# S-Mixing Tuple of Operators on Banach Spaces

Authors: Wei Wang; Yonglu Shu; Xingzhong Wang
Journal: Journal of Function Spaces
 (2016)

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## Abstract

We consider the question: what is the appropriate formulation of Godefroy-Shapiro criterion for tuples of operators? We also introduce a new notion about tuples of operators,S-mixing, which lies between mixing and weakly mixing. We also obtain a sufficient condition to ensure a tuple of operators to be S-mixing. Moreover, we study some new properties of S-mixing operators on several concrete Banach spaces.


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## Body

## 1. Introduction

A continuous linear operatorT acting on a topological vector space X is called hypercyclic, if there exists a vector x∈X such that the orbit of x under T is dense in X; that is, orb(x,T)¯={x,Tx,T2x,…}¯=X. Such a vector x is called a hypercyclic vector for T and the set of hypercyclic vectors for T will be denoted by HC(T).Rolewicz [1] and Kitai [2] proved that there does not exist hypercyclic operator on finite-dimensional space. This implies that hypercyclic phenomenon only appears in infinite dimensional space for a single operator. For more details on this topic, one can refer to [3, 4].In 2006, Feldman provided a new class of hypercyclic operators, tuples of operators, and showed us a different direction from the previous results. This gives us hypercyclic phenomenon in both separable infinite dimensional space and finite-dimensional space. The first example that the author showed to us is that there exists an(n+1)-tuple of diagonal matrices that is hypercyclic on Cn.For anm-tuple of operators we mean a finite sequence of length m of mutually commuting continuous linear operators on a Banach space X. Let T=(T1,T2,…,Tm) be an m-tuple acting on a Banach space X (more general complete topology vector space), and F is the semigroup generated by T; that is, (1)F=T=T1k1T2k2⋯Tmkm:ki∈N,i=1,…,m,which is a finitely generated abelian semigroup.In [5, 6], Feldman extended the definitions of hypercyclic and topologically transitive of single bounded linear operator to tuples of operators. This started a new study in the field of hypercyclicity. In [7, 8], Yousefi gave the definition of mixing tuples of operators, and he obtained some sufficient conditions for a tuple of continuous operators to be hereditarily transitive and investigated the relationships between hypercyclicity and d-dense orbits of a tuple of operators.In the past few years, the authors have studied the following dynamical properties of tuplesT:(1)
hypercyclicity:orb(x,T)¯={Sx:S∈F}¯=X;(2)
chaotic (Devaney) [9]: T is hypercyclic and has a dense set of periodic points;(3)
mixing: for any pairU,V of nonempty open subsets of X, there exists some positive integer N such that (2)T1k1⋯TmkmU∩V≠⌀∀kj≥N,j=1,…,m;(4)
weakly mixing: for a tuple of operatorsT and subsets A,B⊂X define the return set as (3)NTA,B=k1,…,km∈N0m:T1k1⋯TmkmA∩B≠⌀.Then T is called weakly mixing if for every quadruple of nonempty open subsets U1,U2,V1,V2⊂X we have that NT(U1,V1)∩NT(U2,V2)≠⌀.From the above results, one can deduce that(4)mixing⟹weakly mixing⟹hypercyclic.Corresponding to Birkhoff’s transitive theorem of single continuous map which was proposed by Birkhoff in 1920, Feldman [6] and Yousefi and Ershad [10], respectively, showed that hypercyclicity and topologically transitivity of tuples of operators are equivalent on separable infinite dimension Banach space. Here we describe one of the versions of the extended Birkhoff’s transitive theorem as follows.Theorem 1.
LetX be a separable infinite dimensional Banach space X and (T1,T2) the pair of operators T1 and T2. The following assertions are equivalent: (1)
(T1,T2) is hypercyclic;(2)
(T1,T2) is topologically transitive.If one of these conditions holds then the set of points in X with dense orbit is a dense Gδ-set.As we know, for a single continuous linear operator, we can use its point spectra and corresponding eigenvalues to judge the dynamical system properties of the operator. In [11], Godefroy and Shapiro gave a criterion. The following theorem we cited was settled by Grosse-Erdmann and Peris in their book [4].Theorem 2 ((Godefroy-Shapiro criterion) [4, Theorem  3.1]).
