- Email: [email protected]
Strong mixing measures for linear operators and frequent hypercyclicity
J. Math. Anal. Appl. 398 (2013) 462–465
Contents lists available at SciVerse ScienceDirect
Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa
Strong mixing measures for linear operators and frequent hypercyclicity M. Murillo-Arcila a , A. Peris b,∗ a
IUMPA, Universitat Politècnica de València, Edifici 8G, Camí Vera S/N, 46022 València, Spain
b
IUMPA, Universitat Politècnica de València, Departament de Matemàtica Aplicada, Edifici 7A, 46022 València, Spain
article
abstract
info
Article history: Received 9 June 2012 Available online 7 September 2012 Submitted by R.M. Aron Keywords: Hypercyclic operators Strongly mixing measures
We construct strongly mixing invariant measures with full support for operators on F -spaces which satisfy the Frequent Hypercyclicity Criterion. For unilateral backward shifts on sequence spaces, a slight modification shows that one can even obtain exact invariant measures. © 2012 Elsevier Inc. All rights reserved.
1. Introduction We recall that an operator T on a topological vector space X is called hypercyclic if there is a vector x in X such that its orbit Orb(x, T ) = {x, Tx, T 2 x, . . .} is dense in X . The recent books [1,2] contain the theory and most of the recent advances on hypercyclicity and linear dynamics, especially in topological dynamics. Here we are concerned with measure theoretic properties. Let (X , B, µ) be a probability space, where X is a topological space and B denotes the σ -algebra of Borel subsets of X . We will say that a Borel probability measure µ has full support if for each non-empty open set U ⊂ X we have µ(U ) > 0. A measurable map T : (X , B, µ) → (X , B, µ) is called a measurepreserving transformation if µ(T −1 (A)) = µ(A) for all A ∈ B. T is said to be strongly mixing with respect to µ if lim µ(A ∩ T −n (B)) = µ(A)µ(B)
n→∞
(A, B ∈ B),
−n and it is exact if given A ∈ B then either µ(A) = 0 or µ(A) = 1. The interested reader is referred to [3,4] for a n =0 T detailed account on the above properties. Ergodic theory was first used for the dynamics of linear operators by Rudnicki [5] and Flytzanis [6]. During the last few years it has been given special attention thanks to the work of Bayart and Grivaux [7,8]. The papers [9–14], for instance, contain recent advances on the subject. The concept of frequent hypercyclicity was introduced by Bayart and Grivaux [8], inspired by Birkhoff’s ergodic theorem. They also gave the first version of a Frequent Hypercyclicity Criterion, although we will consider the formulation of Bonilla and Grosse-Erdmann [15] for operators on separable F -spaces. Another (probabilistic) version of it was given by Grivaux [16]. We derive under the hypothesis of Bonilla and Grosse-Erdmann a stronger result by showing that a T -invariant mixing measure can be obtained. Recently, Bayart and Matheron gave very general conditions expressed on eigenvector fields associated with unimodular eigenvalues under which an operator T admits a T -invariant mixing measure [11]. Actually, on the one hand our results can be deduced from [11] in the context of complex Fréchet spaces, and on the other hand we only need rather elementary tools. From now on, T will be an operator defined on a separable F -space X .
∞
∗
Corresponding author. E-mail addresses: [email protected] (M. Murillo-Arcila), [email protected] (A. Peris).
