Some Notes on Complex Symmetric Operators

Abstract: In this paper we show that every conjugation $C$ on the Hardy-Hilbert space $H^{2}$ is of type $C=T^{*}C_{1}T$, where $T$ is an unitary operator and $C_{1}f\left(z\right)=\overline{f\left(\overline{z}\right)}$, with $f\in H^{2}$. In the sequence, we extend this result for all separable Hilbert space $\mathcal H$ and we prove some properties of complex symmetry on $\mathcal H$. Finally, we prove some relations of complex symmetry between the operators $T$ and $\left|T\right|$, where $T =U\left|T\right|$ is the polar decomposition of bounded operator $T\in\mathcal L\left(\mathcal H\right)$ on the separable Hilbert space $\mathcal H$.