Let (T, d) be an unbounded and locally finite metric space with a distinguished element o, and let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> be a positive function on T. Denote by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L_{\mu }(T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mi>μ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> the discrete Banach space and by <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L_{\mu }^0(T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mrow> <mi>μ</mi> </mrow> <mn>0</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> the little discrete Banach space. In this paper, without imposing additional conditions on the weight function <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>, we first establish an equivalent characterization for the boundedness of weighted composition operators acting on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L_{\mu }^0(T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mrow> <mi>μ</mi> </mrow> <mn>0</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Subsequently, we provide detailed characterizations of the hypercyclic, weakly mixing, mixing, chaotic and supercyclic behavior of weighted composition operators on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L_{\mu }^0(T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mrow> <mi>μ</mi> </mrow> <mn>0</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In particular, we conclude that the mixing property is equivalent to chaos, and the hypercyclicity is equivalent to weak mixing for weighted composition operators.

Dynamics of Weighted Composition Operators on Discrete Weighted Banach spaces

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发表日期：

2026年1月26日

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Li Zhang,
Cui Chen,
Xiang Chen,
Ze Hua Zhou

摘要

Let (T, d) be an unbounded and locally finite metric space with a distinguished element o, and let $\mu $ $μ$ be a positive function on T. Denote by $L_{\mu }(T)$ $L_{μ} (T)$ the discrete Banach space and by $L_{\mu }^0(T)$ $L_{μ}^{0} (T)$ the little discrete Banach space. In this paper, without imposing additional conditions on the weight function $\mu $ $μ$ , we first establish an equivalent characterization for the boundedness of weighted composition operators acting on $L_{\mu }^0(T)$ $L_{μ}^{0} (T)$ . Subsequently, we provide detailed characterizations of the hypercyclic, weakly mixing, mixing, chaotic and supercyclic behavior of weighted composition operators on $L_{\mu }^0(T)$ $L_{μ}^{0} (T)$ . In particular, we conclude that the mixing property is equivalent to chaos, and the hypercyclicity is equivalent to weak mixing for weighted composition operators.