<p>We introduce the <i>Fejér-Riesz composition operator</i> <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10229_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$F_\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mi>φ</mi> </msub> </math></EquationSource> </InlineEquation> induced by a holomorphic self-map <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10229_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> of the unit disk <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10229_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{D}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">D</mi> </math></EquationSource> </InlineEquation>, and defined on the space of holomorphic functions on <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10229_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{D}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">D</mi> </math></EquationSource> </InlineEquation> by <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10229_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="137" /> </InlineMediaObject> <EquationSource Format="TEX">$F_\varphi f = \chi _{(-1,1)}f\circ \varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>F</mi> <mi>φ</mi> </msub> <mi>f</mi> <mo>=</mo> <msub> <mi>χ</mi> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mi>f</mi> <mo>∘</mo> <mi>φ</mi> </mrow> </math></EquationSource> </InlineEquation>. Denote by <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10229_Article_IEq6.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$A^p_\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>A</mi> <mi>α</mi> <mi>p</mi> </msubsup> </math></EquationSource> </InlineEquation> the standard weighted Bergman space of holomorphic functions on <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10229_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{D}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">D</mi> </math></EquationSource> </InlineEquation> and denote the Hardy space <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10229_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$H^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> by <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10229_Article_IEq9.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">$A^p_{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>A</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> <mi>p</mi> </msubsup> </math></EquationSource> </InlineEquation>. We study the operators <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10229_Article_IEq10.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="138" /> </InlineMediaObject> <EquationSource Format="TEX">$F_\varphi :\,A^p_\alpha \rightarrow L^p(m_\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>F</mi> <mi>φ</mi> </msub> <mo>:</mo> <mspace width="0.166667em" /> <msubsup> <mi>A</mi> <mi>α</mi> <mi>p</mi> </msubsup> <mo stretchy="false">→</mo> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mi>α</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10229_Article_IEq11.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$m_\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>m</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation> is the weighted Lebesgue measure <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10229_Article_IEq12.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="177" /> </InlineMediaObject> <EquationSource Format="TEX">$dm_\alpha (x) = (1-x^2)^{\alpha +1}dx$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <msub> <mi>m</mi> <mi>α</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mi>d</mi> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10229_Article_IEq13.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">$x\in (-1,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. For <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10229_Article_IEq14.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$0<p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10229_Article_IEq15.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \ge -1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>≥</mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, every such <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10229_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$F_\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mi>φ</mi> </msub> </math></EquationSource> </InlineEquation> is bounded, which in the case <InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10229_Article_IEq17.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha =-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> is related to the classical Fejér-Riesz Inequality. We provide characterizations for when <InlineEquation ID="IEq18"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10229_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$F_\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mi>φ</mi> </msub> </math></EquationSource> </InlineEquation> is compact and for when <InlineEquation ID="IEq19"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10229_Article_IEq19.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">$F_\varphi -F_\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>F</mi> <mi>φ</mi> </msub> <mo>-</mo> <msub> <mi>F</mi> <mi>ψ</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> is compact.</p>

Compact Difference of Fejér-Riesz Composition Operators

期刊论文
发表日期：

2025年5月29日

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Boo Rim Choe,
Hyungwoon Koo,
Wayne Smith

摘要

We introduce the Fejér-Riesz composition operator $F_\varphi $ $F_{φ}$ induced by a holomorphic self-map $\varphi $ $φ$ of the unit disk $\textbf{D}$ $D$ , and defined on the space of holomorphic functions on $\textbf{D}$ $D$ by $F_\varphi f = \chi _{(-1,1)}f\circ \varphi $ $F_{φ} f = χ_{(- 1, 1)} f \circ φ$ . Denote by $A^p_\alpha $ $A_{α}^{p}$ the standard weighted Bergman space of holomorphic functions on $\textbf{D}$ $D$ and denote the Hardy space $H^p$ $H^{p}$ by $A^p_{-1}$ $A_{- 1}^{p}$ . We study the operators $F_\varphi :\,A^p_\alpha \rightarrow L^p(m_\alpha )$ $F_{φ} : A_{α}^{p} \to L^{p} (m_{α})$ , where $m_\alpha $ $m_{α}$ is the weighted Lebesgue measure $dm_\alpha (x) = (1-x^2)^{\alpha +1}dx$ $d m_{α} (x) = {(1 - x^{2})}^{α + 1} d x$ , $x\in (-1,1)$ $x \in (- 1, 1)$ . For $0<p<\infty $ $0 < p < \infty$ and $\alpha \ge -1$ $α \geq - 1$ , every such $F_\varphi $ $F_{φ}$ is bounded, which in the case $\alpha =-1$ $α = - 1$ is related to the classical Fejér-Riesz Inequality. We provide characterizations for when $F_\varphi $ $F_{φ}$ is compact and for when $F_\varphi -F_\psi $ $F_{φ} - F_{ψ}$ is compact.