Keywords: Weighted composition operators; Continuous function spaces; \(L^p(\mu)\) spaces; Expansivity; Dissipative systems; Chaos.

2020 Mathematics Subject Classification: Primary 47A16, 47B33, 37B05; Secondary 46E10, 46E15, 46E30.

1 Introduction↩︎

This work is dedicated to the study of the dynamics of weighted composition operators \[C_{w,f}(\varphi) = w \cdot (\varphi \circ f)\] on certain general classes of function spaces and can be considered as a continuation of our recent article [1]. Due to the important role played by weighted composition operators in operator theory and its applications, the dynamics of these operators has been extensively investigated by various authors in different contexts over the last few decades. We refer the reader to the recent articles [1]–[6], where many additional references can be found.

Our goal is to investigate various notions of expansivity for weighted composition operators on a very general class of continuous function spaces and on \(L^p(\mu)\) spaces, as well as several dynamical properties of weighted composition operators on \(L^p(\mu)\) spaces in the dissipative case.

Given a metric space \(M\) with metric \(d\), recall that a homeomorphism \(h : M \to M\) is said to be expansive if it admits an expansivity constant \(c > 0\), which means that for any pair \(x,y\) of distinct points in \(M\), there exists \(n \in \mathbb{Z}\) such that \[d(h^n(x),h^n(y)) > c.\] This concept was introduced by Utz [7] and is one of the fundamental concepts in the modern theory of dynamical systems. In the setting of linear dynamics, Eisenberg [8] proved that an invertible operator on \(\mathbb{C}^n\) is expansive if and only if it has no eigenvalue on the unit circle. In the early 1970s, Eisenberg and Hedlund [9] and Hedlund [10] studied expansivity for operators on Banach spaces. They introduced the notion of uniform expansivity and proved that an invertible operator on a Banach space is uniformly expansive if and only if its approximate point spectrum does not intersect the unit circle. In 2000, Mazur [11] proved that an invertible normal operator \(T\) on a Hilbert space is expansive if and only if \(T^*T\) has no eigenvalue on the unit circle. This subject was revisited by Bernardes et al. [12] in 2018, where many results were obtained, including complete characterizations of the various notions of expansivity for weighted shifts on classical Banach sequence spaces. Since then, several works on expansivity for operators have been developed and interesting results have been obtained. For instance, it was proved that the equalities \[\begin{align} \text{hyperbolicity } &= \text{ expansivity } + \text{ shadowing}\\ &= \text{ average expansivity } + \text{ average shadowing} \end{align}\] hold for invertible operators on any Banach space [13]–[15]. The notions of expansivity have been extended to operators on arbitrary Hausdorff locally convex spaces in the recent papers [16], [17], where investigations on these concepts in this much more general setting have been initiated. Additional references on expansivity in linear dynamics include [2], [18]–[21].

Let us now describe the organization of the article.

In Section 2, we will present the necessary prerequisites for the development of this work.

In Section 3, we will establish complete characterizations of various notions of expansivity for weighted composition operators on a very general class of locally convex spaces of continuous functions, namely, the spaces \(C_\mathfrak{S}(X;Y)\) of all continuous functions \(\varphi\) from a completely regular space \(X\) into a Hausdorff locally convex space \(Y\) such that \(\varphi(B)\) is a bounded subset of \(Y\) for each \(B \in \mathfrak{S}\) (\(\mathfrak{S}\) being a given bornology on \(X\)), endowed with the topology of \(\mathfrak{S}\)-convergence. In fact, our characterizations are also valid for certain subspaces of \(C_\mathfrak{S}(X;Y)\) that we call admissible. Applications to several classical classes of continuous function spaces, including the Banach spaces \(C_0(X)\) and the locally convex spaces \(C(X)_c\), will be presented. As specific examples, we will consider weighted shifts and weighted translation operators.

In Section 4, we will establish complete characterizations of various notions of expansivity for weighted composition operators on \(L^p(\mu)\) spaces, thereby complementing and extending previous characterizations due to Maiuriello [21] in the unweighted case. We will illustrate the general results by presenting applications to weighted shifts and weighted translation operators.

