2.1 Subsets of non-negative integers

By $\mathbb {N}$ , $\mathbb {N}_0$ and $\mathbb {Z}$ , we denote respectively the set of positive integers, the set of non-negative integers and the set of integers.

Before proceeding further, we briefly recall the relevant notation concerning a Furstenberg family (detailed discussions can be found in [Reference Akin1, Reference Furstenberg16]). Let $\mathcal {F}$ be a collection of subsets of $\mathbb {N}_0$ . We say that $\mathcal {F}$ is a Furstenberg family (or family for short) provided that it is hereditary upward, that is, for any $A\in \mathcal {F}$ and $B\subset \mathbb {N}_0$ with $A\subset B$ , we have $B\in \mathcal {F}$ . Denote the family of all infinite subsets of $\mathbb {N}_0$ by $\mathcal {F}_{\mathrm {inf}}$ and the family of all cofinite subsets of $\mathbb {N}_0$ by $\mathcal {F}_{\mathrm {cf}}$ . For a family $\mathcal {F}$ , we say that a collection $\mathcal {F}'$ of subsets of $\mathbb {N}_0$ is a subfamily of $\mathcal {F}$ if $\mathcal {F}'$ is a family with $\mathcal {F}'\subset \mathcal {F}$ .

Throughout this paper, for each Furstenberg family $\mathcal {F}$ , we adopt the conventions that $\emptyset \notin \mathcal {F}$ and $\mathcal {F}\neq \emptyset $ .

For a subset F of $\mathbb {Z}$ and $i\in \mathbb {Z}$ , let $F\pm i=\{m\pm i\colon m\in F\}$ . We say that a family $\mathcal {F}$ is:

• • shift $_+$ -invariant if $F+i\in \mathcal {F}$ for each $i\in \mathbb {N}_0$ and $ F\in \mathcal {F}$ ;

• • shift $_-$ -invariant if $(F-i)\cap \mathbb {N}_0\in \mathcal {F}$ for each $i\in \mathbb {N}_0$ and $ F\in \mathcal {F}$ ;

• • shift-invariant if $(F+i)\cap \mathbb {N}_0 \in \mathcal {F}$ for each $i\in \mathbb {Z}$ and $F\in \mathcal {F}$ .

Definition 2.1. Let $\mathcal {F}$ be a family. The dual family $\mathcal {F}^*$ is given by

$$ \begin{align*} \mathcal{F}^*=\{A\subset \mathbb{N}_0: A\cap B\neq\emptyset \text{ for all }B\in\mathcal{F}\}. \end{align*} $$

We say that $\mathcal {F}$ is a full family if $F\cap F'$ is infinite for any $F\in \mathcal {F}$ and $F'\in \mathcal {F}^*$ ; $\mathcal {F}$ is a filter if $F_1\cap F_2\in \mathcal {F}$ for any $F_1, F_2\in \mathcal {F}$ .

We recall some standard families.

• • $\mathcal {F}_{\mathrm {t}}$ : the family of subsets of $\mathbb {N}_0$ containing intervals of arbitrary length, which are said to be thick.

• • $\mathcal {F}_{\mathrm {s}}$ : the family of subsets of $\mathbb {N}_0$ with bounded gaps, which are said to be syndetic.

• • $\mathcal {F}_{\mathrm {ps}}$ : the family of subsets of $\mathbb {N}_0$ with the form $F_1\cap F_2$ , where $F_1\in \mathcal {F}_{\mathrm {s}}$ and $F_2\in \mathcal {F}_{\mathrm {t}}$ , and these are said to be piecewise syndetic.

• • $\mathcal {F}_{\mathrm {ts}}$ : the family of subsets A of $\mathbb {N}_0$ such that for each $k\in \mathbb {N}_0$ , there exists $F\in \mathcal {F}_{\mathrm {s}}$ with $\bigcup _{i=0}^k (F+i)\subset A$ , and these are said to be thickly syndetic.

It is easy to see that $\mathcal {F}_{\mathrm {s}}^*=\mathcal {F}_{\mathrm {t}}$ and $ \mathcal {F}_{\mathrm {ps}}^*=\mathcal {F}_{\mathrm {ts}}$ .

We turn to a property of families (see [Reference Akin1, Reference Akin and Glasner2]) which plays a significant role in our later discussion.

Definition 2.3. Let $\mathcal {F}$ be a family. We say that $\mathcal {F}$ is a thick family if for any $F\in \mathcal {F}$ and $k\in \mathbb {N}_0$ ,

$$ \begin{align*} \bigcap_{i=0}^k(F-i)\cap\mathbb{N}_0\in\mathcal{F}. \end{align*} $$

Following [Reference Bès, Menet, Peris and Puig10], for a given family $\mathcal {F}$ , we denote

$$ \begin{align*} \widetilde{\mathcal{F}}=\bigg\{F\in\mathcal{F}: \text{for any } k\in\mathbb{N}_0, \bigcap_{i=-k}^k(F+i)\cap\mathbb{N}_0\in\mathcal{F} \bigg\}. \end{align*} $$

Then, $\widetilde {\mathcal {F}}$ is the maximal shift-invariant thick subfamily of $\mathcal {F}$ provided that it is not empty. Note that if $\mathcal {F}$ is full, then $\mathcal {F}_{\mathrm {cf}}\subset \widetilde {\mathcal {F}}$ .

2.2 Topological dynamics

For a topological dynamical system, we mean a pair $(X,T)$ , where X is a metric space and T is a continuous map of X into X. If in addition X is a compact metric space, then we say that $(X,T)$ is a compact dynamical system. See [Reference Furstenberg15, Reference Furstenberg16] for details of compact dynamics.

A topological dynamical system $(X,T)$ is said to be topologically transitive (or transitive for short) if for any pair $U, V$ of non-empty open subsets of X, the return time set

$$ \begin{align*} N_T(U,V)=N(U,V):=\{n\in\mathbb{N}_0: T^nU\cap V\neq\emptyset \} \end{align*} $$

is non-empty. When the operator is fixed and there is no risk of ambiguity, we shall write $N(U,V)$ in place of $N_T(U,V)$ . A topological dynamical system $(X,T)$ is said to be weakly mixing if the product system $(X\times X,T\times T)$ is transitive, and strongly mixing if for any pair $U, V$ of non-empty open subsets of X, $N(U,V)\in \mathcal {F}_{\mathrm {cf}}$ .

Given a family $\mathcal {F}$ , we say that a topological dynamical system $(X,T)$ is:

• • $\mathcal {F}$ -transitive if for any non-empty open subsets $U,V$ of X, $N(U,V)\in \mathcal {F}$ ;

• • $\mathcal {F}$ -mixing if the product system $(X\times X,T\times T)$ is $\mathcal {F}$ -transitive.

For each family $\mathcal {F}$ , since $\emptyset \notin \mathcal {F}$ , the $\mathcal {F}$ -transitivity is stronger than topological transitivity. It can be verified that a topological dynamical system is transitive if and only if it is $\mathcal {F}_{\mathrm {inf}}$ -transitive. A topological dynamical system is said to be topologically ergodic if it is $\mathcal {F}_{\mathrm {s}}$ -transitive.

We have the following characterization of weak mixing of topological dynamical systems, which is essentially contained in the proof of [Reference Furstenberg15, Proposition II.3].

Proposition 2.5. Let $(X,T)$ be a topological dynamical system. Then, $(X,T)$ is weakly mixing if and only if it is $\mathcal {F}_{\mathrm {t}}$ -transitive, and if and only if

$$ \begin{align*} \mathcal{N}_T=\{A\subset\mathbb{N}_0\colon N(U,V)\subset A\text{ for some non-empty open subsets }U,V\text{ of }X\} \end{align*} $$

is a filter.

