#### 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, -mixing, which lies between mixing and weakly mixing. We also obtain a sufficient condition to ensure a tuple of operators to be -mixing. Moreover, we study some new properties of -mixing operators on several concrete Banach spaces.

#### 1. Introduction

A continuous linear operator acting on a topological vector space is called hypercyclic, if there exists a vector such that the orbit of under is dense in ; that is, . Such a vector is called a hypercyclic vector for and the set of hypercyclic vectors for will be denoted by .

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 -tuple of diagonal matrices that is hypercyclic on .

For an -tuple of operators we mean a finite sequence of length of mutually commuting continuous linear operators on a Banach space . Let be an -tuple acting on a Banach space (more general complete topology vector space), and is the semigroup generated by ; that is, 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 -dense orbits of a tuple of operators.

In the past few years, the authors have studied the following dynamical properties of tuples :(1)hypercyclicity: ;(2)chaotic (Devaney) [9]: is hypercyclic and has a dense set of periodic points;(3)mixing: for any pair of nonempty open subsets of , there exists some positive integer such that (4)weakly mixing: for a tuple of operators and subsets define the return set as Then is called weakly mixing if for every quadruple of nonempty open subsets we have that .

From the above results, one can deduce that

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. *Let be a separable infinite dimensional Banach space and the pair of operators and . The following assertions are equivalent: *(1)* is hypercyclic;*(2)* is topologically transitive.**If one of these conditions holds then the set of points in with dense orbit is a dense -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]). *Let be an operator. Suppose that the subspaces are dense in . Then is mixing and in particular hypercyclic.**Moreover, if is a complex space and also the subspace is dense in , then 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 be a pair of commuting continuous linear operators on a separable infinite Banach space . Suppose that the subspaces are dense in . Then is mixing and in particular hypercyclic.**Moreover, if is a complex space and also the subspace is dense in , then is chaotic.*

Comparing the Godefroy-Shapiro criterion for a single operator and Theorem 3 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 generated by the operator . To do so, we need to introduce the following definition.

*Definition 4. *The -tuple of commuting operators on is -mixing if for any nonempty open sets , and for any -tuple of sequences of positive integers such that each sequence is strictly increasing, , and there exists such that

*Remark 5. *We will give some examples which are related to the above definition in Section 3. According to the definition of mixing of a single operator , we have that for any nonempty open sets , is cofinite. However, if is mixing or -mixing, then is infinite. Obviously, if is mixing, then it must be -mixing, and an -mixing tuple of operators must be hypercyclic.

From the definition of -mixing and the remark above, one may ask the following question.

*Question 1. *Is the -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 be a pair of commuting continuous linear operators on a separable infinite Banach space . Suppose that the subspaces are dense in . Then is -mixing and in particular hypercyclic.**Moreover, if is a complex space and also the subspace is dense in , then 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 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 Question 1.

Theorem 7. *Let be a pair of commuting continuous linear operators on a separable infinite Banach space . If is -mixing, then must be weakly mixing. However, the inverse is not true.*

Based on Theorem 7, we obtain a new notion of a dynamic system between mixing and weakly mixing. Let be a dynamic system; then we have

#### 2. Proofs of Main Results

Throughout this paper, let denote the unit disk and denote the unit circle . denotes the entire function space on .

Although the proof of Theorem 6 is simple and similar to the proof of single mixing operator, we still give a brief proof for the sake of completeness.

*Proof of Theorem 6. *Let be a pair of nonempty open subsets of ; by the hypothesis, we can find and . Hence these vectors can be expressed in the form where , , , and , for certain scalars with , .

For any pair of sequences of positive integers such that each sequence is strictly increasing, , and as and for all , then one has It follows that is -mixing and also hypercyclic.

Moreover, if is a complex space, then iterations of the pair of operators show to us that Since , there exists such that Hence is precisely the set of periodic points of the operator . Thus is chaotic whenever is also dense in . This completes the proof.

*Remark 8. *For a pair of operators , if it is -mixing, one considers for any ; then the bounded sup condition is exactly the fact that the sequence (and thus ) 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 Theorem 6, but not mixing.

*Example 9. *Let be a pair of multiple of backward shift operators acting on any space , or , where and . Then is -mixing and chaotic, but not mixing.

One can easily determine the eigenvectors of as being the nonzero multiples of the sequence with corresponding eigenvalue , and ensures that . Hence, is an eigenvector of to the eigenvalue , and this also ensures that is nonempty.

According to [14, Sublemma 7], for any nonempty subset of the unit disk , the set is dense in . Now we consider the following set: contains Since and , we have is a nonempty open subset of and thus is dense in . Since contains and contains , one can show that and are both dense in . Therefore, is -mixing and chaotic.

Now we are going to show that is not mixing. To do this, it sufficient for us to construct a counterexample. By the hypothesis, , let , ; there exist large enough positive integers such that Assume on the contrary that is mixing, according to the definition of mixing tuple of operators, for any pair of nonempty open subsets of , there exists some positive integer such that for all , Let ; one has that Then ; that is, ; this is a contradiction.

The following corollary follows easily from Theorem 6, which tells us that -mixing is stronger than weakly mixing.

Corollary 10. *Let be a pair of commuting bounded linear operators on a separable infinite Banach space . If satisfies Theorem 6, then must be weakly mixing.*

Before giving the proof of Theorem 7, we need another theorem.

Theorem 11. *Let be an -tuple on a separable infinite Banach space . If is -mixing, then the abelian semigroup generated by contains a mixing operator. Moreover, the -tuple is -mixing if and only if the operator is mixing.*

*Proof. *If is -mixing, then for any nonempty open subsets and for any tuple of sequences of positive integers such that each sequence is strictly increasing, , and there exists a positive integer such that for all . Take ; then one has for all . This implies that is mixing.