In Section 5, we will study the dynamics of weighted composition operators on \(L^p(\mu)\) spaces in the dissipative case. In the unweighted case, dissipative composition operators have been studied in the recent papers [1], [6], [19], [21]–[26]. These operators behave very much like weighted shifts. In fact, it was shown in [19] that any dissipative composition operator satisfying a certain technical condition, called bounded distortion, has a certain specific weighted shift as a factor and for many dynamical properties, the composition operator has the property if and only if so does the associated weighted shift. For this reason, these composition operators were called shift-like in [19]. It is well-known that weighted shifts on \(\ell^p\) spaces are conjugate to unweighted shifts on weighted \(\ell^p\) spaces [27]. In a certain sense, one can transfer the weights from the shift to the space. Our main goal in Section 5 is to prove that something similar holds for composition operators on \(L^p(\mu)\) spaces in the dissipative case. More precisely, we will prove that any dissipative weighted composition operator \(C_{w,f}\) on a space \(L^p(\mu)\) is conjugate to an unweighted composition operator \(C_f\) on a space \(L^p(\nu)\) by means of an isometric isomorphism, where the measure \(\nu\) is given by a specific formula involving the measure \(\mu\) and the weight function \(w\). This allows us to transfer known results about dissipative composition operators to the weighted case, and we will illustrate this process with several dynamical properties, including frequent hypercyclicity, Devaney chaos and the shadowing property.

2 Preliminaries↩︎

Throughout the article, \(\mathbb{K}\) denotes either the field \(\mathbb{R}\) of real numbers or the field \(\mathbb{C}\) of complex numbers, \(\mathbb{Z}\) denotes the ring of integers, \(\mathbb{N}\) denotes the set of all positive integers, and \(\mathbb{N}_0 = \mathbb{N}\cup \{0\}\).

Given a seminorm \(\|\cdot\|\) on a vector space \(Y\), the unit sphere of \(\|\cdot\|\) is defined by \[S_{\|\cdot\|} = \{y \in Y : \|y\| = 1\}.\] If \(Y\) is a normed space with norm \(\|\cdot\|\), then we also write \(S_Y\) instead of \(S_{\|\cdot\|}\).

Let \(Y\) be a Hausdorff locally convex space over \(\mathbb{K}\). By an operator on \(Y\), we mean a continuous linear map \(T: Y \to Y\). Such an operator is said to be invertible if it has a continuous inverse. We denote by \(L(Y)\) (resp.\(GL(Y)\)) the set of all operators (resp.invertible operators) on \(Y\). Let \((\|\cdot\|_\alpha)_{\alpha \in I}\) be a family of seminorms that induces the topology of \(Y\). Recall that \(T \in GL(Y)\) is said to be topologically expansive if, for each \(y \in Y \backslash \{0\}\), there exists \(\alpha \in I\) such that \[\sup_{n \in \mathbb{Z}} \|T^n y\|_\alpha = \infty.\] Moreover, \(T\) is said to be uniformly topologically expansive if, for each \(\alpha \in I\), there exists \(\beta \in I\) such that we can write \(S_{\|\cdot\|_\alpha} = S^+_\alpha \cup S^-_\alpha\), where \[\|T^n y\|_\beta \to \infty \text{ uniformly on } S^+_\alpha \text{ as } n \to \infty,\] \[\|T^{-n} y\|_\beta \to \infty \text{ uniformly on } S^-_\alpha \text{ as } n \to \infty.\]

These concepts of expansivity were introduced in [16] as generalizations of the classical notions of expansivity and uniform expansivity for invertible operators on Banach spaces [9]. The word “topologically” was used to distinguish the above notion of expansivity from the usual one in the metric space setting, because they may differ in the case of a metrizable locally convex space (or even a Fréchet space) endowed with a compatible invariant metric [16], although they always coincide in normed spaces. However, as in [17], we will omit the word “topologically” and simply write “expansive” and “uniformly expansive” for the above notions.