LetT be an operator. Suppose that the subspaces (5)
X
0
≔
span
⁡
x
∈
X
:
T
x
=
λ
x
f
o
r
s
o
m
e
λ
∈
K
w
i
t
h
λ
<
1
,
Y
0
≔
span
⁡
x
∈
X
:
T
x
=
λ
x
f
o
r
s
o
m
e
λ
∈
K
w
i
t
h
λ
>
1are dense in X. Then T is mixing and in particular hypercyclic.
Moreover, ifX is a complex space and also the subspace (6)Z0≔span⁡x∈X:Tx=eαπixforsomeα∈Qis dense in X, then T is chaotic.In our another unpublished paper, we established a generalization version of Godefroy-Shapiro criterion for the tuples of operators. Now we state this result as follows.Theorem 3.
Let(T1,T2) be a pair of commuting continuous linear operators on a separable infinite Banach space X. Suppose that the subspaces (7)
X
0
≔
span
⁡
x
∈
X
:
T
1
x
=
λ
1
x
,
T
2
x
=
λ
2
x
for some
λ
1
,
λ
2
∈
K
with
λ
1
<
1
,
λ
2
<
1
,
Y
0
≔
span
⁡
x
∈
X
:
T
1
x
=
λ
1
x
,
T
2
x
=
λ
2
x
for some
λ
1
,
λ
2
∈
K
with
λ
1
>
1
,
λ
2
>
1are dense in X. Then (T1,T2) is mixing and in particular hypercyclic.
Moreover, ifX is a complex space and also the subspace (8)
Z
0
≔
span
⁡
x
∈
X
:
T
1
x
=
λ
1
e
α
π
i
x
,
T
2
x
=
λ
2
e
β
π
i
x
for some
λ
1
,
λ
2
∈
K
with
λ
1
λ
2
=
1
,
α
,
β
∈
Qis dense in X, then (T1,T2) is chaotic.Comparing the Godefroy-Shapiro criterion for a single operator and Theorem3 above, it is natural to try to extend G-S criterion via the product of the joint point spectra of the tuple. However, when considering the product of the joint point spectra that satisfies the conditions in Theorem 3, we find that our idea mentioned above does not work; see Example 9 for the details. Therefore, we turn to consider some special joint point spectra, namely, some special subgroup of the abelian semigroup F generated by the operator T. To do so, we need to introduce the following definition.Definition 4.
Them-tuple (T1,…,Tm) of commuting operators on X is S-mixing if for any nonempty open sets U,V⊂X, and for any m-tuple of sequences of positive integers (k1,n,…,km,n)n∈N such that each sequence (kj,n)n∈N is strictly increasing, j=1,…,m, and (9)sup⁡ki,n-kj,n:n∈N,1≤i,j≤m<∞,there exists N∈N such that (10)Tk1,n⋯Tkm,nU∩V≠⌀∀n≥N.Remark 5.
We will give some examples which are related to the above definition in Section3. According to the definition of mixing of a single operator T, we have that for any nonempty open sets U,V, N(U,V) is cofinite. However, if T is mixing or S-mixing, then N02∖NT(U,V) is infinite. Obviously, if T is mixing, then it must be S-mixing, and an S-mixing tuple of operators T must be hypercyclic.From the definition ofS-mixing and the remark above, one may ask the following question.Question 1.
Is theS-mixing (for the tuple of operators) stronger than weakly mixing (for the tuple of operators) or is the inverse true?The following two theorems are our main results in the present paper.Theorem 6.
Let(T1,T2) be a pair of commuting continuous linear operators on a separable infinite Banach space X. Suppose that the subspaces (11)
X
0
≔
span
⁡
x
∈
X
:
T
1
x
=
λ
1
x
,
T
2
x
=
λ
2
x
for some
λ
1
,
λ
2
∈
K
with
λ
1
λ
2
<
1
,
Y
0
≔
span
⁡
x
∈
X
:
T
1
x
=
λ
1
x
,
T
2
x
=
λ
2
x
for some
λ
1
,
λ
2
∈
K
with
λ
1
λ
2
>
1are dense in X. Then (T1,T2) is S-mixing and in particular hypercyclic.
