(m, C)-Isometric Toeplitz operators with rational symbols

ABSTRACT

An operator $T\in \mathcal{L}(\mathcal{H})$ is said to be (m, C)-isometric if there exists a conjugation C such that $\sum _{j=0}^m(-1{)}^{m-j}\left(\begin{array}{c}m\\ j\end{array}\right)T^{\ast j}C{T}^{j}C=0$for some positive integer m. In this paper, we study (m, C)-isometric Toeplitz operators T_φ with rational symbols φ. We characterize (m, C)-isometric Toeplitz operators T_φ by properties of the rational symbols φ. Moreover, we provide a concrete description of the $(m,\mathcal{C})$-isometric block Toeplitz operators.

Keywords:

• (m, C)-isometric operators
• Toeplitz operators
• block Toeplitz operators

2010 Mathematics Subject Classifications:

• 47A05
• 47A35
• 47A62
• 47B20

Data availability

No data were used to support this study.

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) [grant number 2019R1F1A1058633] and and Basic Science Research program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education [grant number 2019R1A6A1A11051177]. The second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) [grant number 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 [grant number 2018R1D1A1B07048620].