A concept of average expansivity for invertible operators on Banach spaces was introduced and investigated in [13]. It was extended to the context of Hausdorff locally convex spaces in the recent paper [17], by means of the following definition: \(T \in GL(Y)\) is said to be average expansive if, for each \(y \in Y \backslash \{0\}\), there exists \(\alpha \in I\) such that \[\sup_{n \in \mathbb{N}} \Big(\frac{1}{2n+1} \sum_{j=-n}^n \|T^j y\|_\alpha \Big) = \infty.\]

It was observed in [17] that \[\text{uniform expansivity} \;\;\Longrightarrow \;\;\text{average expansivity} \;\;\Longrightarrow \;\;\text{expansivity}.\] However, the converses are false even in the Hilbert space setting, as it was shown in [13] and [13].

For operators that are not necessarily invertible, we have the following variations of the above notions of expansivity [16], [17]: \(T \in L(Y)\) is said to be

• positively expansive if, for each \(y \in Y \backslash \{0\}\), there exists \(\alpha \in I\) such that \[\sup_{n \in \mathbb{N}} \|T^n x\|_\alpha = \infty.\]

• average positively expansive if, for each \(y \in Y \backslash \{0\}\), there exists \(\alpha \in I\) such that \[\sup_{n \in \mathbb{N}} \Big(\frac{1}{n} \sum_{j=0}^{n-1} \|T^j y\|_\alpha \Big) = \infty.\]

• uniformly positively expansive if, for each \(\alpha \in I\), there exists \(\beta \in I\) such that \[\|T^n x\|_\beta \to \infty \text{ uniformly on } S_{\|\cdot\|_\alpha} \text{ as } n \to \infty.\]

All the above notions of expansivity do not depend on the choice of the family of seminorms inducing the topology of \(Y\), so they are completely determined by the topology of \(Y\).

3 Expansive weighted composition operators on locally convex spaces of continuous functions↩︎

Throughout this section, we fix a completely regular space \(X\) and a bornology \(\mathfrak{S}\) on \(X\), that is, a collection of subsets of \(X\) that contains the singletons and is stable under the passage to subsets and the formation of finite unions. We also fix a Hausdorff locally convex space \(Y\) over \(\mathbb{K}\), \(Y \neq \{0\}\), and a family \((\|\cdot\|_\alpha)_{\alpha \in I}\) of seminorms that induces its topology. \(C(X;Y)\) denotes the vector space over \(\mathbb{K}\) of all continuous functions from \(X\) into \(Y\), and \(C_\mathfrak{S}(X;Y)\) is the vector subspace of \(C(X;Y)\) consisting of all \(\varphi \in C(X;Y)\) such that \(\varphi(B)\) is a bounded subset of \(Y\) for each \(B \in \mathfrak{S}\). We consider \(C_\mathfrak{S}(X;Y)\) endowed with the topology of \(\mathfrak{S}\)-convergence (i.e., the topology of uniform convergence on the sets of \(\mathfrak{S}\)), which is the Hausdorff locally convex topology induced by the family of seminorms given by \[\|\varphi\|_{B,\alpha} = \sup_{x \in B} \|\varphi(x)\|_\alpha \;\;\;(\varphi \in C_\mathfrak{S}(X;Y) \text{ and } (B,\alpha) \in \mathfrak{S}\times I).\] Note that if \(\mathcal{B}\) is a base of the bornology \(\mathfrak{S}\) (i.e., \(\mathcal{B}\subset \mathfrak{S}\) and each element of \(\mathfrak{S}\) is contained in an element of \(\mathcal{B}\)), then the subfamily \((\|\cdot\|_{B,\alpha})_{(B,\alpha) \in \mathcal{B}\times I}\) also induces the topology of \(C_\mathfrak{S}(X;Y)\). For each collection \(\mathscr{C}\) of subsets of \(X\) and each \(B \subset X\), we define \[\mathscr{C}_B = \{A \in \mathscr{C} : A \cap B \neq \varnothing\}.\] For each \(B \subset X\) and each bounded function \(\phi : B \to \mathbb{K}\), we define \[\|\phi\|_B = \sup_{x \in B} |\phi(x)|,\] where we consider this supremum to be \(0\) if \(B = \varnothing\). We also fix a homeomorphism \(f : X \to X\) satisfying \(f(\mathfrak{S}) = \mathfrak{S}\) and a zero-free continuous function \(w : X \to \mathbb{K}\) such that both \(w\) and \(1/w\) are bounded on each set \(B \in \mathfrak{S}\). It is easy to check that the weighted composition operator \[C_{w,f}(\varphi) = w \cdot (\varphi \circ f)\] is a well-defined invertible operator on \(C_\mathfrak{S}(X;Y)\) with \[(C_{w,f})^{-1} = C_{\widetilde{w},f^{-1}}, \;\;\text{ where } \widetilde{w} = \frac{1}{w \circ f^{-1}}\cdot\] We associate to \(w\) and \(f\) the following bilateral sequence of continuous functions from \(X\) into \(\mathbb{K}\): \(w^{(0)} = 1\), \(w^{(1)} = w\), \(w^{(-1)} = \widetilde{w}\) and, for each \(n \geq 2\), \[w^{(n)} = w \cdot (w \circ f) \cdots (w \circ f^{n-1}) \;\;\text{ and } \;\; w^{(-n)} = \widetilde{w} \cdot (\widetilde{w} \circ f^{-1}) \cdots (\widetilde{w} \circ f^{-(n-1)}).\] Finally, we say that a subspace \(M\) of \(C_\mathfrak{S}(X;Y)\) is admissible if the following conditions hold:

