Recurrence for weighted translations on groups

Acta Mathematica Scientia 2016,36B(2):443–452 http://actams.wipm.ac.cn

RECURRENCE FOR WEIGHTED TRANSLATIONS ON GROUPS∗

¥A)

Chung-Chuan CHEN (

Department of Mathematics Education, National Taichung University of Education, Taiwan E-mail : [email protected] Abstract Let G be a locally compact group, and let 1 ≤ p < ∞. We characterize topologically multiply recurrent weighted translation operators on Lp (G) in terms of the Haar measure and the weight function. We also show that there do not exist any recurrent weighted translation operators on L∞ (G). Key words

Topologically multiple recurrence; recurrence; hypercyclicity; locally compact group; Lp -space

2010 MR Subject Classification

1

37B20; 47A16; 43A15

Introduction

Recently, we gave sufficient and necessary conditions for weighted translation operators on groups to be hypercyclic and chaotic in [1–4], which subsumes some works in [5–8]. The notion of hypercyclicity in linear dynamics is close to, but stronger than the notion of recurrence in topological dynamics in [9]. It is well known that every hypercyclic operator is recurrent on separable Banach spaces in [10]. However, this is not the case for topologically multiple recurrence, which is a stronger notion than recurrence. There exists a hypercyclic weighted backward shift on ℓ2 (Z) in [10], which is not topologically multiply recurrent. In this note, we will give a sufficient and necessary condition for weighted translation operators on the Lebesgue space Lp (1 ≤ p < ∞) of a locally compact group to be topologically multiply recurrent in terms of the Haar measure and the weight function, and show, on the L∞ space, there are no recurrent weighted translation operators. In linear dynamics, we first recall that an operator T on a Banach space X is called hypercyclic if there exists a vector x ∈ X such that its orbit under T denoted by Orb(x, T ) := {x, T x, T 2 x, · · · } is dense in X in which x is said to be a hypercyclic vector of T . It is known that hypercyclicity is equivalent to topological transitivity. An operator T is topologically transitive if given two nonempty open subsets U, V ⊂ X, there is some n ∈ N such that T n U ∩ V 6= ∅. If T n U ∩ V 6= ∅ from some n onwards, then T is called topologically mixing. Hypercyclicity and transitivity have been studied by many authors. We refer to these books [11, 12] on this subject. In topological dynamics, an operator T is topologically multiply ∗ Received May 9, 2014; revised June 17, 2015. The author is supported by MOST of Taiwan (MOST 104-2115-M-142-002-).

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recurrent if for every positive integer N and every nonempty open set U in X, there is some n ∈ N such that U ∩ T −n U ∩ T −2n U ∩ · · · ∩ T −N n U 6= ∅. If N = 1, then T is called recurrent, that is, the condition U ∩ T −n U 6= ∅ is satisfied. The motivation to connect hypercyclicity with recurrence is inspired by the works in [9, 10]. In [10], Costakis and Parissis characterized topologically multiply recurrent weighted shifts on ℓp (Z) in terms of the weight sequence. On the other hand, that the space ℓ∞ (Z) does not support any recurrent weighted shifts was shown in [9]. We note that the weighted shifts on ℓp (Z) and ℓ∞ (Z) are a special case of the weighted translation operators on the Lebesgue space of a locally compact group. In this article, we will extend some results in [9, 10] to the setting of translations on groups. In what follows, let G be a locally compact group with identity e and a right-invariant Haar measure λ. We denote by Lp (G) (1 ≤ p ≤ ∞) the complex Lebesgue space, with respect to λ. A bounded function w : G → (0, ∞) is called a weight on G. Let a ∈ G and let δa be the unit point mass at a. A weighted translation on G is a weighted convolution operator Ta,w : Lp (G) −→ Lp (G) defined by Ta,w (f ) = wTa (f )

(f ∈ Lp (G)),

where w is a weight on G and Ta (f ) = f ∗ δa ∈ Lp (G) is the convolution: Z (f ∗ δa )(x) := f (xy −1 )dδa (y) = f (xa−1 ) (x ∈ G). G

−1

∞

If w ∈ L (G), then the weighted translation operator Ta−1 ,w−1 ∗δa−1 is the inverse of Ta,w . We write Sa,w for Ta−1 ,w−1 ∗δa−1 to simplify notation. We assume w, w−1 ∈ L∞ (G) throughout.