By Proposition 2.5, it is easy to see that a topological dynamical system $(X,T)$ is $\mathcal {F}$ -mixing if and only if it is $\mathcal {F}$ -transitive and weakly mixing. We have the following folklore result on the return time sets of $\mathcal {F}$ -mixing systems, see e.g. [Reference Bès, Menet, Peris and Puig10, Lemma 2.3] in the setting of linear dynamics.

Proposition 2.6. Let $(X,T)$ be a topological dynamical system, where X is not a singleton, and $\mathcal {F}$ be a family. Then, $(X,T)$ is $\mathcal {F}$ -mixing if and only if

$$ \begin{align*} \mathcal{N}_T=\{A\subset\mathbb{N}_0\colon N(U,V)\subset A\text{ for some non-empty open subsets }U,V\text{ of }X\} \end{align*} $$

is a shift-invariant thick subfamily of $\mathcal {F}$ .

Proof. On the one hand, suppose that T is $\mathcal {F}$ -mixing. It is clear that $\mathcal {N}_T$ is a subfamily of $\mathcal {F}$ . By Proposition 2.5, $\mathcal {N}_T$ is a filter. By Remark 2.4, it suffices to show that $\mathcal {N}_T$ is a shift-invariant family. Let U and V be non-empty open subsets of X and $i\in \mathbb {N}_0$ . Since T is transitive, it has a dense range, and then $T^{-i}U\neq \emptyset $ and $T^{-i}V\neq \emptyset $ . Since T is weakly mixing and X is not a singleton, we have that X has no isolated points. Then, there are non-empty open subsets $W_1\subset T^{-i}U, W_2\subset V$ with $N(W_1,W_2)\subset [i,+\infty )$ . Note that $N(T^{-i}U,V)\cap [i,+\infty )\subset N(U,V)+i$ . So, we have

$$ \begin{align*} N(W_1,W_2)\subset N(T^{-i}U,V)\cap [i,+\infty)\subset N(U,V)+i \end{align*} $$

and

$$ \begin{align*} N(U,T^{-i}V)\subset (N(U,V)-i)\cap\mathbb{N}_0. \end{align*} $$

By the fact that $\mathcal {N}_T$ is a filter, $\mathcal {N}_T$ is a shift-invariant thick family.

On the other hand, suppose that $\mathcal {N}_T$ is a shift-invariant thick subfamily of $\mathcal {F}$ . Then, we have $\mathcal {N}_T\subset \mathcal {F}_{\mathrm {t}}$ . By Proposition 2.5, T is weakly mixing. Since T is $\mathcal {F}$ -transitive, we have that T is $\mathcal {F}$ -mixing.

In 2000, Weiss found in [Reference Weiss28] that the compact dynamical systems that are weakly disjoint from all weakly mixing compact dynamical systems are exactly the topologically ergodic ones. In 2001, Akin and Glasner generalized Weiss’ result to more classes of compact dynamical systems in [Reference Akin and Glasner2], and the generalized result is known as the Weiss–Akin–Glasner theorem, see e.g. [Reference Huang and Ye24, Reference Huang and Ye25]. In this theorem, they characterized the weak disjointness of $\mathcal {F}$ -transitive compact dynamical systems, where $\mathcal {F}$ is a shift-invariant thick family, and connected the study of topological dynamics and combinatorial number theory.

The key ingredient of the proof of the Weiss–Akin–Glasner theorem is the following lemma.

Lemma 2.10. (Weiss–Akin–Glasner lemma, [Reference Akin and Glasner2, Lemma 4.16])

Suppose that $\mathcal {F}$ is a shift-invariant thick family and $F\in \mathcal {F}$ . Then, there exists an $\mathcal {F}$ -transitive compact dynamical system $(X,T)$ and a non-empty open subset U of X such that

$$ \begin{align*} N(U,U)=F\cup\{0\}. \end{align*} $$

Recall that $\widetilde {\mathcal {F}}$ is the maximal shift-invariant thick subfamily of a full $\mathcal {F}$ . Then, we can restate the Weiss–Akin–Glasner lemma as follows.

Lemma 2.12. Suppose that $\mathcal {F}$ is a full family and $F\subset \mathbb {N}_0$ . Then, $F\in \widetilde {\mathcal {F}}$ if and only if there exists an $\mathcal {F}$ -mixing compact dynamical system $(X,T)$ and a non-empty open subset U of X such that

$$ \begin{align*} N(U,U)=F\cup\{0\}. \end{align*} $$

We also restate the Weiss–Akin–Glasner theorem as follows.

The concept of mild mixing was originally introduced by Furstenberg and Weiss in the context of ergodic theory [Reference Furstenberg and Weiss17], and was later introduced independently in topological dynamics by Glasner [Reference Glasner18] and by Huang and Ye [Reference Huang and Ye25], inspired by its ergodic-theoretic counterpart.

The return time sets of mild mixing systems are related to IP-sets. For a sequence $(x_n)_{n\in \mathbb {N}}$ in $\mathbb {N}$ , the finite sum set of $(x_n)_{n\in \mathbb {N}}$ is given by

$$ \begin{align*} \mathrm{FS}((x_n)_{n\in\mathbb{N}})=\{x_{i_1}+x_{i_2}+\cdots+x_{i_n} \colon i_1<i_2<\cdots <i_n, n\in \mathbb{N}\}.\end{align*} $$

A set $A\subset \mathbb {N}_0$ is called an IP-set if there is a sequence $(x_n)_{n\in \mathbb {N}}$ in $\mathbb {N}_0$ with infinitely many non-zero terms such that $\mathrm {FS}((x_n)_{n\in \mathbb {N}})\subset A$ . Denote by $\mathcal {F}_{\mathrm {ip}}$ the family of all IP-sets and by $\Delta _{\mathrm {ip}}$ the family of the sets of the form $(F-F)\cap \mathbb {N}_0$ , where $F\in \mathcal {F}_{\mathrm {ip}}$ and $F-F=\{m-n:m\neq n\in F\}$ . We denote the dual family of $\Delta _{\mathrm {ip}}$ by $\Delta ^*_{\mathrm {ip}}$ . For a topological dynamical system $(X,T), x\in X$ and a subset U of X, we denote

$$ \begin{align*} N_T(x,U)=N(x,U):=\{n\in\mathbb{N}_0: T^nx\in U\}. \end{align*} $$

Recall that a point $x\in X$ is recurrent if for every neighbourhood U of x, $N(x,U)$ is infinite. In 1981, Furstenberg proved the following result for compact dynamical systems, which is in fact true for topological dynamical systems.

The following characterization of mild mixing was obtained by Glasner [Reference Glasner18], and by Huang and Ye [Reference Huang and Ye25] independently.

2.3 Linear dynamics

In recent years, there has been growing interest in the study of linear dynamical systems, see [Reference Bayart and Matheron6, Reference Grosse-Erdmann and Peris Manguillot22] for further information.

Let X be an infinite-dimensional separable Fréchet space over $\mathbb {R}$ or $\mathbb {C}$ . For an operator (i.e. a continuous linear self-map) T on X, the pair $(X,T)$ is called a linear dynamical system. Usually, we simply call the operator T or $T\colon X\to X$ a linear dynamical system. If, in addition, T is an invertible map such that $T^{-1}$ is also an operator on X, then we say that $(X,T)$ (or T for short) is invertible. In this paper, for a Fréchet space X, the symbol $\|\cdot \|_X=\|\cdot \|$ denotes an F-norm defining the topology of X. If, in particular, X is a Banach space, then the symbol $\|\cdot \|_X=\|\cdot \|$ denotes a norm defining the topology of X.

While infinite-dimensional Fréchet spaces are lacking of compactness, many results in linear dynamical systems remain similar to their counterparts from compact dynamical systems, despite being obtained by different methods.

A well-known result to imply the weak mixing of a given operator is the so-called hypercyclicity criterion.

A remarkable feature of the hypercyclicity criterion is that it is actually equivalent to an operator being weakly mixing. In [Reference Bès and Peris11], Bès and Peris obtained this result using the notion of hereditarily hypercyclic operators.