• \(C_{w,f}(M) = M\);

• For each point \(a \in X\), each neighborhood \(O\) of \(a\) in \(X\) and each vector \(y \in Y\), there exists a continuous function \(\phi : X \to [0,1]\) such that \[\phi(a) = 1, \;\;\;\phi(X \backslash O) = \{0\} \;\;\text{ and } \;\;\phi \otimes y \in M,\] where \(\phi \otimes y : X \to Y\) is given by \((\phi \otimes y)(x) = \phi(x)y\).

We now establish the main result of this section, namely, characterizations of the various notions of expansivity for invertible weighted composition operators on admissible subspaces of \(C_\mathfrak{S}(X;Y)\). Note that, since the topological space \(X\) is completely regular, it is clear that \(C_\mathfrak{S}(X;Y)\) is an admissible subspace of itself.

In the next result, we will see that certain additional hypotheses imply that it is sufficient to test the conditions in (a) and (b) of Theorem 1 for a smaller collection of sets \(O\) in \(\mathcal{O}\).

4 Expansive weighted composition operators on \(L^p(\mu)\)↩︎

Throughout this section, we fix a real number \(p \in [1,\infty)\) and an arbitrary positive measure space \((X,\mathfrak{M},\mu)\), unless otherwise specified. We denote by \(\mathfrak{F}\) the set of all measurable sets \(B \in \mathfrak{M}\) satisfying \(0 < \mu(B) < \infty\). \(L^p(\mu)\) denotes the Banach space over \(\mathbb{K}\) of all \(\mathbb{K}\)-valued \(p\)-integrable functions on \((X,\mathfrak{M},\mu)\) endowed with the \(p\)-norm \[\|\varphi\|_p = \Big(\int_X |\varphi|^p d\mu\Big)^{\frac{1}{p}}.\] We fix a weight function \(w : X \to \mathbb{K}\), that is, a measurable map satisfying \[\varphi \cdot w \in L^p(\mu) \;\text{ for all } \varphi \in L^p(\mu).\] We also fix a bimeasurable map \(f : X \to X\) (i.e., \(f(B) \in \mathfrak{M}\) and \(f^{-1}(B) \in \mathfrak{M}\) whenever \(B \in \mathfrak{M}\)) for which the weighted composition operator \[C_{w,f}(\varphi) = w \cdot (\varphi \circ f)\] is a well-defined bounded linear operator on \(L^p(\mu)\). It is not difficult to show that this is the case if and only if there is a constant \(c > 0\) such that \[\int_B |w|^p d\mu \leq c\, \mu(f(B)) \;\text{ for every } B \in \mathfrak{M}.\]