0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.08.050
M. Murillo-Arcila, A. Peris / J. Math. Anal. Appl. 398 (2013) 462–465
463
2. Invariant measures and the Frequent Hypercyclicity Criterion
We recall that a series n xn in X converges unconditionally if it converges and, for any 0-neighborhood U in X , there exists some N ∈ N such that n∈F xn ∈ U for every finite set F ⊂ {N , N + 1, N + 2, . . .}. We are now ready to present our main result. The idea behind the proof is to construct a ‘‘model’’ probability space (Z , µ) and a (Borel) measurable map Φ : Z → X , where Z ⊂ NZ is such that σ (Z ) = Z for the Bernoulli shift σ (. . . , n−1 , n0 , n1 , . . .) = (. . . , n0 , n1 , n2 , . . .), µ is a σ −1 -invariant strongly mixing measure, Y := Φ (Z ) is a T -invariant dense subset of X , Φ σ −1 = T Φ , and then the Borel probability measure µ on X defined by µ(A) = µ(Φ −1 (A)), A ∈ B(X ), is T -invariant and strongly mixing. We will use the slight generalization of the Frequent Hypercyclicity Criterion for operators given in [2, Remark 9.10]. Theorem 1. Let T be an operator on a separable F -space X . If there is a dense subset X0 of X and a sequence of maps Sn : X0 → X such that, for each x ∈ X0 ,
∞
T n x converges unconditionally, (i) n∞=0 (ii) n=0 Sn x converges unconditionally, and (iii) T n Sn x = x and T m Sn x = Sn−m x if n > m, then there is a T -invariant strongly mixing Borel probability measure µ on X with full support. Proof. We suppose that X0 = {xn ; n ∈ N} with x1 = 0 and Sn 0 = 0 for all n ∈ N. Let (Un )n be a basis of balanced open 0-neighborhoods in X such that Un+1 + Un+1 ⊂ Un , n ∈ N. By (i) and (ii), there exists an increasing sequence of positive integers (Nn )n with Nn+2 − Nn+1 > Nn+1 − Nn for all n ∈ N such that
T k xmk ∈ Un+1
and
k>Nn
Sk xmk ∈ Un+1 ,
if mk ≤ 2l, for Nl < k ≤ Nl+1 , l ≥ n.
(1)
k>Nn
Actually, this is a consequence of the completeness of X and the fact that, for each 0-neighborhood U and for all l ∈ N, there is N ∈ N such that k∈F T k x ∈ U and k∈F Sk x ∈ U for any finite subset F ⊂]N , +∞[ and for each x ∈ {x1 , . . . , x2l }. 1. The model probability space (Z , µ) We define K = k∈Z Fk where Fk = {1, . . . , m}
if Nm < |k| ≤ Nm+1 , m ∈ N,
and Fk = {1},
if |k| ≤ N1 .
Let K (s) := σ (K ), s ∈ Z, where σ : N → N is the backward shift. K (s) is a compact space when endowed with the product topology inherited from NZ , s ∈ Z. ∞ We consider in NZ the product measure µ = k∈Z µk , where µk ({n}) = pn for all n ∈ N and µk (N) = n=1 pn = 1, k ∈ Z. The values of pn ∈]0, 1[ are selected such that, if s
βj :=
j
Z
Z
Nj+1 −Nj ,
pi
j ∈ N, then
i =1
Let Z =
s∈Z
∞
βj > 0.
j =1
K (s). We have
µ(Z ) ≥ µ(K ) =
|k|≤N1
µk ({1})
∞ l =1
Nl <|k|≤Nl+1
µk ({1, . . . , l}) =
2N +1 p1 1
∞
2 βl
> 0.
l=1
It is well-known [3] that µ is a σ −1 -invariant strongly mixing Borel probability measure. Since σ (Z ) = Z and it has positive measure, then µ(Z ) = 1. 2. The map Φ Given s ∈ Z we define the map Φ : K (s) → X by
Φ ((nk )k∈Z ) =
k<0
S−k xnk + xn0 +
T k xnk .
(2)
k>0
Φ is well-defined since, given (nk )k∈Z ∈ K (s) and for l ≥ |s|, we have nk ≤ 2l if Nl < |k| ≤ Nl+1 , which shows the convergence of the series in (2) by (1). Φ is also continuous. Indeed, let (α(j))j be a sequence of elements of K (s) that converges to α ∈ K (s) and fix any n ∈ N with n > |s|. We will find n0 ∈ N such that Φ (α(j)) − Φ (α) ∈ Un for n ≥ n0 . To do this, by definition of the topology in K (s) there exists n0 ∈ N such that
α(j)k = αk if |k| ≤ Nn+1 and j ≥ n0 .