2

Recurrence on Lp (G)

In this section, we will prove the result on Lp (G)(1 ≤ p < ∞) for translations by aperiodic elements in G. An element a in a group G is called a torsion element if it is of finite order. In a locally compact group G, an element a ∈ G is called periodic (or compact) in [4] if the closed subgroup G(a) generated by a is compact. We call an element in G aperiodic if it is not periodic. For discrete groups, periodic and torsion elements are identical. It is proved in [4] that an element a ∈ G is aperiodic if and only if for any compact set K ⊂ G, there exists some N ∈ N such that K ∩ Ka±n = ∅ for all n > N . We will make use of the aperiodic condition to obtain the result below. We note that [4] in many familiar non-discrete groups, including the additive group Rd , the Heisenberg group, and the affine group, all elements except the identity are aperiodic. Theorem 2.1 Let G be a locally compact group and let a be an aperiodic element in G. Let 1 ≤ p < ∞ and Ta,w be a weighted translation on Lp (G). The following conditions are equivalent. (i) Ta,w is topologically multiply recurrent; (ii) For each N ∈ N and each compact subset K ⊂ G with λ(K) > 0, there is a sequence of Borel sets (Ek ) in K such that λ(K) = limk→∞ λ(Ek ) and both sequences (for 1 ≤ l ≤ N ) !−1 ln ln−1 Y Y s s ϕln := w ∗ δa−1 and ϕ eln := w ∗ δa s=1

s=0

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admit, respectively, subsequences (ϕlnk ) and (ϕ elnk ) satisfying

lim kϕlnk |Ek k∞ = lim kϕ elnk |Ek k∞ = 0.

k→∞

k→∞

Proof (i) ⇒ (ii). Let Ta,w be topologically multiply recurrent. Let K ⊂ G be a compact set with λ(K) > 0. Let ε ∈ (0, 1). By aperiodicity of a, there is some M such that K ∩Ka−n = ∅ for n > M . Let χK ∈ Lp (G) be the characteristic function of K. ε and let U = {g ∈ Lp (G) : kg − χK kp < δ 2 }. Given some N ∈ N, there Choose 0 < δ < 1+ε exists m > M such that −m −2m −N m U ∩ Ta,w U ∩ Ta,w U ∩ · · · ∩ Ta,w U 6= ∅

by the topologically multiply recurrent assumption. Hence, there exists f ∈ Lp (G) such that kf − χK kp < δ 2

lm and kTa,w f − χK k p < δ 2

for l = 1, 2, . . . , N . Let A = {x ∈ K : |f (x) − 1| ≥ δ} and B = {x ∈ G \ K : |f (x)| ≥ δ}. Then, we have |f (x)| > 1 − δ

(x ∈ K \ A)

and |f (x)| < δ

for x ∈ (G \ K) \ B.

Moreover, by the inequality below,

Z δ 2p > kf − χK kpp = |f (x) − χK (x)|p dλ(x) G Z ≥ |f (x) − 1|p dλ(x) ≥ δ p λ(A), A

p

we have λ(A) < δ . Similarly, λ(B) < δ p . In contrast, let Cl,m = {x ∈ K : |ϕ elm (x)−1 f (xa−lm ) − 1| ≥ δ} and

Dl,m = {x ∈ K : |ϕlm (x)f (x)| ≥ δ}. Then,

and

ϕ elm (x)−1 |f (xa−lm )| > 1 − δ ϕlm (x)|f (x)| < δ

Again, by applying the inequality

(x ∈ K \ Cl,m )

(x ∈ K \ Dl,m ).

