The orthogonal Lie algebra of operators: ideals and derivations

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 $CTC=-T^{\ast }$, 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 $\mathcal{B}(\mathcal{H})$ of all bounded linear operators on $\mathcal{H}$.