A Class of $m$-Complex Symmetric Operators on Hardy Space

Abstract:

      In this paper, we study the necessary and sufficient condition that the Toeplitz operators with respect to the conjugations of one permutation are $2$-complex symmetric. Firstly, we introduce a class of conjugations called the conjugations of one permutations on the classical Hardy space. Secondly, Toeplitz operators are completely characterized as $2$-complex symmetric structure under this class of conjugations. The matrix representation of Toeplitz operators in the classical regular orthogonal basis on Hardy space is used to describe this class of $2$-complex symmetric Toeplitz operators. Finally, we add two preconditions $ f_n=-f_{-n}$ and $ f_n=f_{-n}$ respectively to the Toeplitz operators, and we get more simplified results. Under the second condition, we study the $3$-complex symmetry of Toeplitz operators, and we get the same result for $T_f$ is a 3-$CSO$ with the conjugation $C_{(i,j)}$ and 2-$CSO$'s.