# Dynamical properties of weighted translation operators on the Schwartz space \(\mathcal {S}(\mathbb {R})\)

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## Abstract

In this paper we investigate the dynamical properties of weighted translation operators acting on the Schwartz space \(\mathcal {S}(\mathbb {R})\) of rapidly decreasing functions, i.e., operators of the form \({T_w}:{\mathcal {S}(\mathbb {R})}\rightarrow {\mathcal {S}(\mathbb {R})},~f(\cdot )\mapsto w(\cdot )f(\cdot +1)\). We characterize when those operators are hypercyclic, weakly mixing, mixing and chaotic. Several examples illustrate our results and show which of those classes are different.

## Keywords

Linear dynamics Schwartz space Hypercyclicity Weighted translation operator## Mathematics Subject Classification

47B33 47A16 46E10## 1 Introduction

*w*belongs to the space of smooth functions of moderate growth, i.e., the space \(\mathcal {O}_M(\mathbb {R})\) consisting of all smooth functions \({f}:{\mathbb {R}}\rightarrow {\mathbb {C}}\) such that for every \(k\ge 0\) there is \(l\in \mathbb {N}\) with \(\left| w^{(k)}(x)\right| <l\left( 1+x^2\right) ^l\) for every \(x\in \mathbb {R}\). The space \(\mathcal {O}_M(\mathbb {R})\) is also known as the space of multipliers of \(\mathcal {S}(\mathbb {R})\) since \(w\in \mathcal {O}_M(\mathbb {R})\) if and only if for every \(f\in \mathcal {S}(\mathbb {R})\) also \(w\cdot f\in \mathcal {S}(\mathbb {R})\).

*X*be a Fréchet space and let \(T\in L(X)\) be a continuous linear operator. For every \(n\ge 1\) the operator \(T^n:X\rightarrow X\) is defined as the

*n*th iterate of

*T*, i.e,

*X*. Such an element

*x*is called a hypercyclic vector of

*T*. By the famous Birkhoff’s Transitivity Theorem, an operator

*T*acting on a separable Fréchet space

*X*is hypercyclic if and only if it is topologically transitive, i.e., for every two nonempty open sets \(U,V\subset X\) there is \(n\in \mathbb {N}\) such that \(T^{n}(U)\cap V\not =\emptyset \). The operator

*T*is called weakly mixing if the operator \(T\times T\) is topologically transitive, i.e., for every four nonempty open sets \(U_1,U_2,V_1,V_2\subset X\) there is \(n\in \mathbb {N}\) such that \(T^n(U_1)\cap V_1\not =\emptyset \text { and } T^{n}(U_2)\cap V_2\not =\emptyset .\) The operator

*T*is called mixing if for every two nonempty open sets \(U,V\subset X\) there is \(N\in \mathbb {N}\) such that \(T^n(U)\cap V\not =\emptyset \text { for every }n\ge N.\) Finally,

*T*is called chaotic if it is hypercyclic and has a dense set of periodic points (a point \(x\in X\) is called a periodic point of

*T*if \(T^kx=x\) for some \(k\in \mathbb {N}\)). From the very definitions it is clear that every mixing operator is weakly mixing and every weakly mixing operator is hypercyclic. For a detailed exposition of the subject of linear dynamics we refer to the monographs [1, 9].

The first example of a hypercyclic operator goes back to Birkhoff who proved that the translation operator \({T}:{H(\mathbb {C})}\rightarrow {H(\mathbb {C})}, f(\cdot )\mapsto f(\cdot +1)\) is hypercyclic (see [2]). In fact Birkhoff’s operator is an example of a composition operator and the dynamics of this kind of operators was later on studied by various authors (see [8] for composition operators on spaces of holomorphic functions, see [3] for composition operators on spaces of real analytic functions, see [10] for weighted composition operators on Banach spaces of continuous functions and \(L_p\) spaces, see [11] for weighted composition operators on the space of smooth functions). Surprisingly, as shown below, composition operators are never hypercyclic when acting on the space \(\mathcal {S}(\mathbb {R})\). Let us note that recently (see [6]) a description was found of those smooth functions \({\psi }:{\mathbb {R}}\rightarrow {\mathbb {R}}\) for which the composition operator \({C_\psi }:{\mathcal {S}(\mathbb {R})}\rightarrow {\mathcal {S}(\mathbb {R})}\), \(f\mapsto f\circ \psi \) is well defined.