Moreover, ifX is a complex space and also the subspace (12)
Z
0
≔
span
⁡
x
∈
X
:
T
1
x
=
λ
1
e
α
π
i
x
,
T
2
x
=
λ
2
e
β
π
i
x
for some
λ
1
,
λ
2
∈
K
with
λ
1
λ
2
=
1
,
α
,
β
∈
Qis dense in X, then (T1,T2) is chaotic.In the preceding theorem, we consider the product of the joint point spectra to extend the G-S criterion, and it is not hard to find that the pair of operators(T1,T2) is not mixing. We will give an example (Example 18) below. In the view of preserving the property of mixing, Theorem 3 is more suitable than Theorem 6 as a generalization of G-S criterion for tuples of operators.The following theorem gives a positive answer to Question1.Theorem 7.
Let(T1,T2) be a pair of commuting continuous linear operators on a separable infinite Banach space X. If (T1,T2) is S-mixing, then (T1,T2) must be weakly mixing. However, the inverse is not true.Based on Theorem7, we obtain a new notion of a dynamic system between mixing and weakly mixing. Let ((T1,T2),X) be a dynamic system; then we have (13)mixing⟹S-mixing⟹weakly mixing⟹hypercyclic.


## 2. Proofs of Main Results

Throughout this paper, letD denote the unit disk D=λ∈C:λ<1 and T denote the unit circle T=λ∈C:λ=1. H(C) denotes the entire function space on C.Although the proof of Theorem6 is simple and similar to the proof of single mixing operator, we still give a brief proof for the sake of completeness.Proof of Theorem6.
LetU,V be a pair of nonempty open subsets of X; by the hypothesis, we can find x∈X0∩U and y∈Y0∩V. Hence these vectors can be expressed in the form (14)x=∑j=1majxj,y=∑j=1mbjyj,where T1xj=λj,1xj, T2xj=λj,2xj, T1yj=μj,1yj, and T2yj=μj,2yj, for certain scalars aj,bj,λj,1,λj,2,μj,1,μj,2∈K with λj,1λj,2<1,μj,1μj,2>1, j=1,…,m.
For any pair of sequences of positive integers(k1,n,k2,n)n∈N such that each sequence (kj,n)n∈N is strictly increasing, j=1,2, and (15)sup⁡k1,n-k2,n:n∈N<∞,T1k1,nT2k2,nx=∑j=1majλj,1k1,nλj,2k2,nxj=∑j=1majλj,1k1,n-k2,nλj,1λj,2k2,nxj⟶0,uk1,n,k2,n≔∑j=1mbj1μj,1k1,nμj,2k2,nyj=∑j=1mbj1μj,1k1,n-k2,nμj,1μj,2k2,nyj⟶0as n→∞ and (16)T1k1,nT2k2,nuk1,k2=∑j=1mbjyj=yfor all k1,n,k2,n≥0, then one has (17)x+uk1,k2∈U,T1k1,nT2k2,nx+uk1,n,k2,n∈V.It follows that (T1,T2) is S-mixing and also hypercyclic.
Moreover, ifX is a complex space, then iterations of the pair of operators show to us that (18)T1k1,nT2k2,nx=λ1k1,nek1,nαπiλ2k2,nek2,nβπix=λ1k1,nλ2k2,nek1,nα+k2,nβπix.Since α,β∈Q, there exists k1,n=k2,n such that T1k1,nT2k2,n=ek1,n(α+β)πi. Hence Z0 is precisely the set of periodic points of the operator (T1,T2). Thus (T1,T2) is chaotic whenever Z0 is also dense in X. This completes the proof.Remark 8.
For a pair of operators(T1,T2), if it is S-mixing, one considers k2,n=k1,n+1 for any n∈N; then the bounded sup condition is exactly the fact that the sequence (k1,n)n (and thus (k2,n)n) is syndetic. For single operators, the notion of syndetical hypercyclicity was studied in [12, 13].Next, we give an example that satisfies the conditions of Theorem6, but not mixing.Example 9.