Since we are primarily interested in invertible operators, we will also assume that \(f\) is bijective and that \(\widetilde{w} = \frac{1}{w \circ f^{-1}}\) is a well-defined measurable map on \(X\) satisfying \[\varphi \cdot \widetilde{w} \in L^p(\mu) \;\text{ for all } \varphi \in L^p(\mu)\] and \[\int_B |\widetilde{w}|^p d\mu \leq \tilde{c}\, \mu(f^{-1}(B)) \;\text{ for every } B \in \mathfrak{M},\] where \(\tilde{c} > 0\) is a constant. In this way, \(C_{w,f}\) is an invertible operator on \(L^p(\mu)\) with \[(C_{w,f})^{-1} = C_{\widetilde{w},f^{-1}}.\]

We associate to \(p\), \(w\) and \(f\) the following bilateral sequence of positive measures on \((X,\mathfrak{M})\): \[\mu_n(B) = \int_B |w^{(n)}|^p d\mu \;\;\;\;\;(B \in \mathfrak{M}, n \in \mathbb{Z}),\] where \(w^{(0)} = 1\), \(w^{(1)} = w\), \(w^{(-1)} = \widetilde{w}\), \[w^{(n)} = (w \circ f^{n-1}) \cdots (w \circ f) \cdot w \;\text{ and } \; w^{(-n)} = (\widetilde{w} \circ f^{-(n-1)}) \cdots (\widetilde{w} \circ f^{-1}) \cdot \widetilde{w} \;\;\;(n \geq 2).\]

We now establish the main result of this section, namely, characterizations of the various notions of expansivity for invertible weighted composition operators on \(L^p(\mu)\).

We mention that the proof of item (c) presented below is an adaptation of the original proof given by Maiuriello [21] in the unweighted case.

5 Dissipative weighted composition operators on \(L^p(\mu)\)↩︎

Throughout this section, we fix a real number \(p \in [1,\infty)\).

Note that condition (c) implies that the weighted composition operator \[C_{w,f}(\varphi) = w \cdot (\varphi \circ f)\] is a well-defined bounded linear operator on \(L^p(\mu)\).

In the particular case in which \(w\) is the constant function \(1\), we have the known concept of a measurable system \((X,\mathfrak{M},\mu,f)\). Recall that such a system (or the map \(f\)) is said to be:

• conservative if for each measurable set \(B\) of positive \(\mu\)-measure, there exists \(n \in \mathbb{N}\) such that \(\mu(B \cap f^{-n}(B)) > 0\);

• dissipative if there is a measurable set \(W\) (called a wandering set) such that the sets \(f^n(W)\), \(n \in \mathbb{Z}\), are pairwise disjoint and \(X = \bigcup_{n \in \mathbb{Z}} f^n(W)\).

Recall also that a set \(A \subset X\) is said to be \(f\)-invariant if \(f^{-1}(A) = A\). The following classic result can be found in [28]:

Hopf Decomposition Theorem. If \((X,\mathfrak{M},\mu)\) is a \(\sigma\)-finite measure space and \(f : X \to X\) is a non-singular measurable map, then \(X\) can be written as the union of two disjoint \(f\)-invariant sets \(\mathcal{C}(f)\) and \(\mathcal{D}(f)\), called the conservative* and the dissipative parts of \(f\), respectively, such that \(f|_{\mathcal{C}(f)}\) is conservative and \(f|_{\mathcal{D}(f)}\) is dissipative.*

As mentioned in the Introduction, the dynamics of composition operators \(C_f\) on \(L^p(\mu)\) associated to dissipative measurable systems has been investigated in a series of recent papers.

In this case, \(C_{w,f}\) is an invertible operator on \(L^p(\mu)\) with \[(C_{w,f})^{-1} = C_{\widetilde{w},f^{-1}}.\]

For the remainder of this section, we fix a dissipative invertible weighted \(p\)-measurable system \((X,\mathfrak{M},\mu,f,w)\) with wandering set \(W\) satisfying \(0 < \mu(W) < \infty\), unless otherwise specified. Moreover, the functions \(w^{(n)}\) and the measures \(\mu_n\) are defined exactly as in the previous section.

