调和Hardy空间上的Toeplitz算子的酉等价性

数学物理学报 ›› 2023, Vol. 43 ›› Issue (1): 27-34.

调和Hardy空间上的Toeplitz算子的酉等价性

丁宣浩^1,²(),黄雨浩¹(),李永宁^1,^2,^*()

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

Ding Xuanhao^1,²(),Huang Yuhao¹(),Li Yongning^1,^2,^*()

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)

摘要/Abstract

摘要：

记$H^{2}$是单位圆盘${\Bbb D}=\{\xi\in{\Bbb C}:|\xi|<1\}$上的经典Hardy空间. 设$u$和$v$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.

关键词: 调和Hardy空间, 调和Toeplitz算子, 酉等价, 对偶截断Toeplitz算子, 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.

Key words: Harmonic Hardy space, Harmonic Toeplitz operator, Unitary equivalence, Dual truncated Toeplitz operator, Essential spectrum

中图分类号:

O177.1

丁宣浩, 黄雨浩, 李永宁. 调和Hardy空间上的Toeplitz算子的酉等价性[J]. 数学物理学报, 2023, 43(1): 27-34.

Ding Xuanhao, Huang Yuhao, Li Yongning. The Unitary Equivalence of the Toeplitz Operators on the Harmonic Hardy Space[J]. Acta mathematica scientia,Series A, 2023, 43(1): 27-34.

参考文献 20

[1]	Sarason D. Algebras of functions on the unit circle. Bulletin of the American Mathematical Society, 1973, 79(2): 286-299 doi: 10.1090/S0002-9904-1973-13144-1
[2]	Böttcher A, Silbermann B. Analysis of Toeplitz Operators. Heidelberg: Springer-Verlag, 2005
[3]	Peller V. Hankel Operators and Their Applications. New York: Springer, 2002
[4]	Upmeier H. Toeplitz Operators and Index Theory in Several Complex Variables. Boston: Birkhäuser, 1996
[5]	刘元, 丁宣浩. 重调和Hardy空间上的Toeplitz算子. 中国科学: 数学, 2013, 43(7): 675-690 doi: 10.1360/012012-292
	Liu Y, Ding X. Toeplitz operators on the reharmonic Hardy spaces. Scientia Sinica Mathematica, 2013, 43(7): 675-690 (in Chinese) doi: 10.1360/012012-292
[6]	Guediri H. Dual Toeplitz operators on the sphere. Acta Mathematica Sinica, 2013, 29: 1791-1808
[7]	陈泳, 于涛. 单位球上本质交换的对偶Toeplitz 算子. 数学进展, 2009, 38(4): 453-464
	Chen Y, Yu T. Dual Toeplitz operators with essential exchange on the unit ball. Advances In Mathematics, 2009, 38(4): 453-464 (in Chinese)
[8]	Sarason D. A remark on the Volterra operator. Journal of Mathematical Analysis and Applications, 1965, 12: 244-246 doi: 10.1016/0022-247X(65)90035-1
[9]	Sang Y Q, Qin Y S, Ding X H. Dual truncated Toeplitz $C^{*}$-algebras. Banach Journal of Mathematical Analysis, 2019, 13(2): 275-292 doi: 10.1215/17358787-2018-0030
[10]	Sarason D. Algebraic properties of truncated Toeplitz operators. Operators and Matrices, 2007, 1: 491-526
[11]	Baranov A, Bessonov R, Kapustin V. Symbols of truncated Toeplitz operators. Journal of Functional Analysis, 2011, 261: 3437-3456 doi: 10.1016/j.jfa.2011.08.005
[12]	Nagy B, Foias C, Bercovici H, Kérchy L. Harmonic Analysis of Operators on Hilbert Space. New York: Springer, 2010
[13]	Yang J Y, Lu Y F. Commuting dual Toeplitz operators on the harmonic Bergman space. Science China Mathematics, 2015, 58(7): 1461-1472 doi: 10.1007/s11425-014-4940-x
[14]	Lakhdar B, Hocine G. Properties of dual Toeplitz operators with applications to haplitz products on the Hardy space of the polydisk. Taiwanese Journal of Mathematics, 2015, 19(1): 31-49
[15]	Ding X H, Sang Y Q. Dual truncated Toeplitz operators. Journal of Mathematical Analysis and Applications, 2018, 461(1) : 929-946 doi: 10.1016/j.jmaa.2017.12.032
[16]	Ding X H, Qin Y S, Sang Y Q. Harmonic Hardy space and their operators. Operators and Matrices, 2020, 14(4): 837-855
[17]	孙顺华. Toeplitz算子酉等价的某些问题. 科学通报, 1985, 8: 570-572
	Sun S. Some problems on unitary equivalence of Toeplitz operators. Chinese Science Bulletin, 1985, 8: 570-572 (in Chinese)
[18]	Cima J, Matheson A, Ross W, Wogen W. Truncated Toeplitz operators: spatial isomorphism, unitary equivalence, and similarity. Indiana University Mathematics Journal, 2010, 59(2): 595-620 doi: 10.1512/iumj.2010.59.4097
[19]	Sang Y Q, Qin Y S, Ding X H. A theorem of Brown-Halmos type for dual truncated Toeplitz operators. Annals of Function Analysis, 2020, 11: 271-284 doi: 10.1007/s43034-019-00002-7
[20]	Li Y N, Ding X H, Sang Y Q. The commutant and invariant subspaces for dual truncated Toeplitz operators. Banach Journal of Mathematical Analysis, 2021, 15: 224-250