464
M. Murillo-Arcila, A. Peris / J. Math. Anal. Appl. 398 (2013) 462–465
By (1) we have
Φ (α(j)) − Φ (α) =
S−k (xα(j)k − xαk ) +
T k (xα(j)k − xαk ) ∈ Un
k>Nn+1
k<−Nn+1
for all j ≥ n0 . This shows the continuity of Φ : K (s) → X for every s ∈ Z. The map Φ is then well-defined on Z , and Φ : Z → X is measurable (i.e., Φ −1 (A) ∈ B(Z ) for every A ∈ B(X )). 3. The measure µ on X L(s) := Φ (K (s)) is compact in X , s ∈ Z, and Y := s∈Z L(s) is a T -invariant Borel subset of X because Φ σ −1 = T Φ . We then define in X the measure µ(A) = µ(Φ −1 (A)) for all A ∈ B(X ). Obviously, µ is well-defined and it is a T -invariant strongly mixing Borel probability measure. The proof is completed by showing that µ has full support. Given a non-empty open set U in X , we pick n ∈ N satisfying xn + Un ⊂ U. Thus
µ(U ) ≥ µ
x = xn +
k
T xmk +
k>Nn
≥ µ0 ({n})
Sk xmk ; mk ≤ 2l for Nl < k ≤ Nl+1 , l ≥ n
k>Nn
µk ({1})
0<|k|≤Nn
∞
l=n
µk ({1, . . . , 2l}) >
2N pn p1 n
Nl <|k|≤Nl+1
∞
2 βl
> 0.
l =n
As we mentioned in Section 1, Theorem 1 can be deduced from [11, Corollary 1.3] when dealing with operators on separable complex Fréchet spaces. Indeed, the argument of É. Matheron is the following (we thank Grivaux for letting us know about it): Let T : X → X be an operator on a separable complex Fréchet space X satisfying the hypothesis of the Frequent Hypercyclicity Criterion given in Theorem 1, and suppose that X0 = {xn ; n ∈ N}. We define the following family of continuous T-eigenvector fields for T : Em (λ) =
λ−n T n xm +
λn Sn xm ,
λ ∈ T, m ∈ N .
n∈N
n≥0
They span X since, for any functional x∗ that vanishes on Em (λ) for each λ ∈ T and m ∈ N, the equality ⟨x∗ , Em (λ)⟩ = 0 for fixed m and for all λ ∈ T implies that ⟨x∗ , T n xm ⟩ = 0 for every n ≥ 0. Thus, ⟨x∗ , xm ⟩ = 0 for each m ∈ N, and by density, x∗ = 0. The previous theorem can be applied to different classes of operators. A distinguished one is the class of weighted shifts on sequence F -spaces. By a sequence space we mean a topological vector space X which is continuously included in ω, the countable product of the scalar field K. A sequence F -space is a sequence space that is also an F -space. Given a sequence w = (wn )n of non-zero weights, the associated unilateral (respectively, bilateral) weighted backward shift Bw : KN → KN is defined by Bw (x1 , x2 , . . .) = (w2 x2 , w3 x3 , . . .) (respectively, Bw : KZ → KZ is defined by Bw (. . . , x−1 , x0 , x1 , . . .) = (. . . , w0 x0 , w1 x1 , w2 x2 , . . .)). When a sequence F -space X is invariant under certain weighted backward shift T , then T is also continuous on X by the closed graph theorem. We refer the reader to, e.g., Chapter 4 of [2] for more details about hypercyclic and chaotic weighted shifts on Fréchet sequence spaces. In particular, Theorems 4.6 and 4.12 in [2] (we refer the reader to [17] for the original results) remain valid for F -spaces, and a bilateral (respectively, unilateral) weighted backward shift T : X → X on a sequence F -space X in which the canonical unit vectors ( e ) (respectively, ( e ) ) form an unconditional basis is chaotic if, and only if, n n ∈ Z n n ∈ N n∈Z en (respectively, n∈N en ) converges unconditionally. Corollary 2. Let T : X → X be a chaotic bilateral weighted backward shift on a sequence F -space X in which (en )n∈Z is an unconditional basis. Then there exists a T -invariant strongly mixing Borel probability measure on X with full support. Remark 3. The preceding result can be improved if T is a unilateral backward shift operator on a sequence F -space. In that case, there exists a T -invariant exact Borel probability measure on X with full support. Proof. Let M = {zn ; n ∈ N} be a countable dense set in K with ∞z1 = 0. Let (Un )n be a basis of balanced open 0-neighborhoods in X such that Un+1 + Un+1 ⊂ Un , n ∈ N. Since T is chaotic, n=1 en converges unconditionally, so there exists an increasing sequence of positive integers (Nn )n with Nn+2 − Nn+1 > Nn+1 − Nn for all n ∈ N such that
αk ek ∈ Un+1 ,
if αk ∈ {z1 , . . . , z2m }, for Nm < k ≤ Nm+1 , m ≥ n.
k>Nn
We define K =
k∈N
Fk where
Fk = {z1 , . . . , zm }
if Nm < k ≤ Nm+1 , m ∈ N,
and Fk = {z1 },
if k ≤ N1 .
