调和Hardy空间上调和Toeplitz算子的若干代数性质  

Some Algebraic Properties of Harmonic Toeplitz Operators on Harmonic Hardy Space

作　　者：魏晓东 石岩月[1] Wei Xiaodong;Shi Yanyue(School of Mathematical Sciences,Ocean University of China,Qingdao 266100,China)

机构地区：[1]中国海洋大学数学科学学院,山东青岛266100

出　　处：《中国海洋大学学报（自然科学版）》2022年第12期147-154,共8页Periodical of Ocean University of China

基　　金：中央高校基本科研业务费专项项目(201964007);山东省自然科学基金项目(ZR2020MA009)资助。

摘　　要：为刻画调和Hardy空间uH^2⊕zvH^(2)上调和Toeplitz算子的结构,本文主要研究了此类算子满足的方程,推广了uH^2⊕zH^(2)空间上的相关结论,其中u和v为Hardy空间H 2中的内函数。设T_(φ)为调和Hardy空间上以φ∈L∞为符号的调和Toeplitz算子。本文首先证明了T_(z)与移位算子至多相差一个秩为1的算子;然后给出了T_(φ)-T_(z)-T_(φ)T_(z)的表达式,进而得到T_(φ)的分块矩阵表示并给出T_(φ)和T_(z)交换的充要条件;最后证明了T_(φ)均是复对称算子。研究结果表明,内函数u和v的变化对T_(φ)的结构不产生本质的影响。In order to characterize the structure of harmonic Toeplitz operators on harmonic Hardy space uH^2⊕zvH^(2),we mainly study the equations satisfied by such operators,and generalize some results obtained on space uH^2⊕zH^(2),where u and v are inner functions in Hardy space H 2.Letφbe harmonic Toeplitz operator with symbolφ∈L∞.Firstly,we prove that the difference between z and the shift operator is at most a rank-one operator.Secondly,we give the expression ofφ-z-φz.It implies the block matrix representation ofφ.By this block matrix representation,we give a necessary and sufficient condition for the commutativity ofφand z.Finally,it is proved that everyφis a complex symmetric operator.The results show that the variation of inner functions u and v has no significant impacts on the structure ofφ.

关 键 词：调和Hardy空间 调和Toeplitz算子 复对称算子 换位子 

分 类 号：O177.1[理学—数学]