### Fact 1

Let \({\psi }:{\mathbb {R}}\rightarrow {\mathbb {R}}\) be a smooth function such that the composition operator \({C_\psi }:{\mathcal {S}(\mathbb {R})}\rightarrow {\mathcal {S}(\mathbb {R})}\), \(f\mapsto f\circ \psi \) is well defined. Then \(C_\psi \) is not hypercyclic.

### Proof

Let \(f\in \mathcal {S}(\mathbb {R})\) be arbitrary. There exists \(M>0\) such that \(|f(x)|<M\) for every \(x\in \mathbb {R}\). If \(n\in \mathbb {N}\), then \(\left| \left( C^n_\psi f\right) (x)\right| <M\) for every \(x\in \mathbb {R}\). This clearly implies that \(C_\psi \) is not hypercyclic. \(\square \)

The second important example of a hypercyclic operator goes back to MacLane who proved that the differential operator \(D:H(\mathbb {C})\rightarrow H(\mathbb {C})\), \(f\mapsto f'\) is hypercyclic (this was later on extended even to infinite order differential operators on \(H(\mathbb {C})\), see [4]). By easy arguments from linear dynamics, differential operators are also hypercyclic on the space of smooth functions on the real line. This is not the case if we consider these operators on \(\mathcal {S}(\mathbb {R})\).

### Fact 2

*P*is not hypercyclic.

### Proof

*f*would be a hypercyclic vector for

*P*, then the set \(\{\phi (P^n f):n\in \mathbb {N}\}\) should be dense in \(\mathbb {C}\). This is impossible. \(\square \)

The above considerations show that many classical hypercyclic operators are not hypercyclic when considered on the Schwartz space. Our aim was to find a natural class of operators acting on \(\mathcal {S}(\mathbb {R})\) and to study them from the point of view of linear dynamics. Motivated by weighted bilateral shifts acting on sequence spaces, we decided to study weighted translation operators on \(\mathcal {S}(\mathbb {R})\). In Theorem 1 we give a characterization of hypercyclic weighted translation operators. It is not a surprise that this class is equal to the class of weakly mixing weighted translation operators. In Theorems 2 and 3 we characterize the class of mixing and chaotic weighted translation operators. In fact, in Theorem 4 we show that those classes are equal. In the last sections we present some tools which help to decide if a given function *w* induces an operator \(T_w\) with some dynamical properties. Using those tools we construct examples showing which of the considered classes of weighted translation operators are different.

## 2 Hypercyclic, weakly mixing, mixing and chaotic weighted translation operators

In this section we characterize the dynamical properties of weighted translation operators acting on the Schwartz space \(\mathcal {S}(\mathbb {R})\). In the first theorem we show that hypercyclicity of these operators is equivalent to weak mixing. This is not surprising since for many natural operators the situation is the same (see [11] for weighted composition operators on the space of smooth functions, see [10] for weighted composition operators on Banach spaces of continuous and integrable functions, see [12] and [7] for weighted backward shifts and weighted bilateral shifts on sequence spaces). Let us emphasize that there are hypercyclic operators which are not weakly mixing (see [5]).

### Theorem 1

- (i)
The operator \({T_w}:{\mathcal {S}(\mathbb {R})}\rightarrow {\mathcal {S}(\mathbb {R})}, f(\cdot )\mapsto w(\cdot ) f(\cdot +1)\) is weakly mixing.

- (ii)
The operator \({T_w}:{\mathcal {S}(\mathbb {R})}\rightarrow {\mathcal {S}(\mathbb {R})}, f(\cdot )\mapsto w(\cdot )f(\cdot +1)\) is hypercyclic.

- (iii)The following conditions are satisfied:
- (a)
For every \(x\in \mathbb {R}\) we have \(w(x)\not =0\).

