%0 Journal Article
%T Fock-Sobolev空间上的复对称加权复合算子<br>Complex Symmetric Weighted Composition Operators on the Fock-Sobolev Spaces
%A 喻思琦
%J Pure Mathematics
%P 76-83
%@ 2160-7605
%D 2025
%I Hans Publishing
%R 10.12677/pm.2025.154111
%X 伴随不同函数空间上的复对称加权复合算子得到广泛关注&#65292;本文致力于研究Fock-Sobolev空间上的复对称加权复合算子<math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>&#x03C6;</mi><msub>
   <mi>C</mi>
   <mi>&#x03C4;</mi>
  </msub>
  </mrow>
</math>

。通过引入复对称算子的概念&#65292;运用混合偏导相关公式、分类讨论、数学归纳、反证等方法&#65292;得到了<math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>&#x03C6;</mi><mrow><mo>(</mo>
   <mi>z</mi>
  <mo>)</mo></mrow></mrow>
</math>

恒不为零以及<math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>&#x03C6;</mi><msub>
   <mi>C</mi>
   <mi>&#x03C4;</mi>
  </msub>
  </mrow>
</math>

的核空间只含零向量&#65292;得到了<math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>&#x03C6;</mi><msub>
   <mi>C</mi>
   <mi>&#x03C4;</mi>
  </msub>
  </mrow>
</math>

的特征值都可表示为<math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>&#x03C6;</mi><mrow><mo>(</mo>
   <mi>&#x03BA;</mi>
  <mo>)</mo></mrow><msup>
   <mrow>
    <mrow><mo>(</mo>
     <mrow>
      <msup>
       <mo>&#x2202;</mo>
       <mstyle mathvariant='bold' mathsize='normal'><mn>1</mn></mstyle>
      </msup>
      <mi>&#x03C4;</mi></mrow>
    <mo>)</mo></mrow></mrow>
   <mi>n</mi>
  </msup>
  <mrow><mo>(</mo>
   <mi>&#x03BA;</mi>
  <mo>)</mo></mrow></mrow>
</math>

形式以及其点谱的具体表达式&#65292;并给出了乘法算子<math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>&#x03C6;</mi>
</math>

和复合算子<math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>&#x03C4;</mi>
</math>

关于共轭算子和再生核函数的关系式。这些发现深化了对Fock-Sobolev空间上的复对称加权复合算子的理解&#65292;也为其他函数空间上复对称加权复合算子的研究奠定了理论基础。<br>Complex symmetric weighted composition operators on different function spaces are widely concerned. In this paper, we study complex symmetric weighted composition operators on Fock-Sobolev spaces. By introducing the concept of complex symmetric operator, using the correlation formula of mixed partial derivative, classification discussion, mathematical induction, inverse proof and other methods, we get <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>&#x03C6;</mi><mrow><mo>(</mo>
   <mi>z</mi>
  <mo>)</mo></mrow></mrow>
</math>

 is always non-zero and the kernel space of <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>&#x03C6;</mi><msub>
   <mi>C</mi>
   <mi>&#x03C4;</mi>
  </msub>
  </mrow>
</math>

 only contains zero vector, get the eigenvalues of <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>&#x03C6;</mi><msub>
   <mi>C</mi>
   <mi>&#x03C4;</mi>
  </msub>
%K Fock-Sobolev空间&#65292
%K 加权复合算子&#65292
%K 复对称性<br>Fock-Sobolev Space
%K Weighted Composite Operator
%K Complex Symmetry
%U http://www.hanspub.org/journal/PaperInformation.aspx?PaperID=111193