(3)
M. Murillo-Arcila, A. Peris / J. Math. Anal. Appl. 398 (2013) 462–465
465
Let K (s) := σ s (K ), s ≥ 0. K (s) is a compact space when endowed with the product topology inherited from M N , s ≥ 0. We ∞ N consider in M the product measure µ = k∈N µk , where µk ({zn }) = pn for all n ∈ N and µk (M ) = n=1 pn = 1, k ∈ N. As before, we select the sequence (pn )n of positive numbers such that, if
βj =
j
Nj+1 −Nj ,
pi
then
∞
βj > 0.
j =1
i=1
It is known [3, Section 4.12] that µ is a σ -invariant exact Borel probability measure. By setting Z = µ(Z ) = 1. Now we define the map Φ : K (s) → X given by
Φ ((αk )k∈N ) =
∞
s≥0
K (s), we obtain
αk ek .
k=1
Φ is (well-defined and) continuous, s ≥ 0. We have that Φ : Z → X is measurable. L(s) := Φ (K (s)) is compact in X , s ≥ 0, and Y := s≥0 L(s) = Φ (Z ) is a T -invariant Borel subset of X . We then define on X the measure µ(A) = µ(Φ −1 (A)) for all A ∈ B(X ). As in Theorem 1, we conclude that µ is welldefined, and now it is a T -invariant exact Borel probability measure with full support.
Devaney chaos is therefore a sufficient condition for the existence of strongly mixing measures within the framework of weighted shift operators on sequence F -spaces. For some natural spaces it is even a characterization of this fact. For instance, Bayart and Ruzsa [18] recently proved that weighted shift operators on ℓp , 1 ≤ p < ∞, are frequently hypercyclic if, and only if, they are Devaney chaotic. It turns out that this is equivalent to the existence of an invariant strongly mixing Borel probability measure with full support on ℓp . Also, for the space ω, every weighted shift operator is chaotic [17]. In particular, for the unilateral case we obtain exact measures. Example 4. Every unilateral weighted backward shift operator on ω = KN admits an invariant exact Borel probability measure with full support on ω. We finish the paper by mentioning that a continuous-time version of Theorem 1 can be given by using the Frequent Hypercyclicity Criterion for C0 -semigroups introduced in [19]. This is part of a forthcoming paper [20]. Acknowledgments This work was supported in part by MEC and FEDER, Project MTM2010-14909, and by GV, Project PROMETEO/2008/101. The first author was also supported by a grant from the FPU Program of MEC. We thank the referee whose detailed report led to an improvement in the presentation of this work. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
F. Bayart, É. Matheron, Dynamics of Linear Operators, Cambridge University Press, Cambridge, 2009. K.-G. Grosse-Erdmann, A. Peris Manguillot, Linear Chaos, Universitext, Springer-Verlag London Ltd., London, 2011. P. Walters, An Introduction to Ergodic Theory, Springer, New York, Berlin, 1982. M. Einsiedler, T. Ward, Ergodic Theory with a View Towards Number Theory, in: Graduate Texts in Mathematics, vol. 259, Springer-Verlag London Ltd., London, 2011. R. Rudnicki, Gaussian measure-preserving linear transformations, Univ. Iagel. Acta Math. 30 (1993) 105–112. E. Flytzanis, Unimodular eigenvalues and linear chaos in Hilbert spaces, Geom. Funct. Anal. 5 (1995) 1–13. F. Bayart, S. Grivaux, Hypercyclicity and unimodular point spectrum, J. Funct. Anal. 226 (2005) 281–300. F. Bayart, S. Grivaux, Frequently hypercyclic operators, Trans. Amer. Math. Soc. 358 (2006) 5083–5117. C. Badea, S. Grivaux, Unimodular eigenvalues, uniformly distributed sequences and linear dynamics, Adv. Math. 211 (2007) 766–793. F. Bayart, S. Grivaux, Invariant Gaussian measures for operators on Banach spaces and linear dynamics, Proc. Lond. Math. Soc. (3) 94 (2007) 181–210. F. Bayart, É. Matheron, Mixing operators and small subsets of the circle, preprint. arXiv:1112.1289v1. M. De la Rosa, L. Frerick, S. Grivaux, A. Peris, Frequent hypercyclicity, chaos, and unconditional Schauder decompositions, Israel J. Math. 190 (2012) 389–399. S. Grivaux, A new class of frequently hypercyclic operators, Indiana Univ. Math. J. 60 (2011) 1177–1202. R. Rudnicki, Chaoticity and invariant measures for a cell population model, J. Math. Anal. Appl. 393 (2012) 151–165. A. Bonilla, K.