- (b)For every compact set \(K\subset \mathbb {R}\) there exists an increasing sequence of natural numbers \((n_k)_{k\in \mathbb {N}}\) such that for every \(j,l\ge 0\)and$$\begin{aligned} \sup _{x\in K-n_k} \left( 1+x^2\right) ^j\left| \left( \prod _{n=0}^{n_k-1}w(x +n) \right) ^{(l)}\right| \xrightarrow {k\rightarrow \infty } 0 \end{aligned}$$$$\begin{aligned} \sup _{x\in K+n_k} \left( 1+x^2\right) ^j\left| \left( \frac{1}{\prod _{n=1}^{n_k}w(x -n)} \right) ^{(l)}\right| \xrightarrow {k\rightarrow \infty } 0. \end{aligned}$$

- (a)

### Proof

\((i) \Rightarrow (ii)\) This is obvious since every weakly mixing operator is hypercyclic.

\((ii) \Rightarrow (iii)\) To prove that \((a)\) holds, let us assume to the contrary that there is \(x_0 \in \mathbb {R}\) such that \(w(x_0)=0\). Then for every \(n\in \mathbb {N}\) and \(f\in \mathcal {S}(\mathbb {R})\) we have \(T_w^n(f)(x_0)=0\). This shows that \(T_w\) cannot have a dense orbit.

*f*such that

*C*is a positive constant depending on

*D*,

*N*and

*w*that will be fixed later.

*Claim 1* The set *U* is nonempty.

*Proof of the claim.*Any smooth function supported on \([-D-1,D+1]\) which is equal to the polynomial

*U*. \(\square \)

*Claim 2* The set *U* is open.

*Proof of the claim.*Let \(g\in U\). Then there exists \(\delta >0\) such that

*Claim 3* There is \(k\ge N\) such that the inequalities (1) and (2) hold for \(0\le l\le N\).

*Proof of the claim:*By Claim 1 and Claim 2 the set

*U*is nonempty and open. Since \(T_w\) is hypercyclic, we can find \(k\in \mathbb {N}\) large enough to ensure that

*l*) that there exists a positive constant

*M*such that for \(0\le l\le N\)

*C*sufficiently small we get that the inequality in (1) is true for \(0\le l\le N\).

*K*and let \((n_k)_{k\in \mathbb {N}}\) be an increasing sequence of natural numbers which existence is assumed in condition \({(b)}\). For \(i=1,2\) we define the function \(h_i\) via the formula

*k*is large enough, then the functions \(h_i\) are well defined, compactly supported and smooth. For \(i=1,2\), from the assumptions, using the product rule and the fact that the derivatives of \(g_i\) are bounded we obtain that

*k*is large enough, then for \(i=1,2\) we have that \(h_i\in U_i\) and \(T_w^{n_k}(h_i)\in V_i\). This completes the proof. \(\square \)

The next theorem characterizes the class of mixing weighted translation operators on \(\mathcal {S}(\mathbb {R})\). In Example 3 we will show that this class is strictly smaller than the class of hypercyclic operators (note that for weighted composition operators acting on the space of smooth functions on the real line those classes are equal, see [11]).

### Theorem 2

- (i)
The operator \({T_w}:{\mathcal {S}(\mathbb {R})}\rightarrow {\mathcal {S}(\mathbb {R})}, f(\cdot )\mapsto w(\cdot )f(\cdot +1)\) is mixing.

- (ii)The following conditions are satisfied:
- (a)
For every \(x\in \mathbb {R}\) we have \(w(x)\not =0\)

- (b)For every compact set \(K\subset \mathbb {R}\) and for every \(j,l\ge 0\)and$$\begin{aligned} \sup _{x\in K-k} \left( 1+x^2\right) ^j\left| \left( \prod _{n=0}^{k-1}w(x +n) \right) ^{(l)}\right| \xrightarrow {k\rightarrow \infty } 0 \end{aligned}$$$$\begin{aligned} \sup _{x\in K+k} \left( 1+x^2\right) ^j\left| \left( \frac{1}{\prod _{n=1}^{n_k}w(x -n)} \right) ^{(l)}\right| \xrightarrow {k\rightarrow \infty } 0. \end{aligned}$$

- (a)

The proof of this theorem is similar to the proof of Theorem 1. We include the proof for the convenience of the reader.

