Noncomplex symmetric operators are dense

Abstract

An operator $T\in \mathcal{B}\left(\mathcal{H}\right)$ is complex-symmetric if there exists a conjugate-linear, isometric involution $C:\mathcal{H}\to \mathcal{H}$ so that $CTC=T^{\ast }$. In this note, we prove that on finite-dimensional Hilbert space ${\mathbb{C}}^n$ with $n\ge 3$, noncomplex symmetric operators are dense in $\mathcal{B}\left({\mathbb{C}}^n\right)$.