The orthogonal Lie algebra of operators: ideals and derivations

Abstract: We study in this paper the infinite-dimensional orthogonal Lie algebra $\mathcal{O}_C$ which consists of all bounded linear operators $T$ on a separable, infinite-dimensional, complex Hilbert space $\mathcal{H}$ satisfying $CTC=-T^*$, where $C$ is a conjugation on $\mathcal{H}$. By employing results from the theory of complex symmetric operators and skew-symmetric operators, we determine the Lie ideals of $\mathcal{O}_C$ and their dual spaces. We study derivations of $\mathcal{O}_C$ and determine their spectra. These results complete some results of P. de la Harpe and provide interesting contrasts between $\mathcal{O}_C$ and the algebra $\mathcal{B(H)}$ of all bounded linear operators on $\mathcal{H}$.