### Proof

*U*from the proof of Theorem 1. Since \(T_w\) is mixing, \(T_w^k(U)\cap U\not =\emptyset \) for every

*k*large enough. Repeating the arguments from the previous proof we obtain that for \(0\le l\le N\) and

*k*large enough

*U*and

*V*be nonempty and open subsets of \(\mathcal {S}(\mathbb {R})\). We need to show that \( T_w^k(U)\cap V\not =\emptyset \) for every

*k*large enough. Since compactly supported smooth functions are dense in \(\mathcal {S}(\mathbb {R})\), we can find compactly supported smooth functions

*f*,

*g*, constants \(\varepsilon >0\), \(N\in \mathbb {N}\) such that

*f*,

*g*are contained in a compact set

*K*and let us define the function

*h*via the formula

*k*is large enough, then the function

*h*are well defined, compactly supported and smooth. Using the product rule and the fact that the derivatives of

*g*are bounded we obtain that

*k*is large enough, then \(h\in U\) and \(T_w^{k}(h)\in V\). \(\square \)

The following result gives a description of chaotic weighted translation operators on \(\mathcal {S}(\mathbb {R})\). Surprisingly, for those operators, the existence of a dense set of periodic points already implies that they are chaotic (a similar situation holds for weighted bilateral shifts, see [12]).

### Theorem 3

- (i)
The operator \({T_w}:{\mathcal {S}(\mathbb {R})}\rightarrow {\mathcal {S}(\mathbb {R})}, f(\cdot )\mapsto w(\cdot )f(\cdot +1)\) is chaotic.

- (ii)
The operator \({T_w}:{\mathcal {S}(\mathbb {R})}\rightarrow {\mathcal {S}(\mathbb {R})}, f(\cdot )\mapsto w(\cdot )f(\cdot +1)\) has a dense set of periodic points.

- (iii)The following conditions are satisfied:
- (a)
For every \(x\in \mathbb {R}\) we have \(w(x)\not =0\).

- (b)For every compact set \(K\subset \mathbb {R}\) there exists \(d\in \mathbb {N}\) such that for every \(j,l\ge 0\)and$$\begin{aligned} \sup _{x\in K-kd} \left( 1+x^2\right) ^j\left| \left( \prod _{n=0}^{kd-1}w(x +n) \right) ^{(l)}\right| \xrightarrow {k\rightarrow \infty } 0 \end{aligned}$$$$\begin{aligned} \sup _{x\in K+kd} \left( 1+x^2\right) ^j\left| \left( \frac{1}{\prod _{n=1}^{kd}w(x -n)} \right) ^{(l)}\right| \xrightarrow {k\rightarrow \infty } 0. \end{aligned}$$

- (a)

### Proof

\((i) \Rightarrow (ii)\) This implication is obvious.

\((ii) \Rightarrow (iii)\) First we show that the condition \((a)\) holds. To do this let us assume that there exists \(x_0\in \mathbb {R}\) such that \(w(x_0)=0\). This implies that every periodic point of \(T_w\) vanishes at \(x_0\) which clearly implies that \((ii)\) cannot hold.

*p*must satisfy the following equations

*l*we will show that for every \(l\ge 0\) the following holds: for every \(j\ge 0\)

*Step 1*Let \(l=0\). Since \(p\in \mathcal {S}(\mathbb {R})\), for every \(j\ge 0\)

*Step 2*Assume that the inequalities (9) and (10) are true for \(0, 1, \ldots , l-1\). We will show that they are true for

*l*.

*p*and all its derivatives are bounded we get

\((iii) \Rightarrow (i)\) By Theorem 1 the operator \(T_w\) is hypercyclic and therefore we only need to show that \(T_w\) has a dense set of periodic points. To do this we will prove that for every compactly supported smooth function *f* there is a sequence \((p_s)_{s\in \mathbb {N}}\) of periodic points of \(T_w\) which is convergent to *f* in \(\mathcal {S}(\mathbb {R})\).

*d*be a natural number for which conditions in \((iii)\) are satisfied and such that \(K\pm d\cap K=\emptyset \). For every \(s\in \mathbb {N}\) we define a function \({p_s}:{\mathbb {R}}\rightarrow {\mathbb {C}}\) by the formula

*f*is compactly supported, there exists a constant

*C*such that

*f*in \(\mathcal {S}(\mathbb {R})\). \(\square \)

From Theorems 2 and 3 we immediately obtain the following corollary.

