<p>The differences of integration-composition operators on various analytic function spaces have attracted lots of attention for decades. In this note, we study the differences of mixed products of Volterra operators and composition operators on Bloch spaces. To be specific, we characterize the following four types of differences of mixed products: <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_470_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="96" /> </InlineMediaObject> <EquationSource Format="TEX">$I_gC_{\varphi }-C_{\psi }J_h$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>I</mi> <mi>g</mi> </msub> <msub> <mi>C</mi> <mi>φ</mi> </msub> <mo>-</mo> <msub> <mi>C</mi> <mi>ψ</mi> </msub> <msub> <mi>J</mi> <mi>h</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_470_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="96" /> </InlineMediaObject> <EquationSource Format="TEX">$J_gC_{\varphi }-C_{\psi }I_h$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>J</mi> <mi>g</mi> </msub> <msub> <mi>C</mi> <mi>φ</mi> </msub> <mo>-</mo> <msub> <mi>C</mi> <mi>ψ</mi> </msub> <msub> <mi>I</mi> <mi>h</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_470_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="94" /> </InlineMediaObject> <EquationSource Format="TEX">$I_gC_{\varphi }-C_{\psi }I_h$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>I</mi> <mi>g</mi> </msub> <msub> <mi>C</mi> <mi>φ</mi> </msub> <mo>-</mo> <msub> <mi>C</mi> <mi>ψ</mi> </msub> <msub> <mi>I</mi> <mi>h</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_470_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="98" /> </InlineMediaObject> <EquationSource Format="TEX">$J_gC_{\varphi }-C_{\psi }J_h$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>J</mi> <mi>g</mi> </msub> <msub> <mi>C</mi> <mi>φ</mi> </msub> <mo>-</mo> <msub> <mi>C</mi> <mi>ψ</mi> </msub> <msub> <mi>J</mi> <mi>h</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. One surprising result is that unbounded <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_470_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">$I_gC_{\varphi }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>I</mi> <mi>g</mi> </msub> <msub> <mi>C</mi> <mi>φ</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_470_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$C_{\psi }J_h$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>ψ</mi> </msub> <msub> <mi>J</mi> <mi>h</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> can not induce bounded difference <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_470_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="96" /> </InlineMediaObject> <EquationSource Format="TEX">$I_gC_{\varphi }-C_{\psi }J_h$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>I</mi> <mi>g</mi> </msub> <msub> <mi>C</mi> <mi>φ</mi> </msub> <mo>-</mo> <msub> <mi>C</mi> <mi>ψ</mi> </msub> <msub> <mi>J</mi> <mi>h</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>.</p>

Mixed product differences of composition operators and Volterra operators on Bloch spaces

期刊论文
发表日期：

2025年7月25日

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Xin He,
Cezhong Tong,
Zicong Yang

摘要

The differences of integration-composition operators on various analytic function spaces have attracted lots of attention for decades. In this note, we study the differences of mixed products of Volterra operators and composition operators on Bloch spaces. To be specific, we characterize the following four types of differences of mixed products: $I_gC_{\varphi }-C_{\psi }J_h$ $I_{g} C_{φ} - C_{ψ} J_{h}$ , $J_gC_{\varphi }-C_{\psi }I_h$ $J_{g} C_{φ} - C_{ψ} I_{h}$ , $I_gC_{\varphi }-C_{\psi }I_h$ $I_{g} C_{φ} - C_{ψ} I_{h}$ and $J_gC_{\varphi }-C_{\psi }J_h$ $J_{g} C_{φ} - C_{ψ} J_{h}$ . One surprising result is that unbounded $I_gC_{\varphi }$ $I_{g} C_{φ}$ and $C_{\psi }J_h$ $C_{ψ} J_{h}$ can not induce bounded difference $I_gC_{\varphi }-C_{\psi }J_h$ $I_{g} C_{φ} - C_{ψ} J_{h}$ .