-G. Grosse-Erdmann, Frequently hypercyclic operators and vectors, Ergodic Theory Dynam. Systems 27 (2007) 383–404; Ergodic Theory Dynam. Systems 29 (2009) 1993–1994 (erratum). S. Grivaux, A probabilistic version of the frequent hypercyclicity criterion, Studia Math. 176 (2006) 279–290. K.-G. Grosse-Erdmann, Hypercyclic and chaotic weighted shifts, Studia Math. 139 (2000) 47–68. F. Bayart, I. Z. Ruzsa, Frequently hypercyclic weighted shifts, preprint. E. Mangino, A. Peris, Frequently hypercyclic semigroups, Studia Math. 202 (2011) 227–242. M. Murillo, A. Peris, Strong mixing measures for C0 -semigroups (in preparation).
Contents lists available at SciVerse ScienceDirect
Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa
Strong mixing measures for linear operators and frequent hypercyclicity M. Murillo-Arcila a , A. Peris b,∗ a
IUMPA, Universitat Politècnica de València, Edifici 8G, Camí Vera S/N, 46022 València, Spain
b
IUMPA, Universitat Politècnica de València, Departament de Matemàtica Aplicada, Edifici 7A, 46022 València, Spain
article
abstract
info
Article history: Received 9 June 2012 Available online 7 September 2012 Submitted by R.M. Aron Keywords: Hypercyclic operators Strongly mixing measures
We construct strongly mixing invariant measures with full support for operators on F -spaces which satisfy the Frequent Hypercyclicity Criterion. For unilateral backward shifts on sequence spaces, a slight modification shows that one can even obtain exact invariant measures. © 2012 Elsevier Inc. All rights reserved.
1. Introduction We recall that an operator T on a topological vector space X is called hypercyclic if there is a vector x in X such that its orbit Orb(x, T ) = {x, Tx, T 2 x, . . .} is dense in X . The recent books [1,2] contain the theory and most of the recent advances on hypercyclicity and linear dynamics, especially in topological dynamics. Here we are concerned with measure theoretic properties. Let (X , B, µ) be a probability space, where X is a topological space and B denotes the σ -algebra of Borel subsets of X . We will say that a Borel probability measure µ has full support if for each non-empty open set U ⊂ X we have µ(U ) > 0. A measurable map T : (X , B, µ) → (X , B, µ) is called a measurepreserving transformation if µ(T −1 (A)) = µ(A) for all A ∈ B. T is said to be strongly mixing with respect to µ if lim µ(A ∩ T −n (B)) = µ(A)µ(B)
n→∞
(A, B ∈ B),
−n and it is exact if given A ∈ B then either µ(A) = 0 or µ(A) = 1. The interested reader is referred to [3,4] for a n =0 T detailed account on the above properties. Ergodic theory was first used for the dynamics of linear operators by Rudnicki [5] and Flytzanis [6]. During the last few years it has been given special attention thanks to the work of Bayart and Grivaux [7,8]. The papers [9–14], for instance, contain recent advances on the subject. The concept of frequent hypercyclicity was introduced by Bayart and Grivaux [8], inspired by Birkhoff’s ergodic theorem. They also gave the first version of a Frequent Hypercyclicity Criterion, although we will consider the formulation of Bonilla and Grosse-Erdmann [15] for operators on separable F -spaces. Another (probabilistic) version of it was given by Grivaux [16]. We derive under the hypothesis of Bonilla and Grosse-Erdmann a stronger result by showing that a T -invariant mixing measure can be obtained. Recently, Bayart and Matheron gave very general conditions expressed on eigenvector fields associated with unimodular eigenvalues under which an operator T admits a T -invariant mixing measure [11]. Actually, on the one hand our results can be deduced from [11] in the context of complex Fréchet spaces, and on the other hand we only need rather elementary tools. From now on, T will be an operator defined on a separable F -space X .