### Corollary 1

Let \(w\in \mathcal {O}_M(\mathbb {R})\). If \({T_w}:{\mathcal {S}(\mathbb {R})}\rightarrow {\mathcal {S}(\mathbb {R})}, f(\cdot )\mapsto w(\cdot )f(\cdot +1)\) is mixing then it is chaotic.

We will show now that even more is true.

### Theorem 4

- (i)
The operator \({T_w}:{\mathcal {S}(\mathbb {R})}\rightarrow {\mathcal {S}(\mathbb {R})}, f(\cdot )\mapsto w(\cdot )f(\cdot +1)\) is chaotic.

- (ii)
The operator \({T_w}:{\mathcal {S}(\mathbb {R})}\rightarrow {\mathcal {S}(\mathbb {R})}, f(\cdot )\mapsto w(\cdot )f(\cdot +1)\) is mixing.

### Proof

*K*. By Theorem 3 we know that \(w(x)\not =0\) for every \(x\in \mathbb {R}\) and that there exists \(d\in \mathbb {N}\) such that for every \(j,l\ge 0\)

*Claim 1*For every \(1\le s\le d-1\) and every \(j,l\ge 0\)

*Proof of the claim.*Let us fix \(1\le s\le d-1\) and \(j,l\ge 0\). By the product rule

*C*is a constant which does not depend on

*k*. This gives that

*k*is large enough and \(x\in K-kd\), then

*Claim 2*For every \(1\le s\le d-1\)

*Proof of the claim:*The proof of this claim is very similar to the proof of Claim 1. One needs only to observe that

## 3 Examples

The aim of this section is to illustrate our results and to show which of the considered classes of weighted translation operators on \(\mathcal {S}(\mathbb {R})\) are different. We start with the following proposition which gives a wide class of examples of mixing and chaotic weighted translation operators.

### Proposition 1

- (i)If there exists \(0<c<1\) and \(n_0\in \mathbb {N}\) such that \(|w(x)|<c\) for \(x\le -n_0\), then for every compact set \(K\subset \mathbb {R}\)$$\begin{aligned} \sup _{x\in K-k} \left( 1+x^2\right) ^j\left| \left( \prod _{n=0}^{k-1}w(x +n) \right) ^{(l)}\right| \xrightarrow {k\rightarrow \infty } 0 \quad \text {for every } j,l\ge 0. \end{aligned}$$(17)
- (ii)If there exists \(C>1\) and \(n_0\in \mathbb {N}\) such that \(|w(x)|>C\) for \(x\ge n_0\), then for every compact set \(K\subset \mathbb {R}\)$$\begin{aligned} \sup _{x\in K+k} \left( 1+x^2\right) ^j\left| \left( \frac{1}{\prod _{n=1}^{k}w(x -n)} \right) ^{(l)}\right| \xrightarrow {k\rightarrow \infty } 0\quad \text {for every } j,l\ge 0. \end{aligned}$$(18)
- (iii)
If there exists \(0<\delta <1\) and \(n_0\in \mathbb {N}\) such that \(|w(x)|<\delta \) for \(x\le -n_0\) and \(|w(x)|>\frac{1}{\delta }\) for \(x\ge n_0\), then the operator \({T_w}:{\mathcal {S}(\mathbb {R})}\rightarrow {\mathcal {S}(\mathbb {R})}, f(\cdot )\mapsto w(\cdot )f(\cdot +1)\) is mixing and chaotic.

### Proof

*w*is smooth, for every every \(i\ge 0 \) there exists \(C_i>0\) such that for every \(k\in \mathbb {N}\)

*k*large enough

*k*large enough we get

\((ii)\) The proof of this part is similar to the proof of the first part of the proposition. The only difference is that one needs to use the formula for higher derivatives of the function \(\frac{1}{w(x+n)}\).