Let(μ1B,μ2B) be a pair of multiple of backward shift operators acting on any space X=lp(N),1≤p<∞, or c0(N), where μ1,μ2∈C∖{0} and μ1μ2>1. Then (μ1B,μ2B) is S-mixing and chaotic, but not mixing.
One can easily determine the eigenvectors ofB as being the nonzero multiples of the sequence (19)eλ≔λ,λ2,λ3,…,λ<1with corresponding eigenvalue λ, and λ<1 ensures that eλ∈X. Hence, eλ is an eigenvector of μiB(i=1,2) to the eigenvalue μiλ, and this also ensures that (20)
x
∈
X
:
μ
1
B
x
=
λ
1
x
,
μ
2
B
x
=
λ
2
x
for some
λ
1
,
λ
2
∈
Kis nonempty.
According to [14, Sublemma  7], for any nonempty subset Λ of the unit disk D, the set span{eλ;λ∈Λ} is dense in X. Now we consider the following set: (21)
X
0
=
s
p
a
n
x
∈
X
:
μ
1
B
x
=
λ
1
x
,
μ
2
B
x
=
λ
2
x
for
λ
1
,
λ
2
∈
K
with
λ
1
λ
2
<
1contains span{eλ:λ<1/μ1μ21/2}. Since μ1μ2>1 and λ<1, we have (22)λ∈D:λ<1μ1μ21/2is a nonempty open subset of D and thus X0 is dense in X. Since (23)
Y
0
=
s
p
a
n
x
∈
X
:
μ
1
B
x
=
λ
1
x
,
μ
2
B
x
=
λ
2
x
for
λ
1
,
λ
2
∈
K
with
λ
1
λ
2
>
1contains spaneλ:1/μ1μ21/2<λ<1 and (24)
Z
0
=
s
p
a
n
x
∈
X
:
μ
1
B
x
=
λ
1
e
α
π
i
x
,
μ
2
B
x
=
λ
2
e
β
π
i
x
for some
λ
1
,
λ
2
∈
K
with
λ
1
λ
2
=
1
,
α
,
β
∈
Qcontains spaneλ:λ=1/μ1μ21/2, one can show that Y0 and Z0 are both dense in X. Therefore, (μ1B,μ2B) is S-mixing and chaotic.
Now we are going to show that(μ1B,μ2B) is not mixing. To do this, it sufficient for us to construct a counterexample. By the hypothesis, μ1μ2>1, let μ1=3, μ2=1/2; there exist large enough positive integers α,β such that 3α(1/2)β≤1. Assume on the contrary that (3B,1/2B) is mixing, according to the definition of mixing tuple of operators, for any pair U,V of nonempty open subsets of X, there exists some positive integer N such that (25)3Bk112Bk2U∩V≠⌀for all kj≥N, j=1,2. Let α,β≥N; one has that (26)3Bα12BβU∩V≠⌀.Then (3B)α(1/2B)β>1; that is, 3α(1/2)β>1; this is a contradiction.The following corollary follows easily from Theorem6, which tells us that S-mixing is stronger than weakly mixing.Corollary 10.
Let(T1,T2) be a pair of commuting bounded linear operators on a separable infinite Banach space X. If (T1,T2) satisfies Theorem 6, then (T1,T2) must be weakly mixing.Before giving the proof of Theorem7, we need another theorem.Theorem 11.
Let(T1,…,Tm) be an m-tuple on a separable infinite Banach space X. If (T1,…,Tm) is S-mixing, then the abelian semigroup F generated by (T1,…,Tm) contains a mixing operator. Moreover, the m-tuple (T1,…,Tm) is S-mixing if and only if the operator T1⋯Tm is mixing.Proof.
If(T1,…,Tm) is S-mixing, then for any nonempty open subsets U,V⊂X and for any tuple of sequences of positive integers (k1,n,…,km,n)n∈N such that each sequence (kj,n)n∈N is strictly increasing, j=1,…,m, and (27)sup⁡ki,n-kj,n:n∈N,1≤i,j≤m<∞,there exists a positive integer N such that (28)T1k1,n⋯Tmkm,nU∩V≠⌀for all n≥N. Take k1,n=⋯=km,n; then one has (29)T1⋯Tmk1,nU∩V≠⌀for all n≥N. This implies that T1⋯Tm is mixing.
