Volterra算子与Toeplitz算子的乘积

Volterra算子与Toeplitz算子的乘积

The Product of Volterra Operator and Toeplitz Operator

摘要/Abstract

参考文献 14

Metrics

本文评价

推荐阅读 10

数学物理学报 ›› 2024, Vol. 44 ›› Issue (4): 829-836.

丁宣浩^1,²(),邵长慧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:2023-05-25 修回日期:2023-10-12 出版日期:2024-08-26 发布日期:2024-07-26
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;|邵长慧, E-mail: schdliyx@163.com
基金资助:
国家自然科学基金(12101092);重庆市自然科学基金(CSTB2022NSCQ-MSX1045);重庆市教委科学技术研究项目(KJQN202100822)

Ding Xuanhao^1,²(),Shao Changhui¹(),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:2023-05-25 Revised:2023-10-12 Online:2024-08-26 Published:2024-07-26
Supported by:
National Natural Science Foundation of China(12101092);Chongqing Natural Science Foundation(CSTB2022NSCQ-MSX1045);Science and Technology Research Program of Chongqing Municipal Education Commission(KJQN202100822)

摘要：

该文研究了经典 Hardy 空间上的 Volterra 算子 $V$ 与 Toeplitz 算子 $T_\varphi$ 的乘积算子 $L_\varphi= T_\varphi V$ 与 $R_\varphi= VT_\varphi$ , 得到了 $L_\varphi$ 与 $R_\varphi$ 的一些基本性质, 并给出了 $L_\varphi$ 的类似于 Toeplitz 算子的 Coburn 定理的结果. 该文还给出了 $V$ 和 $T_\varphi$ 可交换的充分必要条件, 以及 $L_\varphi$ 及 $R_\varphi$ 以 $z^jH^2$ $(j=1, 2, \cdots )$ 为不变子空间的符号的充要刻画.

关键词: Volterra 算子, Toeplitz 算子, Coburn 定理, 可交换, 不变子空间

Abstract:

In this paper, we study the product operators of Volterra operator $V$ and Toeplitz operator $T_\varphi$ on the classical Hardy space $L_\varphi=T_\varphi V$ and $R_\varphi=VT_\varphi$ , and we obtain some basic properties of $L_\varphi$ and $R_\varphi$ , we also get a result of $L_\varphi$ which is similar to the Coburn theorem of Toeplitz operator. The necessary and sufficient conditions for $V$ and $T_\varphi$ to be commutative are given in this paper, and the symbol of $L_\varphi$ and $R_\varphi$ is characterized when $z^jH^2\ (j=1, 2, \cdots )$ are the common invariant subspaces of $L_\varphi$ and $R_\varphi$ .

Key words: Volterra operator, Toeplitz operator, Coburn theorem, Commutative, Invariant subspace

中图分类号:

O177.1

丁宣浩, 邵长慧, 李永宁. Volterra算子与Toeplitz算子的乘积[J]. 数学物理学报, 2024, 44(4): 829-836.

Ding Xuanhao, Shao Changhui, Li Yongning. The Product of Volterra Operator and Toeplitz Operator[J]. Acta mathematica scientia,Series A, 2024, 44(4): 829-836.

[1]	Bottcher A, Silbermann B. Analysis of Toeplitz Operators. Berlin: Springer, 1990
[2]	Peller V. Hankel Operators and Their Applications. New York: Springer, 2003
[3]	Upmeier H. Toeplitz Operators and Index Theory in Several Complex Variables. Basel: Birkhauser, 1996
[4]	Brown A, Halmos P. Algebraic properties of Toeplitz operators. Journal für die reine und angewandte Mathematik, 1964, 213: 89-102
[5]	Gohberg I, Krein M. Theory and Applications of Volterra Operators in Hilbert Space. Providence: American Mathematical Society, 1970
[6]	Pommerenke C. Schlichte Funktionen und analytische Funktionen von beschranker mittlerer Oszillation. Commentarii Mathematici Helvetici, 1977, 52(1): 591-601
[7]	Aleman A, Siskakis A. Integration operators on Bergman spaces. Indiana University Mathematics Journal, 1997, 46(2): 337-356
[8]	Aleman A, Cima J. An integral operator on $H^p$ and Hardy's inequality. Journal d'Analyse Mathématique, 2001, 85(1): 157-176
[9]	Aleman A, Siskakis A. An integral operator on $H^p$ . Complex Variables and Elliptic Equations, 1995, 28(2): 149-158
[10]	Donoghue W F. The lattice of invariant subspaces of a completely continuous quasi-nilpotent transformation. Pacif J Math, 1957, 7(2): 1031-1035
[11]	Čučković Ž, Paudyal B. Invariant subspaces of the shift plus complex Volterra operator. Journal of Mathematical Analysis and Applications, 2015, 426(2): 1174-1181
[12]	Čučković Ž, Paudyal B. The lattices of invariant subspaces of a class of operators on the Hardy space. Archiv der Mathematik, 2018, 110(5): 477-486
[13]	Lin Q. The invariant subspaces of the shift plus integer multiple of the Volterra operator on Hardy spaces. Archiv der Mathematik, 2018, 111(5): 513-522
[14]	Douglas R G. Banach Algebra Techniques in Operator Theory. New York: Springer-Verlag, 1998

Viewed

Full text

Abstract

Cited

Shared

Discussed

Just accepted

Online first

Just accepted

Online first

[1]	胡珊珊, 贺素香. 非负组稀疏约束优化问题的最优性条件[J]. 数学物理学报, 2024, 44(2): 500 -512 .
[2]	董建祥. 向量值指数型权 Bergman 空间上的 Hankel 算子[J]. 数学物理学报, 2024, 44(3): 513 -524 .
[3]	张杰, 孙艺铭, 刘永平. 加权 Korobov 空间中多元 $\mathbb{L}_{\infty}$ -逼近问题的指数收敛易处理性[J]. 数学物理学报, 2024, 44(3): 525 -538 .
[4]	王瑶瑶, 吕玫钏, 李文明. 高维 Hardy 算子及其交换子的双权不等式[J]. 数学物理学报, 2024, 44(3): 539 -546 .
[5]	陈浩, 邓雪梅, 别群益. 三维稳态向列型液晶方程各向异性的Liouville定理[J]. 数学物理学报, 2024, 44(3): 661 -669 .
[6]	杨佼朋, 梁勇. 广义b方程的孤立波解及周期波解[J]. 数学物理学报, 2024, 44(3): 670 -686 .
[7]	宾茂君, 施翠云. 半线性Riemann-Liouville分数阶发展方程反馈时间最优控制[J]. 数学物理学报, 2024, 44(3): 687 -698 .
[8]	段雪亮, 吴晓凡, 魏公明, 杨海涛. 带临界指数的Kirchhoff型线性耦合方程组正解的多重性[J]. 数学物理学报, 2024, 44(3): 699 -716 .
[9]	李锋杰, 李平. 伪抛物型 p-Kirchhoff 方程的爆破解[J]. 数学物理学报, 2024, 44(3): 717 -736 .
[10]	孙小春, 吴育联, 徐郜婷. 分数阶不可压缩 Navier-Stokes-Coriolis 方程解的整体适定性[J]. 数学物理学报, 2024, 44(3): 737 -745 .