Characterization of C-Symmetric Toeplitz operators for a Class of Conjugations in Hardy Spaces

TL;DR

This paper introduces new conjugations in Hardy spaces and characterizes when Toeplitz operators are complex symmetric with respect to these conjugations, extending to both scalar and vector-valued cases.

Contribution

It provides a novel class of conjugations and characterizes complex symmetric Toeplitz operators relative to these in scalar and vector Hardy spaces.

Findings

Characterization of complex symmetric Toeplitz operators in scalar Hardy spaces.

Extension of the characterization to block Toeplitz operators on vector-valued Hardy spaces.

Introduction of new conjugations in Hardy spaces for operator symmetry analysis.

Abstract

In this article, we introduce a new class of conjugations in the scalar valued Hardy space $H^2_{\mathbb{C}}(\mathbb{D})$ and provide a characterization of a complex symmetric Toeplitz operator $T_{\phi}$ with respect to these newly introduced conjugations in various cases. Moreover, we obtain a characterization of a complex symmetric block Toeplitz operator $T_{\Phi}$ on the vector valued Hardy space $\hdct$ with respect to certain conjugations introduced in \cite{CamaGP,KangKoLee,LeeKo2019}.