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 $C=T^{*}\mathcal{J} T$, where $T$ is an unitary operator and $\mathcal{J} f\left(z\right)=\overline{f\left(\overline{z}\right)}$ with $f\in H^{2}$. Moreover, 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 $\mathcal{H}$.

Keywords

• hardy space
• toeplitz operator
• complex symmetric operator
• aluthge transform