The Jordan Algebra of Complex Symmetric Operators

Abstract

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

Keywords

Complex symmetric operator / Jordan operator algebra / Cartan factor / Jordan ideal / Automorphism / 47B99 / 46L70 / 17C65 / 46K70

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