一类H-Toeplitz算子的复对称性

Complex symmetry for a class of H-Toeplitz operator

为研究Bergman空间上一类H-Toeplitz算子关于给定的某类复共轭的复对称性,提出选用特殊符号的算子来研究。由于H-Toeplitz算子与Toeplitz算子及Hankel算子之间存在紧密联系,因此,首先,借鉴经典Hardy空间上Toeplitz算子中已有的关于某些复共轭的复对称结果,找出具体的复共轭;其次,由于完全刻画一些具体算子的复对称性极其困难,故通过考察调和函数符号或非调和函数符号的H-Toeplitz算子,来研究该算子关于给定的复共轭的复对称性;最后,根据算子复共轭定义中的等距关系,得到一个有关算子符号的等式,并对此等式进行计算以找出规律。结果表明,当符号为调和函数符号或由拟齐次函数的和组成的非调和函数符号时,对应的H-Toeplitz算子关于给定的复共轭为复对称当且仅当该符号为零。

In order to study the complex symmetry for a class of H-Toeplitz operator on Bergman space with regard to a given class of conjugation,a special coincidence operator was proposed.Given the fact that H-Toeplitz operator is closely related to Toeplitz operator and Hankel operator,firstly,the specific conjugation was found according to the existing complex symmetry results of Toeplitz operator on classical Hardy space.Secondly,given the great difficulty to completely characterize the complex symmetry of some specific operators,the H-Toeplitz operator with harmonic or non-harmonic symbols was investigated to study the complex symmetry of the operator with respect to the specific conjugation.Finally,according to the isometry relation defined in the conjugation,a correlation equation about operator symbol was obtained,from which arose some rules through calculation.The results show that the H-Toeplitz operator with harmonic symbol or non-harmonic symbol consisting of the sum of quasihomogeneous functions is complex symmetric with respect to the given conjugation if and only if the symbol is zero.