We establish complete characterizations of various notions of expansivity for weighted composition operators on a very general class of locally convex spaces of continuous functions. This class includes several classical classes of continuous function spaces, such as the Banach spaces <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$C_0(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of continuous scalar-valued functions vanishing at infinity on a Hausdorff locally compact space X, endowed with the sup norm, and the locally convex spaces <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$C(X)_c$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mi>c</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> of continuous scalar-valued functions on a completely regular space X, endowed with the compact-open topology. We also obtain complete characterizations of various notions of expansivity for weighted composition operators on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L^p(\mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> spaces, thereby complementing and extending previously known results in the unweighted case. Finally, we establish a conjugation between weighted and unweighted composition operators in the case of dissipative systems on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^p(\mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> spaces and apply it to the study of several dynamical properties.

On the Dynamics of Weighted Composition Operators II

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2026年6月16日

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Nilson C. Bernardes Jr.,
Antonio Bonilla,
João V. A. Pinto

摘要

We establish complete characterizations of various notions of expansivity for weighted composition operators on a very general class of locally convex spaces of continuous functions. This class includes several classical classes of continuous function spaces, such as the Banach spaces $C_0(X)$ $C_{0} (X)$ of continuous scalar-valued functions vanishing at infinity on a Hausdorff locally compact space X, endowed with the sup norm, and the locally convex spaces $C(X)_c$ $C {(X)}_{c}$ of continuous scalar-valued functions on a completely regular space X, endowed with the compact-open topology. We also obtain complete characterizations of various notions of expansivity for weighted composition operators on $L^p(\mu )$ $L^{p} (μ)$ spaces, thereby complementing and extending previously known results in the unweighted case. Finally, we establish a conjugation between weighted and unweighted composition operators in the case of dissipative systems on $L^p(\mu )$ $L^{p} (μ)$ spaces and apply it to the study of several dynamical properties.