# On $J$-self-adjoint extensions of the Phillips symmetric operator

## TL;DR

This paper investigates the spectral properties and symmetry characteristics of J-self-adjoint extensions of the Phillips symmetric operator, revealing that such extensions have spectra either real or complex, with stability linked to real spectra.

## Contribution

It introduces the concepts of stable and unstable C-symmetry within extension theory and characterizes the spectrum and symmetry of J-self-adjoint extensions of the Phillips operator.

## Key findings

- Extensions have spectrum either real or complex.
- Real spectrum implies stable C-symmetry and similarity to self-adjoint operators.
- No extensions with unstable C-symmetry exist.

## Abstract

$J$-self-adjoint extensions of the Phillips symmetric operator $S$ are studied. The concepts of stable and unstable $C$-symmetry are introduced in the extension theory framework. The main results are the following: if ${A}$ is a $J$-self-adjoint extension of $S$, then either $\sigma({A})=\mathbb{R}$ or $\sigma({A})=\mathbb{C}$; if ${A}$ has a real spectrum, then ${A}$ has a stable $C$-symmetry and ${A}$ is similar to a self-adjoint operator; there are no $J$-self-adjoint extensions of the Phillips operator with unstable $C$-symmetry.