<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation> be the Lipschitz space of a tree and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$C_{\varphi }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>φ</mi> </msub> </math></EquationSource> </InlineEquation> be the composition operator on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>. In this paper, we first investigate compact operators of the form <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$Q:=C_{\varphi _1}-\sum \nolimits _{i=2}^{m} C_{\varphi _i}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Q</mi> <mo>:</mo> <mo>=</mo> <msub> <mi>C</mi> <msub> <mi>φ</mi> <mn>1</mn> </msub> </msub> <mo>-</mo> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>m</mi> </msubsup> <msub> <mi>C</mi> <msub> <mi>φ</mi> <mi>i</mi> </msub> </msub> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>. Particularly, compact differences of composition operators are characterized. As an application, we also show that the compact difference cancellation is possible for linear combinations of two composition operators by constructing an explicit example of a compact difference formed by two distinct noncompact composition operators on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>. Subsequently, we completely characterize the compactness of a linear combination of three composition operators on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>. As one consequence of this characterization, we show that there is no cancellation property of compact double differences induced by three distinct composition operators on <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>.</p>

Cancellation Properties of Composition Operators on the Lipschitz Space of a Tree

期刊论文
发表日期：

2026年5月27日

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Lei Liu,
Junkai Kong

摘要

Let $\mathcal {L}$ $L$ be the Lipschitz space of a tree and $C_{\varphi }$ $C_{φ}$ be the composition operator on $\mathcal {L}$ $L$ . In this paper, we first investigate compact operators of the form $Q:=C_{\varphi _1}-\sum \nolimits _{i=2}^{m} C_{\varphi _i}$ $Q : = C_{φ_{1}} - \sum_{i = 2}^{m} C_{φ_{i}}$ on $\mathcal {L}$ $L$ . Particularly, compact differences of composition operators are characterized. As an application, we also show that the compact difference cancellation is possible for linear combinations of two composition operators by constructing an explicit example of a compact difference formed by two distinct noncompact composition operators on $\mathcal {L}$ $L$ . Subsequently, we completely characterize the compactness of a linear combination of three composition operators on $\mathcal {L}$ $L$ . As one consequence of this characterization, we show that there is no cancellation property of compact double differences induced by three distinct composition operators on $\mathcal {L}$ $L$ .