Open Access

Fixed points and orbits of non-convolution operators

Fixed Point Theory and Applications20142014:221

https://doi.org/10.1186/1687-1812-2014-221

Received: 1 July 2014

Accepted: 14 October 2014

Published: 29 October 2014

Abstract

A continuous linear operator T on a Fréchet space F is hypercyclic if there exists a vector f F (which is called hypercyclic for T) such that the orbit { T n f : n N } 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 ( λ z + b ) ( λ , b C ) 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 z λ z + b (in the case λ 1 ) plays a central role in the proofs.

Keywords

fixed pointDenjoy-Wolf theoremnon-convolution operatorhypercyclic operatorspaceability

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 f F (called hypercyclic vector for T) such that the orbit ( { T n f : n N } ) 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 ( λ z + b ) , λ , b C . In the first section we show that if λ D and b C 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 , b C ) is hypercyclic. Let us denote by H C ( T ) the set of hypercyclic vectors for T. In Section 3 we characterize when H C ( T λ , b ) is spaceable. Namely H C ( 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 z λ z + b ( λ 1 ).

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 λ D and b C and for any f H ( C ) , the sequence T λ , b n f 0 uniformly on compact subsets of . Therefore T λ , b is not hypercyclic on H ( C ) .

Proof Set φ ( z ) = λ z + b , λ D and b C . Since λ 1 , 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 ) = φ φ ( n times ) ,
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 λ D 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 ) ) | C n ! 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 ) | C n ! 2 n | λ | n ( n 1 ) 2 C e n 1 / 2 ( 2 n | λ | ( n 1 ) / 2 e ) n ,

and since 2 n | 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 λ C and b C , 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 H C ( T λ , b ) is lineable but it is not spaceable.

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

Theorem 3.1 For any λ C and b C , H C ( T λ , b ) is spaceable if and only if | λ | = 1 .

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

    p N ( f ) > 0 , f H C ( 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 ) d ξ . 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 H C ( T λ , b ) is not spaceable.

Now, let us suppose that | λ | = 1 , and let us prove that H C ( 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 H C ( D ) is spaceable. If b 0 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 H C ( T 1 , b ) is spaceable.

Now let us consider the case λ D { 1 } . Set z 0 = b 1 λ the fixed point of φ ( z ) = λ z + 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 n 1 , 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 j k + 1 , then n j n k + 1 C n k , therefore by (1) we have
n j n k 2 n j for  j k + 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 f 0 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. □

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematics, University of Cádiz
(2)
Department of Mathematics, IES Sofía

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Copyright

© León-Saavedra and Romero-de la Rosa; licensee Springer. 2014

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