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### 调和Hardy空间上的Toeplitz算子的酉等价性

1. 1重庆工商大学数学与统计学院 重庆 400067
2经济社会应用统计重庆市重点实验室 重庆 400067
• 收稿日期:2021-12-22 修回日期:2022-08-17 出版日期:2023-02-26 发布日期:2023-03-07
• 通讯作者: *李永宁, E-mail: yongningli@ctbu.edu.cn
• 作者简介:丁宣浩, E-mail: dingxuanhao@ctbu.edu.cn|黄雨浩, E-mail: huangyuhaoctbu@163.com
• 基金资助:
国家自然科学基金(11871122);国家自然科学基金(12101092);重庆市自然科学基金(cstc2020jcyj-msxmX0318);重庆工商大学基金(2056008);重庆工商大学校级科研项目(yjscxx2022-112-73)

### The Unitary Equivalence of the Toeplitz Operators on the Harmonic Hardy Space

Xuanhao Ding1,2(),Yuhao Huang1(),Yongning Li1,2,*()

1. 1School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067
2Chongqing Key Laboratory of Social Economy and Applied Statistics, Chongqing 400067
• Received:2021-12-22 Revised:2022-08-17 Online:2023-02-26 Published:2023-03-07
• Supported by:
The NSFC(11871122);The NSFC(12101092);Natural Science Foundation of Chongqing(cstc2020jcyj-msxmX0318);Chongqing Technology and Business University(2056008);Scientific research project in Chongqing Technology and Business University-level(yjscxx2022-112-73)

$H^{2}$是单位圆盘${\Bbb D}=\{\xi\in{\Bbb C}:|\xi|<1\}$上的经典Hardy空间. 设$u$$v$是内函数且至少其中一个是非常值的, 调和Hardy空间$H_{u,v}^{2}$定义为$H_{u,v}^{2}=uH^{2}\oplus\overline{v}(H^{2})^{\perp}=uH^{2}\oplus\overline{vzH^{2}}$. 对任意的$x\in H_{u,v}^{2},$ 定义$H_{u,v}^{2}$上的调和Toeplitz算子 $\widehat{T}_{\varphi}x=Q_{u,v}(\varphi x),$ 其中, $Q_{u,v}:L^{2}\rightarrow H_{u,v}^{2}$为正交投影. 该文刻画了调和Toeplitz算子和对偶截断Toeplitz算子的酉等价性, 并给出了两个调和Toeplitz算子可交换的充要条件, 调和Toeplitz代数的性质以及$\widehat{T}_{z}$的换位子的刻画. 最后, 该文还得到了有限多个连续符号的调和 Toeplitz算子乘积的本质谱.

Abstract:

Let $H^{2}$ be the Hardy space on the unit disk ${\Bbb D}=\{\xi\in{\Bbb C}:|\xi|<1\}$. Suppose $u$ and $v$ are inner functions and at least one of them is nonconstant, the harmonic Hardy space $H_{u,v}^{2}$ is defined by $H_{u,v}^{2}=uH^{2}\oplus\overline{v}(H^{2})^{\bot}=uH^{2}\oplus\overline{vzH^{2}}.$ For any $x\in H_{u,v}^{2},$ define the Toeplitz operator on the $H_{u,v}^{2}$ by $\widehat{T}_{\varphi}x=Q_{u,v}(\varphi x),$ where $Q_{u,v}$ is the orthogonal projection from $L^{2}\rightarrow H_{u,v}^{2}.$ In this paper, the unitary equivalence of the harmonic Toeplitz operator and the dual truncated Toeplitz operator is obtained, moreover, the sufficient and necessary conditions for when two Toeplitz operators commute is given, and the properties of the harmonic Toeplitz algebra and the commutant of $\widehat{T}_{z}$ are described. Finally, the essential spectrum for the product of finitely many harmonic Toeplitz operators with continuous symbols is obtained in this paper.

• O177.1