Unlike the case for compact dynamical systems, a tool for the study of linear dynamical systems is provided by the neighbourhoods of $0$ . We recall the blow up/collapse condition for weak mixing.

Theorem 2.21. [Reference Grosse-Erdmann and Peris Manguillot22, Theorem 2.47]

Let $T\colon X\to X$ be an operator. Then, T is weakly mixing if and only if for any non-empty open sets $U, V\subset X$ and any neighbourhood W of $0$ ,

$$ \begin{align*} N(U, W)\cap N(W, V)\neq\emptyset. \end{align*} $$

Let $X= c_0(\mathbb {N}_0)$ or $\ell ^p(\mathbb {N}_0)$ , $1\leq p<\infty $ , and let $w=(w_i)_{i\in \mathbb {N}_0}$ be a bounded sequence of positive real numbers. Define a map $B_w\colon X\to X$ by

$$ \begin{align*} (x_0, x_1, x_2, \ldots )\mapsto (w_1x_1, w_2x_2, \ldots). \end{align*} $$

The map $B_w$ is clearly an operator on X and is called a unilateral weighted backward shift. Similarly, one can define bilateral weighted backward shift on $c_0(\mathbb {Z})$ and $\ell ^p(\mathbb {Z})$ , ${1\leq p<\infty }$ . In [Reference Bès, Menet, Peris and Puig10], Bès et al characterized the $\mathcal {F}$ -transitivity of such operators and one of their results is restated here.

We denote by $\omega $ the Fréchet space of all sequences in $\mathbb {K}$ , where $\mathbb {K}$ denotes either $\mathbb {R}$ or $\mathbb {C}$ , that is,

$$ \begin{align*} \omega \colon=\mathbb{K}^{\mathbb{N}_0}=\{(x_n)_n\colon x_n\in \mathbb{K}, n\in \mathbb{N}_0\}, \end{align*} $$

with the topology given by the seminorms: $p_n(x)=\mathrm {sup}_{i\leq n}|x_i|, n\in \mathbb {N}_0$ . The canonical basis of $\omega $ is denoted by $e_n=(\delta _{i,n})_{i\in \mathbb {N}_0}$ , $n\in \mathbb {N}_0$ .

Let X be an infinite-dimensional separable Fréchet space. If X can be embedded continuously (by a linear map, $\theta $ say) into $\omega $ , then X is called a unilateral Fréchet sequence space. Roughly speaking, the convergence in a Fréchet sequence space implies the convergence of each coefficient in $\mathbb {K}$ . Since for each $n\in \mathbb {N}_0$ , $E_n=\theta ^{-1}e_n\in X$ (if exists) must be unique, we denote it still by $e_n$ .

Let X be a unilateral Fréchet sequence space in which $(e_n)_{n\in \mathbb {N}_0}$ is a basis (i.e. $\sum _{i=0}^{\infty }x_ie_i$ converges for each $x\in X$ ). Define a linear map $B\colon X\to \omega $ by

$$ \begin{align*} (x_0, x_1, x_2, \ldots )\mapsto (x_1, x_2, \ldots), \end{align*} $$

where $x=\sum _{i=0}^{\infty }x_ie_i\in X.$ If B maps X into X, then it must be continuous and we say that B is the unilateral backward shift on X. Define a map $F\colon \{x\in X\colon \text {there is } l\in \mathbb {N}_0\text { such that }x_i=0\text { for each }i\geq l\}\to X$ by

$$ \begin{align*} (x_0, x_1, x_2, \ldots,x_l,0,\ldots )\mapsto (0,x_0, x_1, x_2, \ldots,x_l,0,\ldots ). \end{align*} $$

The map F is called the unilateral forward shift for finite sequences of X. Bilateral backward (and forward) shifts on bilateral Fréchet sequence spaces are defined similarly. See [Reference Bès, Menet, Peris and Puig9, Reference Bès, Menet, Peris and Puig10, Reference Grosse-Erdmann and Peris Manguillot22, Reference Salas27] for more details of weighted backward shifts.

3.1 The $\mathcal {F}$ -mixing criterion

It is observed that an operator is weakly mixing if and only if it is weakly disjoint from itself. We investigate how a weakly mixing operator can be weakly disjoint from itself. In terms of Furstenberg families, we study the $\mathcal {F}$ -transitivity of the direct sum of an operator and itself when a family $\mathcal {F}$ is given.

Recall that the blow up/collapse condition characterizes the weak mixing of operators (see Theorem 2.21). We formulate here the blow up/collapse condition for $\mathcal {F}$ -mixing.

Lemma 3.8. Let $\mathcal {F}$ be a family. Then, an operator $T\colon X\to X$ is $\mathcal {F}$ -mixing if and only if for any non-empty open subsets $U,V$ of X and neighbourhood W of $0$ , we have

$$ \begin{align*} N(U,W)\cap N(W,V)\in\mathcal{F}. \end{align*} $$

Inspired by the hypercyclicity criterion (Theorem 2.19), we formulate a criterion for $\mathcal {F}$ -mixing in terms of limits, which is especially applicable to backward shifts (see §5). Before this, we need a notion, which requires the specific shape of the indices of convergent subsequences.

Suppose that X is a metric space and $\mathcal {F}$ is a family. Let $(x_n)_{n\in \mathbb {N}_0}$ be a sequence in X. We say that a point $x\in X$ is an $\mathcal {F}$ -limit of $(x_n)_{n\in \mathbb {N}_0}$ , denoted by $\mathcal {F}$ - $\displaystyle \lim _{n\to \infty }x_n=x$ , if for any neighbourhood U of x, we have

$$ \begin{align*} \{n\in\mathbb{N}_0: x_n\in U\}\in\mathcal{F}. \end{align*} $$

In [Reference Bès, Menet, Peris and Puig10], Bès et al gave a criterion [Reference Bès, Menet, Peris and Puig10, Theorem 2.4] for an operator to be $\mathcal {F}$ -mixing, requiring that $\widetilde {\mathcal {F}}$ is a filter. The following criterion has a strong requirement on the limits, but no requirement for the family.

Theorem 3.9. ( $\mathcal {F}$ -mixing criterion)

Let $T\colon X\to X$ be an operator and $\mathcal {F}$ be a family. If there exist dense subsets $X_0$ , $Y_0$ of X and a sequence $(S_n)_{n\in \mathbb {N}_0}$ of maps of $Y_0$ into X such that for any $x_0\in X_0$ , $y_0\in Y_0$ ,

$$ \begin{align*} \mathcal{F}\text{-}\displaystyle\lim_{n\to\infty}(T^{n}x_0, S_{n}y_0, T^{n}S_{n}y_0)=(0, 0, y_0), \end{align*} $$

then T is $\mathcal {F}$ -mixing.

Proof. Let $U, V$ be non-empty open subsets of X and let $W\subset X$ be a neighbourhood of $0$ . Pick $x_0\in X_0\cap U$ and $y_0\in Y_0\cap V$ . Then,

$$ \begin{align*} A=\{n\in\mathbb{N}_0: T^nx_0\in W, S_ny_0\in W, T^nS_ny_0\in V\}\in\mathcal{F}. \end{align*} $$

Since $A\subset N(U, W)\cap N(W, V)$ , we have $N(U,W)\cap N(W,V)\in \mathcal {F}$ . Then, T is $\mathcal {F}$ -mixing, by Lemma 3.8.

3.2 Mild mixing

It is natural to consider the property of operators that are weakly disjoint from a given class of operators. In 2024, Cardeccia characterized in [Reference Cardeccia13] the operators that are weakly disjoint from all weakly mixing operators and the operators that are weakly disjoint from all topologically ergodic operators. We restate these results, in the language of weak disjointness.