∞
∗
Corresponding author. E-mail addresses: [email protected] (M. Murillo-Arcila), [email protected] (A. Peris).
0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.08.050
M. Murillo-Arcila, A. Peris / J. Math. Anal. Appl. 398 (2013) 462–465
463
2. Invariant measures and the Frequent Hypercyclicity Criterion
We recall that a series n xn in X converges unconditionally if it converges and, for any 0-neighborhood U in X , there exists some N ∈ N such that n∈F xn ∈ U for every finite set F ⊂ {N , N + 1, N + 2, . . .}. We are now ready to present our main result. The idea behind the proof is to construct a ‘‘model’’ probability space (Z , µ) and a (Borel) measurable map Φ : Z → X , where Z ⊂ NZ is such that σ (Z ) = Z for the Bernoulli shift σ (. . . , n−1 , n0 , n1 , . . .) = (. . . , n0 , n1 , n2 , . . .), µ is a σ −1 -invariant strongly mixing measure, Y := Φ (Z ) is a T -invariant dense subset of X , Φ σ −1 = T Φ , and then the Borel probability measure µ on X defined by µ(A) = µ(Φ −1 (A)), A ∈ B(X ), is T -invariant and strongly mixing. We will use the slight generalization of the Frequent Hypercyclicity Criterion for operators given in [2, Remark 9.10]. Theorem 1. Let T be an operator on a separable F -space X . If there is a dense subset X0 of X and a sequence of maps Sn : X0 → X such that, for each x ∈ X0 ,
∞
T n x converges unconditionally, (i) n∞=0 (ii) n=0 Sn x converges unconditionally, and (iii) T n Sn x = x and T m Sn x = Sn−m x if n > m, then there is a T -invariant strongly mixing Borel probability measure µ on X with full support. Proof. We suppose that X0 = {xn ; n ∈ N} with x1 = 0 and Sn 0 = 0 for all n ∈ N. Let (Un )n be a basis of balanced open 0-neighborhoods in X such that Un+1 + Un+1 ⊂ Un , n ∈ N. By (i) and (ii), there exists an increasing sequence of positive integers (Nn )n with Nn+2 − Nn+1 > Nn+1 − Nn for all n ∈ N such that
T k xmk ∈ Un+1
and
k>Nn
Sk xmk ∈ Un+1 ,
if mk ≤ 2l, for Nl < k ≤ Nl+1 , l ≥ n.
(1)
k>Nn
Actually, this is a consequence of the completeness of X and the fact that, for each 0-neighborhood U and for all l ∈ N, there is N ∈ N such that k∈F T k x ∈ U and k∈F Sk x ∈ U for any finite subset F ⊂]N , +∞[ and for each x ∈ {x1 , . . . , x2l }. 1. The model probability space (Z , µ) We define K = k∈Z Fk where Fk = {1, . . . , m}
if Nm < |k| ≤ Nm+1 , m ∈ N,
and Fk = {1},
if |k| ≤ N1 .
Let K (s) := σ (K ), s ∈ Z, where σ : N → N is the backward shift. K (s) is a compact space when endowed with the product topology inherited from NZ , s ∈ Z. ∞ We consider in NZ the product measure µ = k∈Z µk , where µk ({n}) = pn for all n ∈ N and µk (N) = n=1 pn = 1, k ∈ Z. The values of pn ∈]0, 1[ are selected such that, if s
βj :=
j
Z
Z
Nj+1 −Nj ,
pi
j ∈ N, then
i =1
Let Z =
s∈Z
∞
βj > 0.
j =1
K (s). We have
µ(Z ) ≥ µ(K ) =
|k|≤N1
µk ({1})
∞ l =1
Nl <|k|≤Nl+1
µk ({1, . . . , l}) =
2N +1 p1 1
∞
2 βl
> 0.
l=1
It is well-known [3] that µ is a σ −1 -invariant strongly mixing Borel probability measure. Since σ (Z ) = Z and it has positive measure, then µ(Z ) = 1. 2. The map Φ Given s ∈ Z we define the map Φ : K (s) → X by
Φ ((nk )k∈Z ) =
k<0
S−k xnk + xn0 +
T k xnk .
