, the operator T λ , b is hypercyclic in H(C) if and only if |λ|1.

3 Spaceability of the set of hypercyclic vectors for T λ , b

As stated in [3], there are few non-trivial examples of subsets M which are lineable (that is, M{0} contains an infinite-dimensional vector space) and are not spaceable. The following result provides the following examples: for |λ|>1, the set HC( T λ , b ) is lineable but it is not spaceable.

Shkarin [7] showed that for the derivative operator D, the set of hypercyclic vectors HC(D) is spaceable.

Theorem 3.1 For any λC and bC, HC( T λ , b ) is spaceable if and only if |λ|=1.

Proof Firstly, let us suppose that |λ|>1, and let us prove that HC( T λ , b ) does not contain a closed infinite dimensional subspace. Let z 0 be the fixed point of φ(z)=λz+b. Then we consider a sequence of norms defining the topology of H(C). Namely, for nN and fH(C), we write

p n (f)= max | z z 0 | | λ | n / 4 | f ( z ) | .

It is easy to see that the above sequence of semi-norms is increasing and defines the original topology on H(C).

Given the sequence of increasing semi-norms { p n }, according to Theorem 10.25 in [2], it is sufficient to find a sequence of subspaces M n H(C) of finite codimension, positive numbers C n and N1 satisfying the following:

  1. (a)

    p N (f)>0, fHC( T λ , b ).

  2. (b)

    p N ( T λ , b n f) C n p n (f), f M n .

Indeed, let us consider the subspaces

M n = { f H ( C ) : f ( z 0 ) = f ( z 0 ) = = f ( n 1 ) ( z 0 ) = 0 } ,

which are clearly of finite codimension.

Notice that φ n (z) z 0 = λ n (z z 0 ), so that φ n (z) maps the disk D( z 0 ,1)={|z z 0 |1} onto D( z 0 ,λ | n ). Hence,

p 0 ( T λ , b n f ) = max | z z 0 | 1 | T λ , b n f ( z ) | = | λ | n ( n 1 ) 2 max | z z 0 | 1 | f ( n ) ( φ n ( z ) ) | = | λ | n ( n 1 ) 2 max | φ n ( z ) z 0 | | λ | n + 1 | f ( n ) ( φ n ( z ) ) | = | λ | n ( n 1 ) 2 max | w z 0 | | λ | n | f ( n ) ( w ) | .

If f M 1 then f( z 0 )=0, so that f(z)= [ z 0 , z ] f (ξ)dξ. Therefore we have

max | z z 0 | R | f ( z ) | R max | z z 0 | R | f ( z ) | ,

and it follows easily by induction that if f M n then

max | z z 0 | R | f ( z ) | R n max | z z 0 | R | f ( n ) ( z ) | .

Thus,

p 0 ( T λ , b n f ) = | λ | n ( n 1 ) 2 max | w z 0 | | λ | n | f ( n ) ( w ) | | λ | n ( n 1 ) 2 max | w z 0 | | λ | n / 4 | f ( n ) ( w ) | | λ | n ( n 1 ) 2 | λ | n 2 / 4 max | w z 0 | | λ | n / 4 | f ( w ) | = | λ | n 2 2 n 4 p n ( f ) ,

and it follows that condition (b) is satisfied with N=0 and C n = | λ | n 2 2 n 4 as n, and therefore HC( T λ , b ) is not spaceable.

Now, let us suppose that |λ|=1, and let us prove that HC( T λ , b ) is spaceable. Indeed, let us suppose first that λ=1. If b=0 then T 1 , 0 =D, and it was proved by Shkarin [7] that HC(D) is spaceable. If b0 then T 1 , b =D e b D , so that T 1 , b =ψ(D), where ψ(z)=z e b z is an entire function of exponential type that is not a polynomial, and according to Example 10.12 in [[2], p.275], the space HC( T 1 , b ) is spaceable.

Now let us consider the case λD{1}. Set z 0 = b 1 λ the fixed point of φ(z)=λz+b. According to Theorem 10.2 in [2], since T λ , b satisfies the hypercyclicity criterion for the full sequence of natural numbers, it suffices to exhibit an infinite dimensional closed subspace M 0 of H(C) on which suitable powers of T λ , b tend to 0. Now the proof mimics some ideas contained in Example 10.13 in [2]. Indeed, for any n1, there is some C n >0 such that

x n 2 x for all x C n .
(1)

Let us consider a strictly increasing sequence of positive integers ( n k ) k satisfying n k + 1 C n k . If jk+1, then n j n k + 1 C n k , therefore by (1) we have

n j n k 2 n j for jk+1.
(2)

Let us consider M 0 the closed subspace of H(C) of all entire functions f of the form

f(z)= k = 1 a k ( z z 0 ) n k 1 ,

and let us prove that T λ , b n k f0 uniformly on compact subsets as k.

We have

( T n k f ) (z)= λ n k ( n k 1 ) 2 ( D n k f ) ( φ n k ( z ) ) .

Notice that |λ|=1 and the map φ n k takes the disc D( z 0 ,R) onto itself, so that

max | z z 0 | R | ( T n k f ) ( z ) | = max | z z 0 | R | ( D n k f ) ( φ n k ( z ) ) | = max | w z 0 | R | ( D n k f ) ( w ) | .

Finally, we have

max | w z 0 | R | ( D n k f ) ( w ) | = max | w z 0 | R | j = k + 1 a j D n k ( w z 0 ) n j 1 | j = k + 1 | a j | ( n j 1 ) ( n j 2 ) ( n j n k ) R n j n k 1 j = k + 1 | a j | n j n k R n j j = k + 1 | a j | ( 2 R ) n j 0 as  k .

In the last step we used inequality (2). This completes the proof of Theorem 3.1. □

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Acknowledgements

The research was supported by Junta de Andalucía FQM-257. The authors would like to thank the referee for reading our manuscript carefully and for giving such constructive comments, which helped improving the quality of the paper substantially.

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Correspondence to Fernando León-Saavedra.

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León-Saavedra, F., Romero-de la Rosa, P. Fixed points and orbits of non-convolution operators. Fixed Point Theory Appl 2014, 221 (2014). https://doi.org/10.1186/1687-1812-2014-221

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Keywords

  • fixed point
  • Denjoy-Wolf theorem
  • non-convolution operator
  • hypercyclic operator
  • spaceability