Z lm lm δ 2p > kTa,w f − χK kpp = |Ta,w f (x) − χK (x)|p dλ(x) G Z ≥ |w(x)w(xa−1 ) · · · w(xa−(lm−1) )f (xa−lm ) − 1|p dλ(x) Cl,m

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=

Z

Cl,m

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|ϕ elm (x)−1 f (xa−lm ) − 1|p dλ(x) ≥ δ p λ(Cl,m ),

we have λ(Cl,m ) < δ p . Using the similar argument and the right invariance of Haar measure, we have λ(Dl,m ) < δ p . As K ∩ Ka−lm = ∅, we have ϕ elm (x) <

and

δ |f (xa−lm )| < < ε for 1−δ 1−δ

ϕlm (x) <

δ δ < <ε |f (x)| 1−δ

Let Em = (K \ A) \

N [

x ∈ K \ (Cl,m ∪ Balm )

for x ∈ K \ (Dl,m ∪ A).

(Balm ∪ Cl,m ∪ Dl,m ).

l=1 p

Then, we have λ(K \ Em ) < 4N δ and kϕlm |Em k∞ < ε, kϕ elm |Em k∞ < ε, which implies condition (ii). (ii) ⇒ (i). We show that Ta,w is topologically multiply recurrent. Let U be a non-empty open subset of Lp (G). As the space Cc (G) of continuous functions on G with compact support is dense in Lp (G), we can pick f ∈ Cc (G) with f ∈ U . Let K be the compact support of f . Given some N , let Ek ⊂ K and the sequences (ϕln ), (ϕ eln ) satisfy condition (ii). By aperiodicity of a, there exists M ∈ N such that K ∩ Ka±n = ∅ for any n > M . Let ε > 0. There exists M ′ ∈ N such that nk > M and ϕplnk < kfεkpp for k > M ′ . Hence, Z lnk kTa,w f χEk kpp = |w(x)w(xa−1 ) · · · w(xa−(lnk −1) )|p |f (xa−lnk )|p dλ(x) Ek alnk Z = |w(xalnk )w(xalnk −1 ) · · · w(xa)|p |f (x)|p dλ(x) Ek Z = ϕplnk (x)|f (x)|p dλ(x) ≤ kϕlnk |Ek kp∞ kf kpp → 0 Ek

lnk as k → ∞ for 1 ≤ l ≤ N . By the sequence (ϕ elnk ), we have lim kSa,w f χEk kp = 0 for k→∞

l = 1, 2, · · · , N . Now, we are ready to achieve our goal. For each k ∈ N, let

nk 2nk N nk vk = f χEk + Sa,w f χEk + Sa,w f χEk + · · · + Sa,w f χEk ∈ Lp (G).

Then, by the aperiodicity of a, we have kvk − f kpp ≤ kf kp∞ λ(K \ Ek ) +

N X

lnk kSa,w f χEk kpp

l=1

and lnk lnk (l−1)nk nk kTa,w vk − f kpp ≤ kTa,w f χEk kpp + kTa,w f χEk kpp + · · · + kTa,w f χEk kpp nk (N −l)nk +kf χEk − f kpp + kSa,w f χEk kpp + · · · + kSa,w f χEk kpp ,

which implies −nk −2nk −N nk U ∩ Ta,w U ∩ Ta,w U ∩ · · · ∩ Ta,w U 6= ∅.



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Example 2.2 Let G = Z, a = −1 ∈ Z which is aperiodic. Let w ∗ δ−1 be a weight on Z. Then, the weighted translation operator T−1,w∗δ−1 is given by (f ∈ ℓp (Z)).