Next, we need to show that the operatorT1⋯Tm is mixing that implies the m-tuple (T1,…,Tm) is S-mixing. For any m-tuple of sequences of positive integers (k1,n,…,km,n)n∈N such that each sequence (kj,n)n∈N is strictly increasing, j=1,…,m, and (30)sup⁡ki,n-kj,n:n∈N,1≤i,j≤m<∞,we may assume that k1,n≤kj,n,j=2,…,m. Since the operator T1⋯Tm is mixing, for any nonempty open subsets U,V⊂X and for any tuple of sequences of positive integers (k1,n,…,km,n)n∈N above, we have that(31)T1⋯Tmk1,nU∩T2k1,n-k2,n⋯Tmk1,n-km,nV≠⌀for all n≥N. Clearly, (31) is equivalent to (32)U∩T1⋯Tm-k1,nT2k1,n-k2,n⋯Tmk1,n-km,nV≠⌀for all n≥N. Since T1,…,Tm are mutually commutative operators, we obtain that (33)U∩T1-k1,n⋯Tm-km,nV≠⌀for all n≥N. It follows that (34)T1k1,n⋯T1km,nU∩V≠⌀for all n≥N. This completes the proof.Now we are ready to give the proof of Theorem7.Proof of Theorem7.
According to Theorem11, the semigroup F generated by (T1,T2) contains a mixing operator S~=T1T2, for any nonempty open subsets U,V,U1,U2,V1,V2⊂X; there exists a positive integer N~ such that S~n(U)∩V≠⌀ for any n≥N~.
LetS=S~n1,n1≥N~; then S is a continuous map that commutes with (T1,T2) and satisfies S(U)∩V≠⌀ for any nonempty open subsets U,V⊂X. Then we can find nonempty open sets U1′⊂U1,V1′⊂V1 such that S(U1′)⊂U2,S(V1′)⊂V2.
Since(T1,T2) is S-mixing operator, the return set N(U1′,V1′)≠⌀. Suppose that (k1,k2)∈N(U1′,V1′); then there exists x∈U1′ with T1k1T2k2x∈V1′. Therefore, (35)Sx∈U2,T1k1T2k2Sx=ST1k1T2k2x∈V2,which yields that (k1,k2)∈N(U2,V2). It follows that N(U1′,V1′)⊂N(U2,V2). As N(U1′,V1′)⊂N(U1,V1), we have that (36)⌀≠NU1′,V1′⊂NU1,V1∩NU2,V2.Hence (T1,T2) is weakly mixing. For the other direction, we will present a counterexample in Example 18. Thus we complete the proof.


## 3. Applications

In this section, we will give some applications of Theorems6 and 11. For the pair of classical operators, we can easily obtain the following three results via Theorem 11.Proposition 12.
Let(μ1B,μ2B) be a pair of multiple backward shift operators acting on any space lp(N)(1≤p<∞) or c0(N), where μ1,μ2∈C∖0 and μ1≠μ2; then (μ1B,μ2B) is S-mixing if and only if μ1μ2>1.Proposition 13.
Let(Ta1,Ta2):H(C)→H(C) be a pair of translation operators, where Tai:H(C)→H(C),i=1,2, defined by Taif(z)=f(z+ai), for any f∈H(C). Then (Ta1,Ta2) is S-mixing if and only if a1+a2≠0.Proposition 14.
LetH2(D) be a Hardy-Hilbert space of holomorphic function on the complex unit disk D. Let φi(i=1,2) be a nonconstant bounded holomorphic function, Mφi a multiplication operator, and Mφi∗ the corresponding adjoint multiplication operator. Let (Mφ1∗,Mφ2∗) be a pair of adjoint multiplication operators acting on H2(D); then (Mφ1∗,Mφ2∗) is S-mixing if and only if (φ1φ2)(D)∩T≠⌀.Using the G-S criterion (Theorem2), Godefroy and Shapiro in [11] characterized the hypercyclicity of adjoint multipliers, excluding constant multipliers because their adjoint multiplication operators are multipliers of the identity. In [4], the theorem was settled as follows.Theorem 15 (see [4, Theorem  4.42]).