(2)
k>0
Φ is well-defined since, given (nk )k∈Z ∈ K (s) and for l ≥ |s|, we have nk ≤ 2l if Nl < |k| ≤ Nl+1 , which shows the convergence of the series in (2) by (1). Φ is also continuous. Indeed, let (α(j))j be a sequence of elements of K (s) that converges to α ∈ K (s) and fix any n ∈ N with n > |s|. We will find n0 ∈ N such that Φ (α(j)) − Φ (α) ∈ Un for n ≥ n0 . To do this, by definition of the topology in K (s) there exists n0 ∈ N such that
α(j)k = αk if |k| ≤ Nn+1 and j ≥ n0 .
464
M. Murillo-Arcila, A. Peris / J. Math. Anal. Appl. 398 (2013) 462–465
By (1) we have
Φ (α(j)) − Φ (α) =
S−k (xα(j)k − xαk ) +
T k (xα(j)k − xαk ) ∈ Un
k>Nn+1
k<−Nn+1
for all j ≥ n0 . This shows the continuity of Φ : K (s) → X for every s ∈ Z. The map Φ is then well-defined on Z , and Φ : Z → X is measurable (i.e., Φ −1 (A) ∈ B(Z ) for every A ∈ B(X )). 3. The measure µ on X L(s) := Φ (K (s)) is compact in X , s ∈ Z, and Y := s∈Z L(s) is a T -invariant Borel subset of X because Φ σ −1 = T Φ . We then define in X the measure µ(A) = µ(Φ −1 (A)) for all A ∈ B(X ). Obviously, µ is well-defined and it is a T -invariant strongly mixing Borel probability measure. The proof is completed by showing that µ has full support. Given a non-empty open set U in X , we pick n ∈ N satisfying xn + Un ⊂ U. Thus
µ(U ) ≥ µ
x = xn +
k
T xmk +
k>Nn
≥ µ0 ({n})
Sk xmk ; mk ≤ 2l for Nl < k ≤ Nl+1 , l ≥ n
k>Nn
µk ({1})
0<|k|≤Nn
∞
l=n
µk ({1, . . . , 2l}) >
2N pn p1 n
Nl <|k|≤Nl+1
∞
2 βl
> 0.
l =n
As we mentioned in Section 1, Theorem 1 can be deduced from [11, Corollary 1.3] when dealing with operators on separable complex Fréchet spaces. Indeed, the argument of É. Matheron is the following (we thank Grivaux for letting us know about it): Let T : X → X be an operator on a separable complex Fréchet space X satisfying the hypothesis of the Frequent Hypercyclicity Criterion given in Theorem 1, and suppose that X0 = {xn ; n ∈ N}. We define the following family of continuous T-eigenvector fields for T : Em (λ) =
λ−n T n xm +
λn Sn xm ,
λ ∈ T, m ∈ N .
n∈N
n≥0
They span X since, for any functional x∗ that vanishes on Em (λ) for each λ ∈ T and m ∈ N, the equality ⟨x∗ , Em (λ)⟩ = 0 for fixed m and for all λ ∈ T implies that ⟨x∗ , T n xm ⟩ = 0 for every n ≥ 0. Thus, ⟨x∗ , xm ⟩ = 0 for each m ∈ N, and by density, x∗ = 0. The previous theorem can be applied to different classes of operators. A distinguished one is the class of weighted shifts on sequence F -spaces. By a sequence space we mean a topological vector space X which is continuously included in ω, the countable product of the scalar field K. A sequence F -space is a sequence space that is also an F -space. Given a sequence w = (wn )n of non-zero weights, the associated unilateral (respectively, bilateral) weighted backward shift Bw : KN → KN is defined by Bw (x1 , x2 , . . .) = (w2 x2 , w3 x3 , . . .) (respectively, Bw : KZ → KZ is defined by Bw (. . . , x−1 , x0 , x1 , . . .) = (. . . , w0 x0 , w1 x1 , w2 x2 , . . .)). When a sequence F -space X is invariant under certain weighted backward shift T , then T is also continuous on X by the closed graph theorem. We refer the reader to, e.g., Chapter 4 of [2] for more details about hypercyclic and chaotic weighted shifts on Fréchet sequence spaces. In particular, Theorems 4.6 and 4.12 in [2] (we refer the reader to [17] for the original results) remain valid for F -spaces, and a bilateral (respectively, unilateral) weighted backward shift T : X → X on a sequence F -space X in which the canonical unit vectors ( e ) (respectively, ( e ) ) form an unconditional basis