T−1,w∗δ−1 f (j) = w(j + 1)f (j + 1)

In fact, the operator T−1,w∗δ−1 is just the bilateral weighted backward shift T , given by T ej = wj ej−1 with wj = w(j). Here, (ej )j∈Z is the canonical basis of ℓp (Z) and (wj )j∈Z is a sequence of positive real numbers. Hence by Theorem 2.1, the operator T = T−1,w∗δ−1 is topologically multiply recurrent if given ε > 0 and N, q ∈ N, there exists a positive integer n such that for all |j| < q and 1 ≤ l ≤ N , we have

s=1

ln−1 Y

ln−1 Y

ln Y

ϕln (j) =

ln Y

(w ∗ δ−1 ) ∗ δ1s (j) =

w(j − s) < ε

s=0

and −1

ϕ eln (j) =

s (w ∗ δ−1 ) ∗ δ−1 (j) =

s=0

w(j + s) >

s=1

1 , ε

which is the condition in [10, Proposition 5.3]. In contrast, if w−1 ∈ ℓ∞ (Z), then T1,w−1 is the inverse of the operator T−1,w∗δ−1 , and the above weight conditions suffices T1,w−1 to be multiply recurrent. Indeed, the weight conditions in Theorem 2.1 for T1,w−1 is ln Y

s w−1 ∗ δ−1 (j) =

s=1

1 ln Q

<ε

w(j + s)

s=1

and ln−1 Y s=0

w−1 ∗ δ1s (j) =

1 ln−1 Q

>

w(j − s)

1 , ε

s=0

which is the same with the conditions for T−1,w∗δ−1 . If we define w : Z → (0, ∞) as   1 if j < 0; w(j) = 3  3 if j ≥ 0, then both T−1,w∗δ−1 and T1,w−1 are topologically multiply recurrent.

Remark 2.3 In Example 2.2, both T−1,w∗δ−1 and the inverse of T−1,w∗δ−1 could be multiply recurrent simultaneously. In fact, this is also true for general cases. That is, an invertible weighted translation Ta,w generated by an aperiodic element is multiply recurrent on Lp (G), if and only if the inverse of Ta,w is multiply recurrent. Example 2.4 Let G = R, a = 3, and w be a weight on R. Then, the weighted translation T3,w on Lp (R) is given by T3,w f (x) = w(x)f (x − 3)

(f ∈ Lp (R)).

By Theorem 2.1, the operator T3,w is topologically multiply recurrent if given ε > 0, some N ∈ N, and a compact subset K of R, there exists a positive integer n such that for 1 ≤ l ≤ N

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and x ∈ K, we have ln Y

s w ∗ δ−3 (x) =

ln−1 Y

w ∗ δ3s (x) =

ϕln (x) =

s=1

and −1

ϕ eln (x) =

ln Y

w(x + 3s) < ε

s=1

s=0

ln−1 Y

w(x − 3s) >

s=0

1 . ε

We may define w : R → (0, ∞) by

 1   if x ≥ 1;    3 1 w(x) = if − 1 < x < 1;  x  3     3 if x ≤ −1,

which satisfies the above weight condition. Example 2.5 Let

       1 x z    G = H :=  : x, y, z ∈ R 0 1 y       0 0 1

be the Heisenberg group which is neither abelian nor compact. For convenience, an element in G is written as (x, y, z). Let (x, y, z), (x′ , y ′ , z ′ ) ∈ H. Then, the multiplication is given by (x, y, z) · (x′ , y ′ , z ′ ) = (x + x′ , y + y ′ , z + z ′ + xy ′ ) and (x, y, z)−1 = (−x, −y, xy − z). Let a = (1, 0, 2) and w be a weight on H. Then, a−1 = (−1, 0, −2) and the weighted translation T(1,0,2),w on Lp (H) is given by T(1,0,2),w f (x, y, z) = w(x, y, z)f (x − 1, y, z − 2)

(f ∈ Lp (H)).

By Theorem 2.1, the operator T(1,0,2),w is topologically multiply recurrent if given ε > 0, some N ∈ N, and a compact subset K of H, there exists a positive integer n such that for 1 ≤ l ≤ N and x ∈ K, we have ϕln (x, y, z) =

ln Y

s w ∗ δ(1,0,2) (x, y, z) = −1

s=1

and −1

ϕ eln (x, y, z) =

ln−1 Y s=0

ln Y

w(x + s, y, z + 2s) < ε

s=1

s w ∗ δ(1,0,2) (x) =

ln−1 Y s=0

w(x − s, y, z − 2s) >

1 . ε

Similarly, one can obtain the required weight condition by defining w : H → (0, ∞) as follows:  1   if z ≥ 1;   3   1 w(x, y, z) = if − 1 < z < 1;   3z     3 if z ≤ −1.