It is clear that the operators that are weakly disjoint from all strongly mixing operators are exactly the hypercyclic ones. We wonder if there is a non-mixing operator that is weakly disjoint from all hypercyclic operators. We introduce the concept of mild mixing in the framework of linear dynamical systems, and show that the mild mixing of operators is a property that lies strictly between strong mixing and weak mixing.

Next, we deduce a sufficient condition for an operator to be mildly mixing.

Recall from Theorem 2.16, we know that the mild mixing of compact dynamical systems is equivalent to the $\Delta ^*_{\mathrm {ip}}$ -transitivity. In Proposition 3.15, we have shown that each $\Delta ^*_{\mathrm {ip}}$ -transitive operator is mildly mixing, so we ask the following question.

Now, we deduce that for operators, strong mixing is not equivalent to mild mixing. Let $\Delta =\{A\subset \mathbb {N}_0: \text {there is }F\in \mathcal {F}_{\mathrm {inf}}\text { with }(F-F)\cap \mathbb {N}_0\subset A\}$ . Since $\mathcal {F}_{\mathrm {ip}}\subset \mathcal {F}_{\mathrm {inf}}$ , by Proposition 3.15, each $\Delta ^*$ -transitive operator is mildly mixing. Bès et al have constructed a $\Delta ^*$ -transitive weighted backward shift on $c_0(\mathbb {N}_0)$ or $\ell ^p(\mathbb {N}_0), 1\leq p<\infty $ that is not strongly mixing in [Reference Bès, Menet, Peris and Puig10, Proposition 5.5]. So, we have the following.

Salas constructed in [Reference Salas27, Corollary 2.6] two weakly mixing operators $T,S$ on $\ell ^2(\mathbb {Z})$ such that T and S are not weakly disjoint. He also constructed in [Reference Salas27, Corollary 2.3] an operator T on $\ell ^2(\mathbb {Z})$ such that both T and $T^*$ are weakly mixing, while T and $T^*$ can never be weakly disjoint by Remark 3.18. It is illustrated in Example 3.5 that there is a weakly mixing operator T on $\ell ^1(\mathbb {N}_0)$ such that T and $T^2$ are not weakly disjoint. The above examples show that mild mixing is not equivalent to weak mixing for operators.

It is observed that each strongly mixing operator is mildly mixing and each mildly mixing operator is weakly mixing. By Propositions 3.17 and 3.19, the mild mixing of operators lies strictly between strong mixing and weak mixing. In 1993, Ansari showed in [Reference Ansari3] that an operator is hypercyclic if and only if each of its iterates is hypercyclic.

By Ansari’s result, we have the following two more sufficient conditions for mild mixing. Recall that an operator is said to be chaotic in the sense of Devaney (or chaotic for short) if it is hypercyclic and has a dense set of periodic points. In 2007, Badea and Grivaux constructed in [Reference Badea and Grivaux4, Corollary 4.7] a chaotic operator that is not strongly mixing, which shows that chaos does not imply strong mixing. However, we have that chaos does imply mild mixing. We also obtain that an operator is mildly mixing if some iterate of it is mildly mixing.

It is natural to consider the following question.

4.1 The Weiss–Akin–Glasner theorem for operators

We have the following characterization of subsets of non-negative integers that can be realized by the return time sets of $\mathcal {F}$ -mixing operators, which is an analogue of the Weiss–Akin–Glasner lemma (Lemma 2.12) from topological dynamics.

Lemma 4.1. Let $\mathcal {F}$ be a full family and $F\subset \mathbb {N}_0$ . Then, $F\in \widetilde {\mathcal {F}}$ if and only if there is an invertible $\mathcal {F}$ -mixing weighted backward shift $B_w$ on X, and a non-empty open subset W of X such that

$$ \begin{align*} N_{B_w}(W,W)=F\cup\{0\}, \end{align*} $$

where $X=c_0(\mathbb {Z})$ or $ \ell ^p(\mathbb {Z})$ , $1\leq p<\infty $ .

Proof. If such an operator $B_w\colon X\to X$ exists, then by Proposition 2.6, $\mathcal {N}_{B_w}$ is a shift-invariant thick subfamily of $\mathcal {F}$ . Then, $F\cup \{0\} \in \mathcal {N}_{B_w}\subset \widetilde {\mathcal {F}}$ since $\widetilde {\mathcal {F}}$ is the maximal shift-invariant thick subfamily of $\mathcal {F}$ . Since $\widetilde {\mathcal {F}}$ is shift-invariant, it is full. Then, ${(F\cup \{0\})\cap \mathbb {N}\in \widetilde {\mathcal {F}}}$ . It is observed that $(F\cup \{0\})\cap \mathbb {N}\subset F$ . So, $F\in \widetilde {\mathcal {F}}$ .

Conversely, suppose that $F\in \widetilde {\mathcal {F}}$ . Since $\widetilde {\mathcal {F}}$ is hereditary upward, $F\cup \{0\}\in \widetilde {\mathcal {F}}$ . As the desired return time set is $F\cup \{0\}$ , replacing F by $F\cup \{0\}$ if necessary, we may assume that $0\in F$ . We divide the construction of the weighted backward shift into two cases.

Case A: $F\in \mathcal {F}_{\mathrm {cf}}$ . Write $F=I_1\cup I_2\cup \cdots \cup I_t\cup ([L,\infty )\cap \mathbb N_0)$ with $t,L\geq 0$ such that $\{I_k\}_{k=1}^t$ are disjoint maximal finite integer intervals. Write $\mathbb {N}_0\setminus F=J_1\cup J_2\cup \cdots \cup J_t$ such that $\{J_k\}_{k=1}^t$ are disjoint maximal finite integer intervals. Without loss of generality, we assume that $\min I_1<\min I_2<\cdots <\min I_t$ and $\min J_1<\min J_2<\cdots <\min J_t$ . If $F=\mathbb {N}_0$ , then there is no such $I_k$ and $J_k$ . For each $k\in \{1,2,\ldots , t\}$ , let

$$ \begin{align*} \mathcal{I}_k= \begin{cases} \bigg[\underbrace{2,2,\ldots,2}_{s\text{ terms}},\underbrace{\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2}}_{s-1\text{ terms}},1\bigg] & \text{ if }|I_k|=2s\text{ for some }s\in\mathbb{N}, \\ \bigg[\underbrace{2,2,\ldots,2}_{s+1\text{ terms}},\underbrace{\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2}}_{s\text{ terms}}\bigg] & \text{ if }|I_k|=2s+1\text{ for some }s\in\mathbb{N}_0, \end{cases} \end{align*} $$

and

$$ \begin{align*} \mathcal{J}_{k}= \begin{cases} \bigg[\dfrac{1}{4},1,\ldots,1\bigg] & \text{if } k=1,\\[7pt] \bigg[\dfrac{1}{2},1,\ldots,1\bigg] &\text{otherwise}, \end{cases} \end{align*} $$

with $|\mathcal {J}_{k}|=|J_k|$ . We consider the weighted backward shift $B_w\colon X\to X$ defined by the weight

$$ \begin{align*} w= \bigg(\underbrace{\cdots\frac{1}{2},\frac{1}{2},\frac{1}{2}}_{-\mathbb{N}},\underbrace{\mathcal{I}_1,\mathcal{J}_1,\ldots,\mathcal{I}_t, \mathcal{J}_t,2,2, \ldots}_{\mathbb{N}_0}\bigg). \end{align*} $$

Since both $(w_i)_{i\in \mathbb {Z}}$ and $({1}/{w_{i+1}})_{i\in \mathbb {Z}}$ are bounded, $B_w$ is an invertible operator on X. Clearly, $B_w$ is strongly mixing (see [Reference Grosse-Erdmann and Peris Manguillot22, Example 4.15]) and, thus, $\mathcal F$ -mixing.

