<p>In this paper, we study the complex symmetric weighted composition operators on the weighted Bergman space <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1682_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {A}^2_{\alpha }(\mathbb {H})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mi mathvariant="script">A</mi> </mrow> <mi>α</mi> <mn>2</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> defined over the right half-plane <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1682_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {H}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">H</mi> </math></EquationSource> </InlineEquation> with the reproducing kernels <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1682_Article_IEq3.gif" Format="GIF" Height="29" Rendition="HTML" Resolution="72" Type="Linedraw" Width="122" /> </InlineMediaObject> <EquationSource Format="TEX">\(K^{\alpha }_w(z)=\frac{2^{\alpha }(\alpha +1)}{(z+{\bar{w}})^{\alpha +2}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>K</mi> <mi>w</mi> <mi>α</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msup> <mn>2</mn> <mi>α</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>α</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, as well as on the Hilbert space <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1682_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {H}_s\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">H</mi> <mi>s</mi> </msub> </math></EquationSource> </InlineEquation> of analytic functions over the unit ball <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1682_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {B}_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> with the reproducing kernel <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1682_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="167" /> </InlineMediaObject> <EquationSource Format="TEX">\(K_w^s(z)=(1-\langle z, w \rangle )^{-s}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>K</mi> <mi>w</mi> <mi>s</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mrow> <mo stretchy="false">⟨</mo> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">⟩</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mi>s</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1682_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\( \alpha \in \mathbb {N}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1682_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">\( s\in \mathbb {N}^+ \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">N</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>. As a consequence of our investigation, the open question raised in Huang et al. (Complex Anal Oper Theory 17:119, 2023) is answered completely. Moreover, we provide a complete characterization of the complex symmetry exhibited by the generalized weighted composition-partial differentiation operators, denoted as <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1682_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">\(W_{m, \tau , \varphi }\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>W</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>τ</mi> <mo>,</mo> <mi>φ</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, defined as <Equation ID="Equ27"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1682_Article_Equ27.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="330" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} W_{m, \tau , \varphi }f(z):=\tau (z)\partial ^mf(\varphi (z))\;\; \text {for}\;\; f\in \mathcal {H}_s(\mathbb {B}_n) \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>W</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>τ</mi> <mo>,</mo> <mi>φ</mi> </mrow> </msub> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <mi>τ</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>∂</mi> <mi>m</mi> </msup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mspace width="0.277778em" /> <mspace width="0.277778em" /> <mtext>for</mtext> <mspace width="0.277778em" /> <mspace width="0.277778em" /> <mi>f</mi> <mo>∈</mo> <msub> <mi mathvariant="script">H</mi> <mi>s</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">B</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>in relation to the conjugation operator <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1682_Article_IEq10.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {J}_V\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">J</mi> <mi>V</mi> </msub> </math></EquationSource> </InlineEquation>.</p>

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Complex Symmetric Weighted Composition Operators on Weighted Bergman Spaces

  • Molla Basir Ahamed,
  • Taimur Rahman

摘要

In this paper, we study the complex symmetric weighted composition operators on the weighted Bergman space \(\mathcal {A}^2_{\alpha }(\mathbb {H})\) A α 2 ( H ) defined over the right half-plane \(\mathbb {H}\) H with the reproducing kernels \(K^{\alpha }_w(z)=\frac{2^{\alpha }(\alpha +1)}{(z+{\bar{w}})^{\alpha +2}}\) K w α ( z ) = 2 α ( α + 1 ) ( z + w ¯ ) α + 2 , as well as on the Hilbert space \(\mathcal {H}_s\) H s of analytic functions over the unit ball \(\mathbb {B}_n\) B n with the reproducing kernel \(K_w^s(z)=(1-\langle z, w \rangle )^{-s}\) K w s ( z ) = ( 1 - z , w ) - s , where \( \alpha \in \mathbb {N}\) α N and \( s\in \mathbb {N}^+ \) s N + . As a consequence of our investigation, the open question raised in Huang et al. (Complex Anal Oper Theory 17:119, 2023) is answered completely. Moreover, we provide a complete characterization of the complex symmetry exhibited by the generalized weighted composition-partial differentiation operators, denoted as \(W_{m, \tau , \varphi }\) W m , τ , φ , defined as \(\begin{aligned} W_{m, \tau , \varphi }f(z):=\tau (z)\partial ^mf(\varphi (z))\;\; \text {for}\;\; f\in \mathcal {H}_s(\mathbb {B}_n) \end{aligned}\) W m , τ , φ f ( z ) : = τ ( z ) m f ( φ ( z ) ) for f H s ( B n ) in relation to the conjugation operator \(\mathcal {J}_V\) J V .