A Class of m-Complex Symmetric Operators on Hardy Space

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 fn=-f-nand fn=f-nrespectively 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 Tfis a 3-CSO with the conjugation C((i,j)) and 2-CSO’s.