is chaotic if, and only if, n n ∈ Z n n ∈ N n∈Z en (respectively, n∈N en ) converges unconditionally. Corollary 2. Let T : X → X be a chaotic bilateral weighted backward shift on a sequence F -space X in which (en )n∈Z is an unconditional basis. Then there exists a T -invariant strongly mixing Borel probability measure on X with full support. Remark 3. The preceding result can be improved if T is a unilateral backward shift operator on a sequence F -space. In that case, there exists a T -invariant exact Borel probability measure on X with full support. Proof. Let M = {zn ; n ∈ N} be a countable dense set in K with ∞z1 = 0. Let (Un )n be a basis of balanced open 0-neighborhoods in X such that Un+1 + Un+1 ⊂ Un , n ∈ N. Since T is chaotic, n=1 en converges unconditionally, so there exists an increasing sequence of positive integers (Nn )n with Nn+2 − Nn+1 > Nn+1 − Nn for all n ∈ N such that
αk ek ∈ Un+1 ,
if αk ∈ {z1 , . . . , z2m }, for Nm < k ≤ Nm+1 , m ≥ n.
k>Nn
We define K =
k∈N
Fk where
Fk = {z1 , . . . , zm }
if Nm < k ≤ Nm+1 , m ∈ N,
and Fk = {z1 },
if k ≤ N1 .
(3)
M. Murillo-Arcila, A. Peris / J. Math. Anal. Appl. 398 (2013) 462–465
465
Let K (s) := σ s (K ), s ≥ 0. K (s) is a compact space when endowed with the product topology inherited from M N , s ≥ 0. We ∞ N consider in M the product measure µ = k∈N µk , where µk ({zn }) = pn for all n ∈ N and µk (M ) = n=1 pn = 1, k ∈ N. As before, we select the sequence (pn )n of positive numbers such that, if
βj =
j
Nj+1 −Nj ,
pi
then
∞
βj > 0.
j =1
i=1
It is known [3, Section 4.12] that µ is a σ -invariant exact Borel probability measure. By setting Z = µ(Z ) = 1. Now we define the map Φ : K (s) → X given by
Φ ((αk )k∈N ) =
∞
s≥0
K (s), we obtain
αk ek .
k=1
Φ is (well-defined and) continuous, s ≥ 0. We have that Φ : Z → X is measurable. L(s) := Φ (K (s)) is compact in X , s ≥ 0, and Y := s≥0 L(s) = Φ (Z ) is a T -invariant Borel subset of X . We then define on X the measure µ(A) = µ(Φ −1 (A)) for all A ∈ B(X ). As in Theorem 1, we conclude that µ is welldefined, and now it is a T -invariant exact Borel probability measure with full support.
Devaney chaos is therefore a sufficient condition for the existence of strongly mixing measures within the framework of weighted shift operators on sequence F -spaces. For some natural spaces it is even a characterization of this fact. For instance, Bayart and Ruzsa [18] recently proved that weighted shift operators on ℓp , 1 ≤ p < ∞, are frequently hypercyclic if, and only if, they are Devaney chaotic. It turns out that this is equivalent to the existence of an invariant strongly mixing Borel probability measure with full support on ℓp . Also, for the space ω, every weighted shift operator is chaotic [17]. In particular, for the unilateral case we obtain exact measures. Example 4. Every unilateral weighted backward shift operator on ω = KN admits an invariant exact Borel probability measure with full support on ω. We finish the paper by mentioning that a continuous-time version of Theorem 1 can be given by using the Frequent Hypercyclicity Criterion for C0 -semigroups introduced in [19]. This is part of a forthcoming paper [20]. Acknowledgments This work was supported in part by MEC and FEDER, Project MTM2010-14909, and by GV, Project PROMETEO/2008/101. The first author was also supported by a grant from the FPU Program of MEC. We thank the referee whose detailed report led to an improvement in the presentation of this work. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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