The Jordan algebra of complex symmetric operators

TL;DR

This paper explores the structure and properties of the Jordan algebra of complex symmetric operators, including automorphisms, ideals, spectra, and invertibility, extending classical operator algebra results to this specific setting.

Contribution

It provides a detailed analysis of the Jordan algebra of complex symmetric operators, including classification of automorphisms, determination of ideals, and spectral properties, which are new extensions of classical operator algebra theory.

Findings

Jordan automorphisms are classified.

Spectra of Jordan multiplication operators are characterized.

Jordan invertible operators form a dense, path-connected subset.

Abstract

For a conjugation $C$ on a separable, complex Hilbert space $\mathcal{H}$, the set $\mathcal{S}_C$ of $C$-symmetric operators on $\mathcal{H}$ forms a weakly closed, selfadjoint, Jordan operator algebra. In this paper we study $\mathcal{S}_C$ in comparison with the algebra $\mathcal{B(H)}$ of all bounded linear operators on $\mathcal{H}$, and obtain $\mathcal{S}_C$-analogues of some classical results concerning $\mathcal{B(H)}$. We determine the Jordan ideals of $\mathcal{S}_C$ and their dual spaces. Jordan automorphisms of $\mathcal{S}_C$ are classified. We determine the spectra of Jordan multiplication operators on $\mathcal{S}_C$ and their different parts. It is proved that those Jordan invertible ones constitute a dense, path connected subset of $\mathcal{S}_C$.