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It is known that Devaney’s notion of chaos consists of topological transitivity together with periodicity. That is, an operator T on a Banach space is chaotic if T is transitive and the set of periodic elements, {x ∈ X : ∃ n ∈ N with T n x = x}, is dense in X. In [1, Corollary 2.3], we characterize chaotic weighted translation operators on discrete groups. Corollary 2.6 ([1, Corollary 2.3]) Let G be a discrete group and let a be a non-torsion element in G. Let 1 ≤ p < ∞ and Ta,w be a weighted translation on ℓp (G). Then, the following conditions are equivalent. (i) Ta,w is chaotic. (ii) For each finite subset K ⊂ G, both sequences !−1 n n−1 Y Y s s ϕn = w ∗ δa−1 and ϕ en = w ∗ δa s=1

s=0

admit, respectively, subsequences (ϕnk ) and (ϕ enk ) satisfying lim

k→∞

∞ X X

ϕplnk (x)

+

l=1 K

∞ X X l=1 K

!

ϕ eplnk (x)

= 0.

The following corollaries reveal that both chaos on discrete groups and topological mixing on locally compact groups are stronger notions than topologically multiple recurrence. Corollary 2.7 Let G be a discrete group and let a be a non-torsion element in G. Let 1 ≤ p < ∞ and Ta,w be a weighted translation operator on ℓp (G). If Ta,w is chaotic, then Ta,w is topologically multiply recurrent on ℓp (G). Proof Let Ta,w be chaotic on ℓp (G). By the weight condition in Corollary 2.6, Ta,w is topologically multiply recurrent.  Corollary 2.8 Let G be a locally compact group and let a be an aperiodic element in G. Let 1 ≤ p < ∞ and Ta,w be a weighted translation operator on Lp (G). If Ta,w is topologically mixing, then Ta,w is topologically multiply recurrent on Lp (G). ε . Let K be a compact set of G, and let Proof Let ε > 0 and choose 0 < δ < 1+ε p 2 U = {g ∈ L (G) : kg − χK kp < δ }. Given N ∈ N, by the topologically mixing assumption, there exists some m ∈ N such that m Ta,w (U ) ∩ U 6= ∅

from m onwards. Hence, lm Ta,w (U ) ∩ U 6= ∅ lm for l = 1, 2, · · ·, N . Therefore, we can pick, for each l, a function fl ∈ U with Ta,w fl ∈ U which gives

kfl − χK k < δ 2

and

lm kTa,w f l − χK k < δ 2 .

Using this for each fl and repeating the arguments in the proof of Theorem 2.1, we find Borel sets Ek ⊂ K such that λ(K) = lim λ(Ek ) and k→∞

lim kϕlnk |Ek k = lim kϕ elnk |Ek k = 0,

k→∞

k→∞

which says Ta,w is topologically multiply recurrent.



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Example 2.9 Let G = Z, a = −1 ∈ Z and let w ∗ δ−1 be a weight on Z. Then, T−1,w∗δ−1 is the bilateral weighted backward shift on ℓp (Z). By Corollaries 2.7 and 2.8, the operator T−1,w∗δ−1 is topologically multiply recurrent if T−1,w∗δ−1 is topologically mixing or chaotic. Remark 2.10 We note that there exists a weighted shift which is topologically multiply recurrent but is neither chaotic nor topologically mixing in [13]. To end up this section, we show that for weighted translations Ta,w , hypercyclicity is equivalent to recurrence. We, first, recall a work in [3]. Theorem 2.11 ([3, Theorem 2.1]) Let G be a locally compact group and let a be an aperiodic element in G. Let 1 ≤ p < ∞ and Ta,w be a weighted translation on Lp (G). The following conditions are equivalent. (i) Ta,w is hypercyclic. (ii) For each compact subset K ⊂ G with λ(K) > 0, there is a sequence of Borel sets (Ek ) in K such that λ(K) = lim λ(Ek ) and both sequences k→∞

ϕn :=

n Y

w∗

δas−1

s=1

and

ϕ en :=

n−1 Y

w∗

δas

!−1

s=0

admit, respectively, subsequences (ϕnk ) and (ϕ enk ) satisfying

lim kϕnk |Ek k∞ = lim kϕ enk |Ek k∞ = 0.

k→∞

k→∞

Now, we state the result below.

