Hardy空间上一类$m$-复对称Toeplitz算子

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

中文关键词:  $m$-复对称Toeplitz算子  Toeplitz算子  Hardy空间

英文关键词:$m$-complex symmetric operator  Toeplitz operator  Hardy space

中文摘要:

      本文主要研究Toeplitz算子相对于一对置换的共轭算子是2-复对称的充要条件. 首先在经典的Hardy空间上介绍一类被称为一对置换的共轭算子, 其次完整地刻画了在这类共轭算子下Toeplitz算子是2-复对称的结构, 利用Toeplitz算子在Hardy空间的经典正规正交基下的矩阵表示来刻画2-复对称Toeplitz算子. 最后对于Toeplitz算子分别补充前提$f_n=-f_{-n}$和$f_n=f_{-n}$, 得到了更简化的结果. 在第二个前提下, 研究Toeplitz算子的3-复对称性, 得到$T_f$关于$C_{(i,j)}$是3-CSO的结果和是2-CSO相同.

英文摘要:

      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.