Case B: $F\in \widetilde {\mathcal {F}}\setminus \mathcal {F}_{\mathrm {cf}}$ . Write F as the disjoint union of a sequence $(I_k)_{k\in \mathbb {N}}$ of maximal finite integer intervals and write $\mathbb {N}_0\setminus F$ as the disjoint union of a sequence $(J_k)_{k\in \mathbb {N}}$ of maximal finite integer intervals. Without loss of generality, we assume that the sequences $(\min I_k)_{k\in \mathbb {N}}$ and $(\min J_k)_{k\in \mathbb {N}}$ are increasing. Note that $0\in I_1$ . For each $k\in \mathbb {N}$ , let $\mathcal {I}_k$ and $\mathcal {J}_{k}$ be defined by the same formulae as in Case A. Let $B_w$ be the bilateral weighted backward shift on X defined by the weight

$$ \begin{align*} w=\bigg(\underbrace{\cdots\frac{1}{2},\frac{1}{2},\frac{1}{2}}_{-\mathbb{N}},\underbrace{\mathcal{I}_1,\mathcal{J}_1,\ldots,\mathcal{I}_n, \mathcal{J}_n,\mathcal{I}_{n+1}, \mathcal{J}_{n+1}, \ldots}_{\mathbb{N}_0}\bigg). \end{align*} $$

Since both $(w_i)_{i\in \mathbb {Z}}$ and $({1}/{w_{i+1}})_{i\in \mathbb {Z}}$ are bounded, $B_w$ is an invertible operator on X. We first prove the following claim.

Claim 4.2. If $m\in \mathbb {N}_0$ satisfies $\{m-r-1,m-r,\ldots ,m+r+1\}\subset F$ for some $r\in \mathbb {N}_0$ with $r>|I_1|$ , then $w_0w_1\cdots w_m\geq 2^r$ .

Proof of the claim.

Denote $\varphi (j)=w_0w_1\cdots w_j$ for $j\in \mathbb {N}_0$ . Since $(I_k)_{k\in \mathbb {N}}$ are disjoint, there is a unique $k\in \mathbb {N}$ such that $\{m-r-1,m-r,\ldots ,m+r+1\}\subset I_k$ . Denote ${a=\min I_k}$ and $b=\max I_k$ . Since $|I_k|\geq 2r+3>|I_1|$ , we have $k\geq 2$ and $|I_k|\geq 3$ . If $|I_k|=2s$ for some $s\in \mathbb {N}$ , then

$$ \begin{align*} \varphi(a+i)=\begin{cases} 2^{i}, &0\leq i\leq s-1,\\ 2^{2s-i-2}, &s\leq i\leq 2s-2,\\ 1, &i=2s-1. \end{cases} \end{align*} $$

If $|I_k|=2s+1$ for some $s\in \mathbb {N}$ , then

$$ \begin{align*} \varphi(a+i)=\begin{cases} 2^{i}, &0\leq i\leq s,\\ 2^{2s-i}, &s+1\leq i\leq 2s. \end{cases} \end{align*} $$

As $a+r+1\leq m \leq b-r-1$ , it is clear that $\varphi (m)\geq 2^r$ .

To show that $B_w$ is $\mathcal {F}$ -mixing, by Proposition 2.22, it suffices to show that for any $M>0$ and $j\in \mathbb {N}$ , $A_{M,j}\cap \overline {A}_{M,j}\in \mathcal {F}$ , where

$$ \begin{align*} A_{M,j}&=\{n\in\mathbb{N}\colon w_{j+1}w_{j+2}\cdots w_{j+n}>M\}, \\ \overline{A}_{M,j}&=\{n\in\mathbb{N}\colon w_{j}w_{j-1}\cdots w_{j-n+1}<{1}/{M}\}. \end{align*} $$

Now, fix $M>0$ and $j\in \mathbb {N}$ . There is $q\in \mathbb {N}_0$ with $q>|I_1|$ and $2^q>Mw_0w_1w_2\cdots w_j$ . Consider the set

$$ \begin{align*} E=\{m\in\mathbb{N}_0\colon \{m-q-j-1,m-q-j,\ldots, m+q+j+1\}\subset F\}. \end{align*} $$

Recall that $\widetilde {\mathcal {F}}$ is the maximal shift-invariant thick subfamily of $\mathcal {F}$ and $F\in \widetilde {\mathcal {F}}$ , by Remark 2.4, $E\in \widetilde {\mathcal F}\subset \mathcal {F}$ . Fix $n\in E$ . We have $\{(j+n)-q-1,(j+n)-q,\ldots , (j+n)+ q+1\}\subset F$ . By the claim,

$$ \begin{align*} w_0w_1w_2\cdots w_{j+n}\geq 2^q. \end{align*} $$

This implies that

$$ \begin{align*} w_{j+1}w_{j+2}\cdots w_{j+n}=\frac{w_0w_1w_2\cdots w_{j+n}}{w_0w_1w_2\cdots w_{j}}\geq \frac{2^q}{w_0w_1w_2\cdots w_{j}}>M. \end{align*} $$

It is obvious that $n\geq q+j+1$ . Then,

$$ \begin{align*} w_jw_{j-1}\cdots w_{j-n+1}&=(w_jw_{j-1}\cdots w_0)\cdot (w_{-1}\cdots w_{j-n+1})\\ &=(w_0w_1\cdots w_j)\cdot \frac{1}{2^{n-j-1}} \leq (w_0w_1\cdots w_j)\cdot \frac{1}{2^{q}}<\frac{1}{M}. \end{align*} $$

So, $n\in A_{M,j}\cap \overline {A}_{M,j}$ and then $E\subset A_{M,j}\cap \overline {A}_{M,j}$ . This gives that $A_{M,j}\cap \overline {A}_{M,j}\in \mathcal {F}$ . Hence, $B_w$ is $\mathcal {F}$ -mixing, by Proposition 2.22.

So, in both cases, we have constructed an $\mathcal {F}$ -mixing bilateral weighted backward shift $B_w$ on X and for each $n\in \mathbb N_0$ , one has:

1. (1) $w_0w_1w_2\cdots w_n =\frac {1}{2}$ if and only if $n\notin F$ ;

2. (2) $w_0w_1w_2\cdots w_n \geq 1$ if and only if $n\in F$ .

For each $x=\sum _{i\in \mathbb Z}x_ie_i\in X$ , let

$$ \begin{align*} \|x\|_+=\bigg\|\!\sum_{i\in\mathbb N_0}x_ie_i\bigg\|. \end{align*} $$

The function $\|\cdot \|_+$ is clearly a continuous seminorm on X. Consider the following neighbourhood W of $e_0$ :

$$ \begin{align*} W=\bigg\{y\in X\colon \|y-e_0\|_+<\frac{4}{5}\bigg\}. \end{align*} $$

Now, we verify that $N_{B_w}(W,W)=F$ . On the one hand, fix $m\in N_{B_w}(W,W)\setminus \{0\}$ . Then, there is a vector $x=\sum _{i\in \mathbb Z}x_ie_i\in W$ such that $B_w^mx\in W$ , which gives that $|x_m|<\frac 45$ and $|w_1w_2\cdots w_mx_m-1|<\frac 45$ . Then, we have $w_1w_2\cdots w_m|x_m|>\frac {1}{5}$ and

$$ \begin{align*} w_0w_1w_2\cdots w_m=\frac{w_0w_1w_2\cdots w_m|x_m|}{|x_m|}>\frac{w_0}{4}=\frac{1}{2}, \end{align*} $$

which gives that $m\in F$ . On the other hand, fix $m\in F\setminus \{0\}$ . Let $x=e_0+ ({3}/{10w_1w_2\cdots w_m})e_m$ . Then,

$$ \begin{align*} \|x-e_0\|_+=\frac{3}{10w_1w_2\cdots w_m}\|e_m\|=\frac{3w_0}{10w_0w_1w_2\cdots w_m}\leq \frac{3w_0}{10}=\frac{3}{5}<\frac{4}{5} \end{align*} $$

and, hence, $x\in W$ . Moreover,

$$ \begin{align*} \|B_w^mx-e_0\|_+=\bigg\|\frac{3}{10}e_0-e_0 + (w_{-m+1}w_{-m+2}\cdots w_{0}) e_{-m}\bigg\|_+=\frac{7}{10}\|e_0\|=\frac{7}{10}<\frac{4}{5}. \end{align*} $$

So, $B_w^mx\in W$ . Therefore, $m\in N_{B_w}(W,W)$ .