Corollary 2.12 Let G be a locally compact group and let a be an aperiodic element in G. Let 1 ≤ p < ∞ and Ta,w be a weighted translation on Lp (G). Then, Ta,w is hypercyclic if and only if Ta,w is recurrent. Proof As hypercyclicity implies recurrence, we only need to show if Ta,w is recurrent, then Ta,w is hypercyclic. Let N = 1 in Theorem 2.1, then the condition (ii) in Theorem 2.1 suffices Ta,w to be hypercyclic by Theorem 2.11.  Example 2.13 It was shown in [10, Proposition 5.1] that the bilateral weighted shift on ℓ (Z) is hypercyclic if and only if it is recurrent. One can obtain this result in the following way. Let G = Z, a = −1 ∈ Z and let w ∗ δ−1 be a weight on Z. Then, T−1,w∗δ−1 is the bilateral weighted backward shift on ℓp (Z). By Corollary 2.12, hypercyclicity and recurrence occur on T−1,w∗δ−1 simultaneously. p

3

Recurrence on L∞ (G)

It is known in [14, 15] that a complex Banach space admits a hypercyclic operator if and only if it is infinite-dimensional and separable. Therefore, there is no hypercyclic weighted translation operator Ta,w on L∞ (G) by the fact L∞ (G) is non-separable. Moreover, we will show, in this section, that the space L∞ (G) does not support any recurrent weighted translation operators Ta,w . Theorem 3.1 Let G be a locally compact group and a ∈ G be an aperiodic element. Let w ∈ L∞ (G) be a weight on G with w−1 ∈ L∞ (G). Then, there does not exist recurrent

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weighted translation operator Ta,w on L∞ (G). Proof Suppose that Ta,w is recurrent. Let M > 1. Given a non-null compact set K, we define g(x) = M if x ∈ K; otherwise, g(x) = 3. As Ta,w is recurrent, there exist a recurrent vector f and a positive integer n > 1 such that Ka±(n−1) ∩K = ∅ from n onwards, kf −gk∞ < 21 and n−1 1 Y n −s −n kTa,w f − f k∞ = ess sup w(xa )f (xa ) − f (x) < . 2 x∈G s=0 Hence, 2 < |f (x)| < 4 a.e. on G \ K. Moreover, we have n−1 Y M −1< w(xa−s )f (xa−n ) < M + 1

a.e. on K.

s=0

Let E = K \ Ka−1 , then λ(E) > 0. Otherwise, K ⊂ Ka−1 a.e., and then Kan ⊂ Kan−1 ⊂ · · · ⊂ K, which is impossible. Hence, n−1 Y −s −n a.e. on Ea, 2< w(xa )f (xa ) < 4 s=0

which implies

n−1 Y 2< w(xa−(s−1) )f (xa−(n−1) ) < 4

a.e. on E.

s=0

Using Ka±(n−1) ∩ K = ∅, the precious estimates, and n−1 Q −s −n w(xa )f (xa ) w(xa−(n−1) )|f (xa−n )| s=0 = , n−1 Q w(xa)|f (xa−(n−1) )| −(s−1) )f (xa−(n−1) ) w(xa s=0