Letφ be a nonconstant bounded holomorphic function on D and let Mφ∗ be the corresponding adjoint multiplier on H2(D). Then the following assertions are equivalent: (1)
Mφ∗ is hypercyclic;(2)
Mφ∗ is mixing;(3)
Mφ∗ is chaotic;(4)
φ(D)∩T≠⌀.Feldman in [15] focused on pair or tuple of adjoint multiplication operators on the Hardy-Hilbert space; he showed that most pairs of coanalytic Toeplitz operators are hypercyclic. More precisely, the author proved the following theorem.Theorem 16 (see [15, Theorem  3.1]).
Letf,g∈H∞(D)∖{0} satisfy fz>1 for all z∈D and gz>1 for all z∈D, and let Mf and Mg be the corresponding multiplication operators on H2(D). Then neither Mf∗ nor Mg∗ is hypercyclic on H2(D), but the following assertions are equivalent: (1)
the pair(Mf∗,Mg∗) is hypercyclic on H2(D);(2)
the semigroupF generated by the pair (Mf∗,Mg∗) contains a hypercyclic operator;(3)
there existn,k∈N such that fngk is nonconstant and fngk(D)∩T≠⌀;(4)
there does not existp>0 and θ∈R such that f(z)=eiθ/gp(z) for all z∈D.Motivated by Theorems15 and 16, we deduce the following corollary from Theorems 6 and 11.Corollary 17.
LetH2(D) be a Hardy-Hilbert space of holomorphic function on the complex unit disk D. Let φi(i=1,…,m) be a nonconstant bounded holomorphic function, Mφi a multiplication operator, and Mφi∗ the corresponding adjoint multiplication operator. Let (Mφ1∗,…,Mφm∗) be a tuple of adjoint multiplication operators acting on H2(D); the following assertions are equivalent: (1)
(Mφ1∗,…,Mφm∗) is S-mixing;(2)
Mφ1∗⋯Mφm∗ is mixing;(3)
Mφ1∗⋯Mφm∗ is chaotic;(4)
(φ1⋯φm)(D)∩T≠⌀.Next, we give an example to show that a pair of commutative operators is weakly mixing, but notS-mixing.Example 18.
Let(2B,1/3B) be a pair of multiple backward shift operators acting on any space lp(N),1≤p<∞, or c0(N), and then (2B,1/3B) is hypercyclic also weakly mixing, but not S-mixing.Assume on the contrary that(2B,1/3B) is S-mixing; then for any nonempty open subsets U,V⊂X, (37)2k113k2Bk1+k2U∩V≠⌀forsupk1,k2≥N⁡k1-k2<∞.Take m1=m2≥N; we have (38)2m113m2Bm1+m2U∩V≠⌀.Equivalently, (39)23B2m2U∩V≠⌀∀m2≥N.Hence the operator 2/3B2 is mixing, but this is a contradiction.Next, we check that the operator(2B,1/3B) satisfies hypercyclic criterion of tuple of operators [6, Proposition 2.5]. We know that the finite sequences (i.e., sequences of the form (x1,…,xn,0,0,…),n≥1) constitute a dense subset of X.LetX0 be the finite sequences, and let (nj)j≥1,(kj)j≥1 be increasing sequences of positive integers and satisfy (nj-kj)→+∞ as j→∞. Let Sj:X0→X0, defined by Sj=(1/2)nj3kjFnj+kj; then for any x∈X0, (2B)nj(1/3B)kjx→0, (2B)nj(1/3B)kjSj→I, (40)Sjx=12Fnj3Fkjx≤12nj3kjx⟶0as j→∞. Hence (2B,1/3B) is weakly mixing and in particular hypercyclic. Therefore, (2B,1/3B) is hypercyclic and weakly mixing, but not S-mixing.Proposition 19.
Under the same assumptions in Proposition12, if (μ1B,μ2B) is S-mixing, then the semigroup F generated by (μ1B,μ2B) contains a mixing operator. But, the inverse is not true.Proof.
Since(μ1B,μ2B) is an S-mixing operator, according to Proposition 12, we have that μ1μ2>1 and μ1μ2B2 is a mixing operator. Hence the semigroup F generated by (μ1B,μ2B) contains a mixing operator. Based on Example 18, we can find that the inverse of this is not true. Thus we complete the proof.Remark 20.