By Lemma 4.1, for a given full family $\mathcal {F}$ , we characterize the operators that are weakly disjoint from all $\mathcal {F}$ -mixing operators, which is an analogue of the Weiss–Akin–Glasner theorem (Theorem 2.13) from topological dynamics.

For a shift-invariant thick family $\mathcal {F}$ , if every $\mathcal {F}^*$ -transitive operator is weakly mixing, then the characterization of the weak disjointness of $\mathcal {F}$ -transitive operators can be strengthened as follows.

Since $\mathcal {F}_{\mathrm {ts}}$ and $\mathcal {F}_{\mathrm {t}}$ are shift-invariant thick families, and topologically ergodic operators are weakly mixing, by Theorem 4.4 and Proposition 4.5, we can strengthen Cardeccia’s results (Theorems 3.11 and 3.12) as follows.

As we can see, if each member of a family $\mathcal {F}$ contains a return time set of an $\mathcal {F}$ -transitive operator, then the $\mathcal {F}^*$ -transitivity of an operator is equivalent to the weak disjointness from all $\mathcal {F}$ -transitive operators. By Lemma 4.1, every member of a shift-invariant thick family contains a return time set of an $\mathcal {F}$ -mixing operator. However, one can construct a family $\mathcal {F}$ , where every member contains a return time set of an $\mathcal {F}$ -transitive operator but is not thick, using the following result.

Let $T\colon X\to X$ be a hypercyclic operator that is not weakly mixing and let ${\mathcal {N}_T=\{A\subset \mathbb {N}_0\colon N(U,V)\subset A\text { for some non-empty open subsets }U,V\text { of }X\}}$ . It is easy to check that an operator $S\colon Y\to Y$ is $\mathcal {N}_T^*$ -transitive if and only if it is weakly disjoint from all $\mathcal {N}_T$ -transitive operators if and only if it is weakly disjoint from T. However, since T is not weakly mixing, we have $N(U_0,V_0)\not \in \mathcal {F}_{\mathrm {t}}$ for some non-empty open subsets $U_0,V_0$ of X and then $\mathcal {N}_T$ is not a thick family. Lemma 4.1 gives a characterization of subsets of non-negative integers that can be realized by the return time sets of $\mathcal {F}$ -mixing operators. However, the characterization of return time sets remains open for hypercyclic operators that are not weakly mixing. This raises the following question.

By the Weiss–Akin–Glasner theorem (Theorem 2.9), there is no family $\mathcal {F}$ such that the $\mathcal {F}$ -transitivity of compact dynamical systems is equivalent to the weak disjointness from all $\mathcal {F}$ -transitive compact dynamical systems, since there exists a topologically ergodic compact dynamical system that is not weakly mixing. However, things change under the framework of linear dynamical systems, since each topologically ergodic operator must be $\mathcal {F}_{\mathrm {ts}}$ -transitive, thus weakly mixing, so we ask the following question.

In 1995, Salas provided in [Reference Salas27, Corollary 2.3] a hypercyclic operator T on $\ell ^2(\mathbb {Z})$ such that $T^*$ is also hypercyclic. In Lemma 4.1, if we restrict the underlying space X to be $\ell ^2(\mathbb {Z})$ , require that $\mathbb {N}_0\setminus F$ is large enough and modify our construction slightly, then the operator we obtain possesses a hypercyclic adjoint.

Proof. Let $F=F_1$ . It is clear that both F and $\mathbb {N}_0\setminus F$ are not cofinite. Write F as the disjoint union of a sequence $(I_k)_{k\in \mathbb {N}}$ of maximal finite integer intervals. Since ${\{ n\in \mathbb {N}_0\colon \{n-2,n-1,n\}\subset F\}\in \mathcal {F}_1}$ , without loss of generality, we require that $0\notin F,1\notin F$ and $|I_k|\geq 3$ for each $k\in \mathbb {N}$ . In addition, we assume that the sequence $(\mathrm \min I_k)_{k\in \mathbb {N}}$ is increasing. Let $J_1=[1,\mathrm \min I_1-1]\cap \mathbb {N}_0$ . For each $k\in \mathbb {N}$ , let ${J_{k+1}=[\mathrm \max I_{k}+1,\mathrm \min I_{k+1}-1]\cap \mathbb {N}_0}$ . For $n\in \mathbb {N}$ , let

$$ \begin{align*} \mathcal{I}_n=\begin{cases} \bigg[\underbrace{2,2,\ldots,2}_{m\text{ terms}},\underbrace{\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2}}_{m\text{ terms}}\bigg]\quad\text{ if }|I_n|=2m\text{ for some }m\in\mathbb{N}, \\ \bigg[\underbrace{2,2,\ldots,2}_{m\text{ terms}},\underbrace{\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2}}_{m\text{ terms}},1\bigg] \quad\text{if }|I_n|=2m+1\text{ for some }m\in\mathbb{N} \end{cases} \end{align*} $$

and

$$ \begin{align*} \mathcal{J}_n=\begin{cases} \bigg[\underbrace{\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2}}_{m\text{ terms}},\underbrace{2,2,\ldots,2}_{m\text{ terms}}\bigg] \quad\text{if }|J_n|=2m\text{ for some }m\in\mathbb{N}, \\ \bigg[\underbrace{\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2}}_{m\text{ terms}},\underbrace{2,2,\ldots,2}_{m\text{ terms}},1\bigg] \quad\text{if }|J_n|=2m+1\text{ for some }m\in\mathbb{N}, \\ [1] \quad\text{if } |J_n|=1. \end{cases} \end{align*} $$

In addition, let

$$ \begin{align*} \overline{\mathcal{I}_n}=\begin{cases} \bigg[\underbrace{2,2,\ldots,2}_{m\text{ terms}},\underbrace{\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2}}_{m\text{ terms}}\bigg] \quad\text{if }|I_n|=2m\text{ for some }m\in\mathbb{N}, \\ \bigg[1,\underbrace{2,2,\ldots,2}_{m\text{ terms}},\underbrace{\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2}}_{m\text{ terms}}\bigg] \quad\text{if }|I_n|=2m+1\text{ for some }m\in\mathbb{N} \end{cases} \end{align*} $$

and

$$ \begin{align*} \overline{\mathcal{J}_n}=\begin{cases} \bigg[\underbrace{\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2}}_{m\text{ terms}},\underbrace{2,2,\ldots,2}_{m\text{ terms}}\bigg] \quad\text{if }|J_n|=2m\text{ for some }m\in\mathbb{N}, \\ \bigg[1,\underbrace{\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2}}_{m\text{ terms}},\underbrace{2,2,\ldots,2}_{m\text{ terms}}\bigg]\quad \text{if }|J_n|=2m+1\text{ for some }m\in\mathbb{N}, \\ [1] \quad\text{if } |J_n|=1. \end{cases} \end{align*} $$

Let $B_w$ be the bilateral weighted backward shift on X defined by the weight w, where $w=(w_i)_{i\in \mathbb {Z}}$ is given by

$$ \begin{align*} w=(\underbrace{\cdots \overline{\mathcal{I}_n},\overline{\mathcal{J}_n}, \ldots\overline{\mathcal{I}_1},\overline{\mathcal{J}_1}}_{-\mathbb{N}}, \underbrace{[1]}_{0}, \underbrace{\mathcal{J}_1,\mathcal{I}_1,\ldots, \mathcal{J}_n,\mathcal{I}_n, \ldots}_{\mathbb{N}}). \end{align*} $$

For each $i\in -\mathbb {N}$ , we have $w_i={1}/{w_{-i}}$ . Since both $(w_i)_{i\in \mathbb {Z}}$ and $({1}/{w_{i+1}})_{i\in \mathbb {Z}}$ are bounded, $B_w$ is an invertible operator on $\ell ^2(\mathbb {Z})$ . Let $v=(w_{1-i})_{i\in \mathbb {N}_0}$ and $T=B_w$ . It is easy to see that $T^*$ is conjugate to $B_v$ . It remains to show that $B_w$ is $\mathcal {F}_1$ -transitive and $B_v$ is $\mathcal {F}_2$ -transitive.

