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 $C=T^{\ast }{C}_{1}T$, 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 $\mathcal{H}$.

arXiv comment8 pages