Complex symmetric Toeplitz operators on the weighted Bergman space

Abstract

In this paper, we give a characterization of a complex symmetric Toeplitz operator $T_{\phi }$ on the weighted Bergman space $A_{\alpha }^2(\mathbb{D})$. We first give properties of complex symmetric Toeplitz operators $T_{\phi }$ on $A_{\alpha }^2(\mathbb{D})$. Next, we prove that if $T_{\phi }$ is complex symmetric with finite symbol, then $T_{\phi }$ is hyponormal on $A_{\alpha }^2(\mathbb{D})$ if and only if it is hyponormal on the Hardy space $H^2(\mathbb{T})$. Finally, we consider the complex symmetric Toeplitz operator $T_{\phi }$ on $A_{\alpha }^2(\mathbb{D})$ when the conjugation is a special case.

Keywords:

• Complex symmetric operator
• Toeplitz operator
• normal operator
• weighted Bergman space

AMS Subject Classifications:

• Primary 47B35
• 47B15
• Secondary 47A05

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2019R1F1A1058633). and the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2019R1A6A1A11051177). The second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2019R1A2C1002653). The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07048620).