Some Notes on Complex Symmetric Operators

TL;DR

This paper characterizes conjugations on the Hardy-Hilbert space as unitary conjugations of a specific form and extends these results to general separable Hilbert spaces, exploring properties and relations of complex symmetric operators.

Contribution

It provides a new representation for conjugations on Hardy-Hilbert spaces and generalizes the concept of complex symmetry to all separable Hilbert spaces, including relations with polar decomposition.

Findings

Every conjugation on H^2 is of the form T^* C_1 T with T unitary.

Extended the conjugation representation to all separable Hilbert spaces.

Established relations between complex symmetry of T and |T| in polar decomposition.

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 $\mathcal H$.