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<p>Let <i>C</i> be a conjugation on a Hilbert space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {H}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>. A densely defined linear operator <i>A</i> on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {H}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> is called <i>C</i>-symmetric if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(CAC\subseteq A^*\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mi>A</mi> <mi>C</mi> <mo>⊆</mo> <msup> <mi>A</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> and <i>C</i>-self-adjoint if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(CAC=A^*\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mi>A</mi> <mi>C</mi> <mo>=</mo> <msup> <mi>A</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>. Our main results describe all <i>C</i>-self-adjoint extensions of <i>A</i> on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathcal {H}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>. Further, we prove a <i>C</i>-self-adjointness criterion based on quasi-analytic vectors and we characterize <i>C</i>-self-adjoint operators in terms of their polar decompositions.</p>

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On C-Symmetric and C-Self-adjoint Unbounded Operators on Hilbert Space

  • 期刊论文

  • 发表日期：

    2026年3月28日

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  • Yury Arlinskii,
  • Konrad Schmüdgen

摘要

Let C be a conjugation on a Hilbert space \(\mathcal {H}\) H . A densely defined linear operator A on \(\mathcal {H}\) H is called C-symmetric if \(CAC\subseteq A^*\) C A C ⊆ A ∗ and C-self-adjoint if \(CAC=A^*\) C A C = A ∗ . Our main results describe all C-self-adjoint extensions of A on \(\mathcal {H}\) H . Further, we prove a C-self-adjointness criterion based on quasi-analytic vectors and we characterize C-self-adjoint operators in terms of their polar decompositions.