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

Yury Arlinskii, Konrad Schmüdgen

TL;DR

This work advances the extension theory of unbounded operators by providing a complete description of all $C$-self-adjoint extensions of densely defined $C$-symmetric operators via a matrix-trick reduction to symmetric theory and a deficiency-space parametrization. It establishes a bijective correspondence between $C$-self-adjoint extensions and conjugations on a deficiency subspace satisfying an anticommutaion with $A^*C$, with explicit domain and action formulas. A quasi-analytic vectors criterion, based on Nussbaum’s theorem, offers a practical condition for $C$-self-adjointness. Additionally, the paper characterizes $C$-self-adjoint operators through polar decomposition, linking $U_A$, $|A|$, and $|CAC|$, and providing constructive representations $A=CJT$ with a positive $T$ and a partial conjugation $J$.

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.

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

TL;DR

This work advances the extension theory of unbounded operators by providing a complete description of all -self-adjoint extensions of densely defined -symmetric operators via a matrix-trick reduction to symmetric theory and a deficiency-space parametrization. It establishes a bijective correspondence between -self-adjoint extensions and conjugations on a deficiency subspace satisfying an anticommutaion with , with explicit domain and action formulas. A quasi-analytic vectors criterion, based on Nussbaum’s theorem, offers a practical condition for -self-adjointness. Additionally, the paper characterizes -self-adjoint operators through polar decomposition, linking , , and , and providing constructive representations with a positive and a partial conjugation .

Abstract

Let be a conjugation on a Hilbert space . A densely defined linear operator on is called -symmetric if and -self-adjoint if . Our main results describe all -self-adjoint extensions of on . Further, we prove a -self-adjointness criterion based on quasi-analytic vectors and we characterize -self-adjoint operators in terms of their polar decompositions.

Paper Structure

This paper contains 7 sections, 18 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Basic Notions and Some General Results on $C$-Self-adjoint Operators
  3. The Matrix Trick for $C$-Symmetric operators
  4. $C$-Self-adjoint extensions of densely defined $C$-symmetric operators
  5. Orthonormal bases, conjugations, skew-conjugations, von Neumann's formulas, and $C$-selfadjoint extensions
  6. A $C$-Self-adjointness criterion based on quasi-analytic vectors
  7. Polar decomposition

Key Result

Proposition 3

Suppose $A$ is a densely defined $C$-symmetric linear operator on $\mathcal{H}\,$. Let $\widetilde{A}$ be a closed extension of $A$ on $\mathcal{H}\,$. Then $\widetilde{A}$ is $C$-self-adjoint if and only if

Theorems & Definitions (43)

  • Definition 1
  • Definition 2
  • Proposition 3
  • proof
  • Lemma 4
  • proof
  • Corollary 5
  • proof
  • Corollary 6
  • Remark 7
  • ...and 33 more