\((ii)\) The last assertion in the proposition follows immediately from the first two parts, Theorems 2 and 3. \(\square \)

### Example 1

Let \(w\in \mathcal {O}_M(\mathbb {R})\) be such that \(w(x)\not =0\) for \(x\in \mathbb {R}\) and satisfies \(\lim _{x\rightarrow -\infty }|w(x)|=0\) and \(\lim _{x\rightarrow \infty }|w(x)|=\infty \). Then, by Proposition 1, the operator \({T_w}:{\mathcal {S}(\mathbb {R})}\rightarrow {\mathcal {S}(\mathbb {R})}, f(\cdot )\mapsto w(\cdot )f(\cdot +1)\) is mixing and chaotic.

### Example 2

Let \({w}:{\mathbb {R}}\rightarrow {\mathbb {R}}\) be any smooth function such that \(w(x)=\frac{1}{2}\) if \(x\le 0\), \(w(x)=2\) if \(x\ge 1\) and \(w(x)\not =0\) for \(x\in [0,1]\). It is clear that such a function belongs to \(\mathcal {O}_M(\mathbb {R})\). By Proposition 1, the operator \({T_w}:{\mathcal {S}(\mathbb {R})}\rightarrow {\mathcal {S}(\mathbb {R})}, f(\cdot )\mapsto w(\cdot )f(\cdot +1)\) is mixing and chaotic.

The following proposition is an useful tool to construct examples of weighted translation operators with various dynamical properties.

### Proposition 2

- (i)Assume that there exist \(c>0\) and \(n_0\in \mathbb {N}\) such that \(|w(x)|>c\) for \(x\le -n_0\). Let \((n_k)_{k\in \mathbb {N}}\) be an increasing sequence of natural numbers such that for every \(j\ge 0\)Then for every \(j,l\ge 0\)$$\begin{aligned} \sup _{x\in K-n_k} \left( 1+x^2\right) ^j\left| \prod _{n=0}^{n_k-1}w(x +n)\right| \xrightarrow {k\rightarrow \infty } 0. \end{aligned}$$$$\begin{aligned} \sup _{x\in K-n_k} \left( 1+x^2\right) ^j\left| \left( \prod _{n=0}^{n_k-1}w(x +n) \right) ^{(l)}\right| \xrightarrow {k\rightarrow \infty } 0. \end{aligned}$$
- (ii)Assume that there exist \(C>0\) and \(n_0\in \mathbb {N}\) such that \(|w(x)|\le C\) for \(x\ge n_0\). Let \((n_k)_{k\in \mathbb {N}}\) be an increasing sequence of natural numbers such that for every \(j\ge 0\)Then for every \(j,l\ge 0\)$$\begin{aligned} \sup _{x\in K+n_k} \left( 1+x^2\right) ^j\left| \frac{1}{\prod _{n=1}^{n_k}w(x -n)} \right| \xrightarrow {k\rightarrow \infty } 0. \end{aligned}$$$$\begin{aligned} \sup _{x\in K+n_k} \left( 1+x^2\right) ^j\left| \left( \frac{1}{\prod _{n=1}^{n_k}w(x -n)} \right) ^{(l)}\right| \xrightarrow {k\rightarrow \infty } 0. \end{aligned}$$

### Proof

*k*large enough

*k*large enough we have that

*k*goes to infinity. This proves (22).

\((ii)\) The proof of this part is similar to the proof of the first part of the theorem. The only difference is that one needs to use the formula for higher derivatives of the function \(\frac{1}{w(x+n)}\). \(\square \)

In the following example we construct a weighted translation operator on \(\mathcal {S}(\mathbb {R})\) which is hypercyclic and is not mixing (and thus not chaotic).

### Example 3

- (i)
\(w(x)=2\) for \(x\ge -2\) and for \(x \in [-2^{k+1}, -2^k-2^{k-1}-1]\) for \(k\in \mathbb {N}\);

- (ii)
\(w(x)=1\slash 2\) for \(x\in [-2^k-2^{k-1}, -2^k-1]\) for \(k\in \mathbb {N}\)

- (iii)
*w*is decreasing on \([-2^k-2^{k-1}-1, -2^k-2^{k-1}]\) for \(k\in \mathbb {N}\); - (iv)
*w*is increasing on \([-2^k-1, -2^k]\) for \(k\in \mathbb {N}\); - (v)
\(w(x)w(x-2^{k-1})=1\) for \(x\in [-2^k-1,-2^k]\) and \(k\in \mathbb {N}\).

## Notes

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