<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {D}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">D</mi> </math></EquationSource> </InlineEquation> denote the open unit disc in the complex plane. Let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varphi (z)=z^m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>z</mi> <mi>m</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\psi (z)=z^{k_1}+z^{k_2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>z</mi> <msub> <mi>k</mi> <mn>1</mn> </msub> </msup> <mo>+</mo> <msup> <mi>z</mi> <msub> <mi>k</mi> <mn>2</mn> </msub> </msup> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$z \in \mathbb {D}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>z</mi> <mo>∈</mo> <mi mathvariant="double-struck">D</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$k_1, k_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> and <i>m</i> are positive integers. Let <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$C_{\psi ,\varphi }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mrow> <mi>ψ</mi> <mo>,</mo> <mi>φ</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, be the weighted composition operator on the weighted Hardy space <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$H^2(\beta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, induced by <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\psi .$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> In this paper, we completely determine the point spectrum, spectrum and essential spectrum of the operators <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$C^_{\psi ,\varphi }C_{\psi ,\varphi }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>C</mi> <mrow> <mi>ψ</mi> <mo>,</mo> <mi>φ</mi> </mrow> <mo>∗</mo> </msubsup> <msub> <mi>C</mi> <mrow> <mi>ψ</mi> <mo>,</mo> <mi>φ</mi> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$C_{\psi ,\varphi }C^_{\psi ,\varphi }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mrow> <mi>ψ</mi> <mo>,</mo> <mi>φ</mi> </mrow> </msub> <msubsup> <mi>C</mi> <mrow> <mi>ψ</mi> <mo>,</mo> <mi>φ</mi> </mrow> <mo>∗</mo> </msubsup> </mrow> </math></EquationSource> </InlineEquation>, self-commutators of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$C_{\psi ,\varphi }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mrow> <mi>ψ</mi> <mo>,</mo> <mi>φ</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> and anti-self-commutator of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$C_{\psi ,\varphi }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mrow> <mi>ψ</mi> <mo>,</mo> <mi>φ</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>. Additionally, we determine the eigenfunctions corresponding to these operators.</p>

Self-commutators of weighted composition operators with certain symbols on weighted Hardy space

期刊论文
发表日期：

2026年6月15日

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Bahram Khani-Robati,
Samira Mehrangiz

摘要

Let $\mathbb {D}$ $D$ denote the open unit disc in the complex plane. Let $\varphi (z)=z^m$ $φ (z) = z^{m}$ , and $\psi (z)=z^{k_1}+z^{k_2}$ $ψ (z) = z^{k_{1}} + z^{k_{2}}$ for all $z \in \mathbb {D}$ $z \in D$ , where $k_1, k_2$ $k_{1}, k_{2}$ and m are positive integers. Let $C_{\psi ,\varphi }$ $C_{ψ, φ}$ , be the weighted composition operator on the weighted Hardy space $H^2(\beta )$ $H^{2} (β)$ , induced by $\varphi $ $φ$ and $\psi .$ $ψ .$ In this paper, we completely determine the point spectrum, spectrum and essential spectrum of the operators $C^*_{\psi ,\varphi }C_{\psi ,\varphi }$ $C_{ψ, φ}^{*} C_{ψ, φ}$ , $C_{\psi ,\varphi }C^*_{\psi ,\varphi }$ $C_{ψ, φ} C_{ψ, φ}^{*}$ , self-commutators of $C_{\psi ,\varphi }$ $C_{ψ, φ}$ and anti-self-commutator of $C_{\psi ,\varphi }$ $C_{ψ, φ}$ . Additionally, we determine the eigenfunctions corresponding to these operators.