The previous theorem provides us with a method for extending results about dissipative unweighted composition operators to the weighted case. We will illustrate this method with the results below. First, let us recall some definitions. The lower density of a set \(A \subset \mathbb{N}\) is defined by \[\operatorname{\underline{dens}}(A) = \liminf_{n \to \infty} \frac{\operatorname{card}(A \cap [1,n])}{n}\cdot\] Let \(T\) be an operator on a Banach space \(Y\). For any subsets \(U\) and \(V\) of \(Y\), the return set from \(U\) to \(V\) is given by \[N(U,V) = \{n \in \mathbb{N}_0 : T^n(U) \cap V \neq \varnothing\}.\] The operator \(T\) is said to be:

• topologically transitive (resp.) if \(N(U,V)\) is nonempty (resp.co-finite) whenever \(U\) and \(V\) are nonempty open sets in \(Y\).

• Devaney chaotic if it is topologically transitive and has a dense set of periodic points.

• hypercyclic (resp.) if there is a vector \(y \in Y\) such that \(N(\{y\},V)\) is nonempty (resp.has positive lower density) for every nonempty open set \(V\) in \(Y\).

• recurrent (resp.) if there is dense set of vectors \(y \in Y\) with the property that \(N(\{y\},V)\) is nonempty (resp.has positive lower density) for every neighborhood \(V\) of \(y\) in \(Y\).

• Li-Yorke chaotic if there is an uncountable set \(S \subset Y\) such that each pair \((y,z)\) of distinct points in \(S\) satisfies \[\liminf_{n \to \infty} \|T^n y - T^n z\| = 0 \;\;\text{ and } \;\;\limsup_{n \to \infty} \|T^n y - T^n z\| > 0.\]

Moreover, we say that an invertible operator \(T\) on \(Y\) has the shadowing property if, for every \(\varepsilon> 0\), there exists \(\delta > 0\) such that for every bilateral sequence \((y_j)_{j \in \mathbb{Z}}\) in \(Y\) satisfying \[\|T y_j - y_{j+1}\| \leq \delta \;\text{ for all } j \in \mathbb{Z},\] there exists \(y \in Y\) such that \[\|y_j - T^j y\| < \varepsilon\;\text{ for all } j \in \mathbb{Z}.\]

Recall that \((X,\mathfrak{M},\mu,f,w)\) is a dissipative invertible weighted \(p\)-measurable system with wandering set \(W\) satisfying \(0 < \mu(W) < \infty\). Moreover, the measure \(\nu\) is as in Theorem 5.

Let us now extend the concept of bounded distortion [22], [25] to our context.

Given Banach spaces \(Y\) and \(Z\), and given operators \(T : Y \to Y\) and \(S : Z \to Z\), recall that \(S\) is said to be a linear factor of \(T\) if there is a linear factor map (i.e., a bounded linear surjection) \(\Phi : Y \to Z\) for which the following diagram commutes:

\[\xymatrix{ Y \ar[rrr]^T \ar[d]_\Phi & & & Y \ar[d]^{\Phi}\\ Z \ar[rrr]^{S} & & & Z }\]

Declarations↩︎

Conflict of Interest The authors declare that they have no conflict of interest.

Data Availability Not applicable.

Acknowledgements↩︎

The first author was partially supported by CNPq – Project #308238/2021-4, and by CAPES – Finance Code 001. The first and the second authors were partially supported by Project PID2022-139449NB-I00, funded by MCIN/AEI/10.13039/501100011033/FEDER, UE.

Nilson C. Bernardes Jr.

Departamento de Matemática Aplicada, Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, RJ 21941-909, Brazil.
e-mail address: bernardes@im.ufrj.br

Departamento de Análisis Matemático and Instituto de Matemáticas y Aplicaciones (IMAULL), Universidad de La Laguna, C/Astrofı́sico Francisco Sánchez, s/n, 38721 La Laguna, Tenerife, Spain
email address: abonilla@ull.edu.es

João V. A. Pinto

Departamento de Matemática Aplicada, Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, RJ 21941-909, Brazil.
e-mail address: joao.pinto@ufu.br

References↩︎