By the same argument as the one in Case B of the proof of Lemma 4.1, we have the following facts:

• • if $m\in \mathbb {N}_0$ satisfies $\{m-r,m-r+1,\ldots ,m+r+1\}\subset F$ for some $r\in \mathbb {N}_0$ , then $w_1\cdots w_m\geq 2^r$ ;

• • if $m\in \mathbb {N}_0$ satisfies $\{m-r,m-r+1,\ldots ,m+r+1\}\subset \mathbb {N}\setminus F$ for some $r\in \mathbb {N}_0$ , then $w_1\cdots w_m\leq {1}/{2^r}$ .

To show that $B_w$ is $\mathcal {F}_1$ -transitive, by Proposition 2.22, it suffices to show that for any $M>0$ and $j\in \mathbb {N}$ , $A_{M,j}(w)\cap \overline {A}_{M,j}(w)\in \mathcal {F}_1$ , where

$$ \begin{align*} A_{M,j}(w)&=\{n\in\mathbb{N}: w_{j+1}w_{j+2}\cdots w_{j+n}>M\}, \\ \overline{A}_{M,j}(w)&=\bigg\{n\in\mathbb{N}: w_{j}w_{j-1}\cdots w_{j-n+1}<\frac{1}{M}\bigg\}. \end{align*} $$

Fix $j\in \mathbb {N}$ and $M>0$ . There is $q_1\in \mathbb {N}$ with $2^{q_1}>Mw_1\cdots w_j$ . Consider the set

$$ \begin{align*} E_1=\{m\in\mathbb{N}_0: \{m-q_1-j-1, m-q_1-j,\ldots,m+q_1+j+1\}\subset F\}. \end{align*} $$

Since $\mathcal {F}_1$ is a shift-invariant thick family and $F\in \mathcal {F}_1$ , by Remark 2.4, $E_1\in \mathcal {F}_1$ . Fix $n\in E_1$ , we have $\{(n+j)-q_1, (n+j)-q_1+1,\ldots ,(n+j)+q_1+1\}\subset F$ and $\{(n-j-1)-q_1, (n-j-1)-q_1+1,\ldots ,(n-j-1)+q_1+1\}\subset F$ . Then, $w_1w_2\cdots w_{j+n}\geq 2^{q_1}$ and $w_1w_2\cdots w_{n-j-1}\geq 2^{q_1}$ . We have

$$ \begin{align*} w_{j+1}w_{j+2}\cdots w_{j+n}=\frac{w_1w_2\cdots w_{j+n}}{w_1w_2\cdots w_j}\geq \frac{2^{q_1}}{w_1w_2\cdots w_j}>M \end{align*} $$

and

$$ \begin{align*} w_jw_{j-1}\cdots w_0w_{-1}\cdots w_{j-n+1}=w_0w_1\cdots w_j \frac{1}{w_{1}\cdots w_{n-j-1}}\leq \frac{w_1\cdots w_j}{2^{q_1}}<\frac{1}{M}. \end{align*} $$

This gives that $n\in A_{M,j}(w)\cap \overline {A}_{M,j}(w)$ and then $E_1\subset A_{M,j}(w)\cap \overline {A}_{M,j}(w)$ . Then, $A_{M,j}(w)\cap \overline {A}_{M,j}(w)\in \mathcal {F}_1$ and $B_w$ is $\mathcal {F}_1$ -transitive by Proposition 2.22.

To show that $B_v$ is $\mathcal {F}_2$ -transitive, by Proposition 2.22, it suffices to show that for any $M>0$ and $j\in \mathbb {N}$ , $A_{M,j}(v)\cap \overline {A}_{M,j}(v)\in \mathcal {F}_2$ , where

$$ \begin{align*} A_{M,j}(v)&=\{n\in\mathbb{N}: v_{j+1}v_{j+2}\cdots v_{j+n}>M\}, \\ \overline{A}_{M,j}(v)&=\bigg\{n\in\mathbb{N}: v_{j}v_{j-1}\cdots v_{j-n+1}<\frac{1}{M}\bigg\}. \end{align*} $$

Fix $j\in \mathbb {N}$ and $M>0$ . There is $q_2\in \mathbb {N}$ with $2^{q_2}>M\mathrm \max \{w_0w_{-1}\cdots w_{-j+1},\, v_jv_{j-1} \cdots v_2\}$ . Consider the set

$$ \begin{align*} E_2=\{m\in\mathbb{N}_0: \{m-q_2-j,m-q_2-j+1,\ldots, m+q_2+j+1\}\subset \mathbb{N}\setminus F\}. \end{align*} $$

Since $\mathbb {N}_0\setminus F\in \mathcal {F}_2$ and $\mathcal {F}_2$ is full, we have $\mathbb {N}\setminus F\in \mathcal {F}_2$ . Since $\mathcal {F}_2$ is a shift-invariant thick family, by Remark 2.4, $E_2\in \mathcal {F}_2$ . Fix $n\in E_2$ . We have $\{(n+j)-q_2,(n+j)-q_2+1,\ldots , (n+j)+q_2+1\}\subset \mathbb {N}\setminus F$ and $\{(n-j)-q_2,(n-j)-q_2+1,\ldots , (n-~j)+q_2+1\}\subset \mathbb {N}\setminus F$ . By the above analysis, we have $w_1w_2\cdots w_{j+n-1}\leq {1}/{2^{q_2}}$ and ${w_1w_2\cdots w_{n-j}\leq {1}/{2^{q_2}}}$ . Note that

$$ \begin{align*} v_1\cdots v_{j+n}=\frac{1}{w_1w_2\cdots w_{j+n-1}}=\frac{1}{w_0w_1w_2\cdots w_{j+n-1}}. \end{align*} $$

We have

$$ \begin{align*} v_{j+1}\cdots v_{j+n}=\frac{{1}/({w_1w_2\cdots w_{j+n-1}})}{v_1\cdots v_j}\geq\frac{2^{q_2}}{w_0w_{-1}\cdots w_{-j+1}}>M \end{align*} $$

and

$$ \begin{align*} v_jv_{j-1}\cdots v_0 v_{-1}\cdots v_{j-n+1}=v_jv_{j-1}\cdots v_2 w_0w_1 w_2\cdots w_{-j+n}\leq\frac{v_jv_{j-1}\cdots v_2}{2^{q_2}}<\frac{1}{M}. \end{align*} $$

This gives that $n\in A_{M,j}(v)\cap \overline {A}_{M,j}(v)$ and then $E_2\subset A_{M,j}(v)\cap \overline {A}_{M,j}(v)$ . Then, $A_{M,j}(v)\cap \overline {A}_{M,j}(v)\in \mathcal {F}_2$ and $B_v$ is $\mathcal {F}_2$ -transitive by Proposition 2.22.

4.2 Some examples

Applying Lemma 4.1, we can provide a large number of operators with some prescribed properties. We first give two general results and then provide certain examples.

Proof. On the one hand, let $T\colon X\to X$ be an invertible $\mathcal {F}_1$ -transitive operator that is not $\mathcal {F}_2$ -transitive. Then, there are non-empty open subsets $U,V$ of X with $N(U,V)\notin \mathcal {F}_2$ . We are done since $N(U,V)\in \mathcal {F}_1$ .

