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

Yury Arlinskii; Konrad Schmüdgen

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

Functional Analysis 2025-10-10 v2

Authors: Yury Arlinskii , Konrad Schmüdgen

Abstract

Let $C$ be a conjugation on a Hilbert space $\mathcal{H}$ . A densely defined linear operator $A$ on $\mathcal{H}$ is called $C$ -symmetric if $CAC\subseteq A^*$ and $C$ -self-adjoint if $CAC=A^*$ . Our main results describe all $C$ -self-adjoint extensions of $A$ on $\mathcal{H}$ . 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.

Keywords

C-symmetric Operators C-self-adjoint Operators Hilbert Space Conjugation Polar Decomposition Quasi-analytic Vectors Self-adjoint Extensions Unbounded Operators

Cite

@article{arxiv.2507.01847,
  title  = {On $C$-Symmetric and $C$-Self-adjoint Unbounded Operators on Hilbert Space},
  author = {Yury Arlinskii and Konrad Schmüdgen},
  journal= {arXiv preprint arXiv:2507.01847},
  year   = {2025}
}

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22 pages

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

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