we arrive at

w(xa−(n−1) ) >

w(xa)(M − 1) m(M − 1) M −1 2 1 · · · w(xa) = ≥ 4 1 4 8 8

a.e. on E, where m = ess inf{w(xa) : x ∈ K} > 0 by w−1 ∈ L∞ (G). As M can be chosen to arbitrarily large, we conclude w 6∈ L∞ (G). Hence, Ta,w is not recurrent.  Example 3.2 Let G = Z, a = −1 ∈ Z and let w ∗ δ−1 be a weight on Z. Then, T−1,w∗δ−1 is the bilateral weighted backward shift on ℓ∞ (Z). By Theorem above, there is no recurrent bilateral weighted backward shift T−1,w∗δ−1 on ℓ∞ (Z). Hence, a result in [9, Theorem 5.1] is recovered. A well known result of Salas in [8] says that the operator I + T is hypercyclic whenever T is any unilateral weighted backward shift on ℓ2 (N). In the non-separable space L∞ (G), the operator I + Ta,w can never be recurrent on L∞ (G). Theorem 3.3 Let G be a locally compact group and a ∈ G. Let w ∈ L∞ (G) be a weight on G with w−1 ∈ L∞ (G). Let Ta,w be a weighted translation operator on L∞ (G). Then, the operator I + Ta,w is not recurrent on L∞ (G). Proof Suppose that I + Ta,w is recurrent. Let g(x) = 1 for x ∈ G. Then, there exist a recurrent vector f and an arbitrarily large integer N with N m > 2, where m = ess inf{w(x) :

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x ∈ G} > 0, satisfying kf − gk∞ < N

k(I + Ta,w ) f − f k∞

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and

N   X N = ess sup l x∈G l=1

l−1 Y

s=0

w(xa

−s

) f (xa ) < 1.

Taking real parts in the inequalities above, we have Re(f (xa−1 )) > ! N   l−1 X Y N −s w(xa ) Ref (xa−l ) < 1 l s=0

! 1 2

−l

and

l=1

a.e. on G, which implies

1 N m < N w(x)Re(f (xa−1 )) < 1. 2 We conclude that N m < 2, which is a contradiction. Hence, I + Ta,w is not recurrent.



Example 3.4 Let G = Z and a = 0 ∈ Z. Let w be a weight on Z. Then, the operator I + T0,w is not recurrent on ℓ∞ (Z). References [1] Chen C C. Chaotic weighted translations on groups. Arch Math, 2011, 97: 61–68 [2] Chen C C. Supercyclic and Ces` aro hypercyclic weighted translations on groups. Taiwanese J Math, 2012, 16: 1815–1827 [3] Chen C C. Hypercyclic weighted translations generated by non-torsion elements. Arch Math, 2013, 101: 135–141 [4] Chen C C, Chu C H. Hypercyclic weighted translations on groups. Proc Amer Math Soc, 2011, 139: 2839–2846 [5] Costakis G, Sambarino M. Topologically mixing hypercyclic operators. Proc Amer Math Soc, 2004, 132: 385–389 [6] Grosse-Erdmann K G. Hypercyclic and chaotic weighted shifts. Studia Math, 2000, 139: 47–68 [7] Le´ on-Saavedra F. Operators with hypercyclic Ces` aro means. Studia Math, 2002, 152: 201–215 [8] Salas H. Hypercyclic weighted shifts. Trans Amer Math Soc, 1995, 347: 993–1004 [9] Costakis G, Manoussos A, Parissis I. Recurrent linear operators. Complex Anal Oper Th, 2014, 8: 1601– 1643 [10] Costakis G, Parissis I. Szemer` edi’s theorem, frequent hypercyclicity and multiple recurrence. Math Scand, 2012, 110: 251–272 ´ Dynamics of linear operators. Cambridge: Cambridge University Press, 2009 [11] Bayart F, Matheron E. [12] Grosse-Erdmann K G, Peris A. Linear Chaos. London: Springer, 2011 [13] Bayart F, Grivaux S. Invariant Gaussian measures for operators on Banach spaces and linear dynamics. Proc Lond Math Soc, 2007, 94: 181–210 [14] Ansari S I. Existence of hypercyclic operators on topological vector spaces. J Funct Anal, 1997, 148: 384–390 [15] Bernal-Gonz´ alez L. On hypercyclic operators on Banach spaces. Proc Amer Math Soc, 1999, 127: 1003– 1010