Conversely, fix $F\in \mathcal {F}_1\setminus \mathcal {F}_2$ . By Lemma 4.1, there exist an invertible $\mathcal {F}_1$ -transitive operator $T\colon X\to X$ and a non-empty open subset W of X such that

$$ \begin{align*} N(W,W)= F\cup\{0\}, \end{align*} $$

where $X=c_0(\mathbb {Z})$ or $\ell ^p(\mathbb {Z})$ , $1\leq p<\infty $ . Since X has no isolated points, there are non-empty open sets $W', W"\subset W$ with $N(W', W")\subset [1,+\infty )$ . Finally, we have $N(W', W")\subset F$ . Since $F\notin \mathcal F_2$ , we have that $N(W', W")\notin \mathcal F_2$ . Hence, T is not $\mathcal {F}_2$ -transitive.

In 2019, Bès et al constructed several operators to distinguish transitivity with respect to different families [Reference Bès, Menet, Peris and Puig10]. We provide here more invertible operators of this kind. We recall the definitions of density and Banach density of a subset of $\mathbb {N}_0$ .

It is observed that for any $A\subset \mathbb {N}_0$ , we have $\underline {\mathrm {BD}}(A)\leq \underline {\mathrm {D}}(A)\leq \overline {\mathrm {D}}(A)\leq \overline {\mathrm {BD}}(A)$ . We denote:

• • $\mathcal {F}_{\mathrm {pud}}=\{A\subset \mathbb {N}_0: \overline {\mathrm {D}}(A)>0\}$ ; $\mathcal {F}_{\mathrm {pld}}=\{A\subset \mathbb {N}_0: \underline {\mathrm {D}}(A)>0\}$ ;

• • $\mathcal {F}_{\mathrm {ud1}}=\{A\subset \mathbb {N}_0: \overline {\mathrm {D}}(A)=1\}$ ; $\mathcal {F}_{\mathrm {ld1}}=\{A\subset \mathbb {N}_0: \underline {\mathrm {D}}(A)=1\}$ ;

• • $\mathcal {F}_{\mathrm {pubd}}=\{A\subset \mathbb {N}_0: \overline {\mathrm {BD}}(A)>0\}$ ; $\mathcal {F}_{\mathrm {lbd1}}=\{A\subset \mathbb {N}_0: \underline {\mathrm {BD}}(A)=1\}$ .

It can be checked that the above six families are shift-invariant and $\mathcal {F}_{\mathrm {ld1}}, \mathcal {F}_{\mathrm {lbd1}}$ are filters. By Remark 2.4(5), both $\mathcal {F}_{\mathrm {ld1}}$ and $\mathcal {F}_{\mathrm {lbd1}}$ are shift-invariant thick families. In addition, we have $\mathcal {F}^*_{\mathrm {ld1}}=\mathcal {F}_{\mathrm {pud}}$ and $\mathcal {F}^*_{\mathrm {lbd1}}=\mathcal {F}_{\mathrm {pubd}}$ . By Proposition 4.13, once we have found a subset of $\mathbb {N}_0$ with some prescribed properties, we will obtain an operator to distinguish certain classes of hypercyclic operators.

Proof. (1) Let $F=\bigcup _{n\in \mathbb {N}} \{2^n,2^n+1,\ldots ,2^n+n\}$ . It is clear that $F\in \mathcal {F}_{\mathrm {t}}\setminus \mathcal {F}_{\mathrm {pud}}$ . Since $\mathcal {F}_{\mathrm {t}}$ is a shift-invariant thick family, the result follows from Proposition 4.13.

(2) Let $F=\mathbb {N}_0\setminus \mathrm {FS}((10^{n})_{n\in \mathbb {N}})$ . It can be verified that $F\in \mathcal {F}_{\mathrm {ts}}\setminus \mathcal {F}^*_{\mathrm {ip}}$ . Since $\mathcal {F}_{\mathrm {ts}}$ is a shift-invariant thick family, the result follows from Proposition 4.13.

(3) For each $n\in \mathbb {N}_0$ , let $F_n = \{(n 3^n)k+i\colon k\in \mathbb {N}, i\in \{0,1,\ldots ,n-1\}\}$ . Then, for each $n\in \mathbb {N}$ , $F_n$ is syndetic and $\overline {\mathrm {D}}(F_n)=\underline {\mathrm {D}}(F_n)={1}/{3^n}$ . Then, $F: =\bigcup _{n=1}^\infty F_n$ is thickly syndetic and $\overline {\mathrm {D}}(F)\leq \sum _{n=1}^\infty \overline {\mathrm {D}}(F_n)=\frac 23$ . This shows that $F\in \mathcal {F}_{\mathrm {ts}}\setminus \mathcal {F}_{\mathrm {ud1}}$ . Since $\mathcal {F}_{\mathrm {ts}}$ is a shift-invariant thick family, the result follows from Proposition 4.13.

(4) Let $F=\mathbb {N}_0\setminus \bigcup _{n\in \mathbb {N}} \{2^n,2^n+1,\ldots ,2^n+n\}$ . By the proof of (1), $F\in \mathcal {F}_{\mathrm {ld1}}\setminus \mathcal {F}_{\mathrm {s}}$ . Since $\mathcal {F}_{\mathrm {ld1}}$ is a shift-invariant thick family, the result follows from Proposition 4.13.

(5) Let $F=\mathbb {N}_0\setminus \{2^n\colon n\in \mathbb {N}\}$ . It is clear that $F\in \mathcal {F}_{\mathrm {lbd1}}\setminus \mathcal {F}_{\mathrm {cf}}$ . Since $\mathcal {F}_{\mathrm {lbd1}}$ is a shift-invariant thick family, the result follows from Proposition 4.13.

(6) Let $(s_n)_{n\in \mathbb {N}}$ be a sequence of positive integers such that $s_{n+1}>4(\sum _{i=1}^n s_i+n)$ for each $n\in \mathbb {N}$ . Put $S=\{s_n\colon n\in \mathbb {N}\}$ and $A=\mathbb {N}_0\setminus (S-S)$ . The family

$$ \begin{align*} \mathcal{F}_A=\bigg\{B\subset\mathbb{N}_0: \text{there exist }m_1,m_2,\ldots,m_k\in\mathbb{Z}\text{ with }\bigcap_{i=1}^k (A+m_i)\bigg\} \end{align*} $$

is a shift-invariant thick subfamily of $\mathcal {F}^*_{\mathrm {ip}}$ (see [Reference Huang and Ye25, Example B]). Now, we have $A\in \mathcal {F}_A\setminus \Delta ^*$ . Then, the result follows from Proposition 4.13.

(7) Let $(s_n)_{n\in \mathbb {N}}$ be the same sequence as the one in the proof of part (6). Put ${A=\mathbb {N}_0\setminus \mathrm {FS}((s_n)_{n\in \mathbb {N}})}$ . For each $k\in \mathbb {N}$ , let $A_k=\bigcap _{i=-k}^k (A+i)$ . The family

$$ \begin{align*} \mathcal{F}_A=\{B\subset\mathbb{N}_0: \text{there exists }k\in\mathbb{N}\text{ with }A_k\subset B\} \end{align*} $$

is a shift-invariant thick subfamily of $\Delta ^*_{\mathrm {ip}}$ (see [Reference Huang and Ye25, Example C]). Now, we have ${A\in \mathcal {F}_A\setminus \mathcal {F}^*_{\mathrm {ip}}}$ . Then, the result follows from Proposition 4.13.

We provide examples of weakly mixing operators without hypercyclic direct sums.

In §3, we have demonstrated that the mild mixing of operators lies strictly between strong mixing and weak mixing. By Proposition 4.16 and Corollary 4.17, we immediately obtain that the mild mixing of invertible operators lies strictly between strong mixing and weak mixing.

We provide a further example of a weakly mixing operator with a weakly mixing adjoint. By Remark 3.18, this is also an example of a weakly mixing operator $T\colon X\to X$ such that neither T nor $T^*$ is mildly mixing.