The Jordan algebra of complex symmetric operators

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 $\mathcal{B}(\mathcal{H})$ of all bounded linear operators on $\mathcal{H}$, and obtain ${\mathcal{S}}_C$-analogues of some classical results concerning $\mathcal{B}(\mathcal{H})$. 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 ${\mathcal{S}}_C$.

24 pages

References in corpus (3)

• Complex symmetric partial isometries
• Reducible and irreducible approximation of complex symmetric operators
• Involutive operator algebras