Next, we need to show that the operator is mixing that implies the -tuple is -mixing. For any -tuple of sequences of positive integers such that each sequence is strictly increasing, , and we may assume that . Since the operator is mixing, for any nonempty open subsets and for any tuple of sequences of positive integers above, we have thatfor all . Clearly, (31) is equivalent to for all . Since are mutually commutative operators, we obtain that for all . It follows that for all . This completes the proof.

Now we are ready to give the proof of Theorem 7.

*Proof of Theorem 7. *According to Theorem 11, the semigroup generated by contains a mixing operator , for any nonempty open subsets ; there exists a positive integer such that for any

Let ; then is a continuous map that commutes with and satisfies for any nonempty open subsets Then we can find nonempty open sets such that

Since is -mixing operator, the return set . Suppose that ; then there exists with . Therefore, which yields that . It follows that As , we have that Hence 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 Theorems 6 and 11. For the pair of classical operators, we can easily obtain the following three results via Theorem 11.

Proposition 12. *Let be a pair of multiple backward shift operators acting on any space or , where and ; then is -mixing if and only if .*

Proposition 13. *Let be a pair of translation operators, where , defined by , for any Then is -mixing if and only if .*

Proposition 14. *Let be a Hardy-Hilbert space of holomorphic function on the complex unit disk . Let be a nonconstant bounded holomorphic function, a multiplication operator, and the corresponding adjoint multiplication operator. Let be a pair of adjoint multiplication operators acting on ; then is -mixing if and only if .*

Using the G-S criterion (Theorem 2), 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]). *Let be a nonconstant bounded holomorphic function on and let be the corresponding adjoint multiplier on . Then the following assertions are equivalent: *(1)* is hypercyclic;*(2)* is mixing;*(3)* is chaotic;*(4)*.*

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]). *Let satisfy for all and for all , and let and be the corresponding multiplication operators on . Then neither nor is hypercyclic on , but the following assertions are equivalent: *(1)*the pair is hypercyclic on ;*(2)*the semigroup generated by the pair contains a hypercyclic operator;*(3)*there exist such that is nonconstant and ;*(4)*there does not exist and such that for all .*

Motivated by Theorems 15 and 16, we deduce the following corollary from Theorems 6 and 11.

Corollary 17. *Let be a Hardy-Hilbert space of holomorphic function on the complex unit disk . Let be a nonconstant bounded holomorphic function, a multiplication operator, and the corresponding adjoint multiplication operator. Let be a tuple of adjoint multiplication operators acting on ; the following assertions are equivalent: *(1)* is -mixing;*(2)* is mixing;*(3)* is chaotic;*(4)*.*

Next, we give an example to show that a pair of commutative operators is weakly mixing, but not -mixing.

*Example 18. *Let be a pair of multiple backward shift operators acting on any space , or , and then is hypercyclic also weakly mixing, but not -mixing.

Assume on the contrary that is -mixing; then for any nonempty open subsets , Take ; we have Equivalently, Hence the operator is mixing, but this is a contradiction.

Next, we check that the operator satisfies hypercyclic criterion of tuple of operators [6, Proposition ]. We know that the finite sequences (i.e., sequences of the form ) constitute a dense subset of .

Let be the finite sequences, and let be increasing sequences of positive integers and satisfy as . Let , defined by ; then for any , , , as . Hence is weakly mixing and in particular hypercyclic. Therefore, is hypercyclic and weakly mixing, but not -mixing.

Proposition 19. *Under the same assumptions in Proposition 12, if is -mixing, then the semigroup generated by contains a mixing operator. But, the inverse is not true.*

*Proof. *Since is an -mixing operator, according to Proposition 12, we have that and is a mixing operator. Hence the semigroup generated by 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 in which every single operator with , , is mixing (thus the tuple is -mixing), but the corresponding semigroup generated by the tuple is not mixing. Consider the example of translation semigroup on a sector of the complex plane given in Example in [16]. The subsemigroup generated by the tuple of translations of the translation semigroup in this example, where and , 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 between -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 Proposition 12, then the following assertions are equivalent: *(1)* is chaotic;*(2)* is hypercyclic;*(3)* for any nonempty open subsets ;*(4)*;*(5)*the semigroup generated by contains a hypercyclic operator;*(6)*the semigroup generated by contains a chaotic operator;*(7)* has a dense set of periodic points and for any pair of nonempty open subsets .*

*Proof. *, , and are trivial.

: by hypothesis, is hypercyclic. According to Theorem 1, we have that the set of all hypercyclic vectors of is a dense -set in . Let ; we have Then there exists some such that is dense in . Assume on the contrary that ; then we have because for all This is a contradiction. Therefore, .

: Without loss of generality, we may assume that , then is hypercyclic. Hence the semigroup generated by contains a hypercyclic operator. Therefore, we get Based on the proof above, we can easily obtain .

: without loss of generality, we assume that ; then is a chaotic operator. Therefore, the semigroup generated by contains a chaotic operator.

: by hypothesis, the semigroup generated by contains a chaotic operator . Let ; then has a dense set of periodic points and there exists such that is dense in . For any , there exists a positive integer such that ; that is, Hence ; then we have Therefore, has a dense set of periodic points and is dense in . Hence is chaotic.

Since , we have Thus we complete the proof.

*Remark 22. *Based on Propositions 12 and 21, we can see that the operator in Example 18 is chaotic, but not -mixing.

#### Competing Interests

The authors declare that they have no competing interests.

#### Acknowledgments

The authors are grateful to Dr. Xianfeng Zhao for many discussions. This work was partially supported by a NSFC Grant (11271387), Chongqing Natural Science Foundation (cstc 2013jjB0050), and Project in Chongqing (1020709520130059).