There are examples of tuples(T1,T2) in which every single operator T1nT2m with n,m≥0, n+m>0, is mixing (thus the tuple is S-mixing), but the corresponding semigroup generated by the tuple (T1,T2) is not mixing. Consider the example of translation semigroup on a sector of the complex plane given in Example  4.15 in [16]. The subsemigroup generated by the tuple of translations (T1,T2) of the translation semigroup in this example, where T1=T1+i and T2=T1-i, is an example of this type, if we follow the argument given there.For single operator, the relationships between mixing operators and chaotic operators have been studied very clearly; next we give a brief discussion on the relationships betweenS-mixing tuples and chaotic tuples. Firstly, we present a proposition on the pair of multiple backward shift operators.Proposition 21.
Under the same assumptions in Proposition12, then the following assertions are equivalent: (1)
(μ1B,μ2B) is chaotic;(2)
(μ1B,μ2B) is hypercyclic;(3)
N(U,V)≠⌀ for any nonempty open subsets U,V⊂X;(4)
max⁡μ1,μ2>1;(5)
the semigroupF generated by (μ1B,μ2B) contains a hypercyclic operator;(6)
the semigroupF generated by (μ1B,μ2B) contains a chaotic operator;(7)
(μ1B,μ2B) has a dense set of periodic points and N(U,V)≠⌀ for any pair of nonempty open subsets U,V⊂X.Proof.
( 1 ) ⇒ ( 2 ), (2)⇔(3), and (5)⇒(2) are trivial.
( 2 ) ⇒ ( 4 ): by hypothesis, (μ1B,μ2B) is hypercyclic. According to Theorem 1, we have that the set HC((μ1B,μ2B)) of all hypercyclic vectors of (μ1B,μ2B) is a dense Gδ-set in X. Let BX=x∈X:x<1; we have (41)BX∩HCμ1B,μ2B≠⌀.Then there exists some y∈BX∩HC((μ1B,μ2B)) such that orb((μ1B,μ2B),y) is dense in X. Assume on the contrary that max⁡μ1,μ2≤1; then we have (42)orbμ1B,μ2B,y⊆BX,because (43)μ1Bk1μ2Bk2y≤μ1Bk1μ2Bk2y≤μ1k1μ2k2y<1,for all k1,k2∈N. This is a contradiction. Therefore, max{|μ1|,|μ2|}>1.
( 4 ) ⇒ ( 5 ): Without loss of generality, we may assume that μ1>1, then μ1B is hypercyclic. Hence the semigroup F generated by (μ1B,μ2B) contains a hypercyclic operator. Therefore, we get (44)2⟺3⟹4⟹5⟹2.Based on the proof above, we can easily obtain (1)⇒(4).
( 4 ) ⇒ ( 6 ): without loss of generality, we assume that μ1>1; then μ1B is a chaotic operator. Therefore, the semigroup F generated by (μ1B,μ2B) contains a chaotic operator.
( 6 ) ⇒ ( 1 ): by hypothesis, the semigroup F generated by (μ1B,μ2B) contains a chaotic operator T. Let T=μ1α1μ2α2Bα1+α2,α1,α2∈N; then T has a dense set of periodic points and there exists x∈X such that orb(x,T) is dense in X. For any x∈Per⁡(T), there exists a positive integer N such that TNx=x; that is, (45)μ1α1Nμ2α2NBα1N+α2Nx=x.Hence x∈Per⁡((μ1B,μ2B)); then we have (46)Per⁡T⊆Per⁡μ1B,μ2B,orbx,T⊆orbμ1B,μ2B,x.Therefore, (μ1B,μ2B) has a dense set of periodic points and orb((μ1B,μ2B),x) is dense in X. Hence (μ1B,μ2B) is chaotic.
Since(7)⇔(1), we have (47)1⟹4⟹6⟹1⟺7.Thus we complete the proof.Remark 22.
Based on Propositions12 and 21, we can see that the operator (2B,1/3B) in Example 18 is chaotic, but not S-mixing.



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*Source: 9251672-2016-10-20.xml*