Conjugations and Complex Symmetric Toeplitz Operators on the Weighted Hardy Space

Abstract

In this paper, we introduce a new conjugation $$C_{\xi }$$ on the weighted Hardy space $$H_{\rho }(\mathbb {D})$$ , where $$C_{\xi }$$ is given by (2.1) in Theorem 2.2. In particular, we prove that $$C_{\xi }$$ and $$C_{\mu ,\lambda }$$ are unitarily equivalent where $$C_{\mu ,\lambda }$$ is given in Ko and Lee (J Math Anal Appl 434:20–34, 2016). Using this, we investigate a complex symmetric Toeplitz operator $$T_{\varphi }$$ with respect to the conjugation $$C_{\xi }$$ on the weighted Hardy space $$H_{\rho }(\mathbb {D})$$ . Finally, we consider $$C_{\mu ,\lambda }$$ -invariant of Berezin transform.