几类广义Sylvester算子方程的可解性

数学物理学报 ›› 2022, Vol. 42 ›› Issue (6): 1640-1652.

几类广义Sylvester算子方程的可解性

孙晓林,王华*()

^{内蒙古工业大学理学院数学系呼和浩特 010051}

收稿日期:2021-10-26 出版日期:2022-12-26 发布日期:2022-12-16
通讯作者: 王华 E-mail:hrenly@163.com
基金资助:
国家自然科学基金(11961052);内蒙古自然科学基金(2022MS01005);内蒙古自然科学基金(2021LHMS01004);自治区直属高校基本科研业务费项目(JY220083);自治区直属高校基本科研业务费项目(JY20220151);内蒙古工业大学研究生教育教学改革项目(YJG2020027)

Solvability of Several Kinds of Ggeneralized Sylvester Operator Equations

Xiaolin Sun,Hua Wang*()

^{College of Sciences, Inner Mongolia University of Technology, Hohhot 010051}

Received:2021-10-26 Online:2022-12-26 Published:2022-12-16
Contact: Hua Wang E-mail:hrenly@163.com
Supported by:
the NSFC(11961052);the NSF of Inner Mongolia(2022MS01005);the NSF of Inner Mongolia(2021LHMS01004);the Basic Science Research Fund in the University Directly under the Inner Mongolia Autonomous Region(JY220083);the Basic Science Research Fund in the University Directly under the Inner Mongolia Autonomous Region(JY20220151);Tenching Reform of Postgraduate Education in Inner Mongolia University of Technology(YJG2020027)

摘要/Abstract

摘要：

该文讨论Hilbert空间上几类广义Sylvester算子方程和算子方程组的可解性.首先在正常算子的情况下, 给出两类算子方程AXB-XD=EB和AXB-CX=AE的解存在的充要条件; 其次利用算子对的一致等价性讨论三类算子方程的可解性; 最后给出当三角算子矩阵与对角算子矩阵相似时, 相应的算子方程组是可解的.

关键词: Sylvester方程, 算子方程, 正常算子, Putnam-Fuglede定理

Abstract:

In this paper, we discuss the solvability of some kinds of generalized Sylvester operator equations on Hilbert Spaces. Firstly, the necessary and sufficient conditions for the existence of solutions of AXB-XD=EB and AXB-CX = AE are given in the case of normal operators. Secondly, the solvability of three kinds of operator equations is discussed by using the uniform equivalence of operator pairs. Finally, when trigonometric operator matrix is similar to diagonal operator matrix, the corresponding operator equations are solvable.

Key words: Sylvester equation, Operator equation, Normal operator, Putnam-Fuglede's theorem

中图分类号:

O151.1

孙晓林,王华. 几类广义Sylvester算子方程的可解性[J]. 数学物理学报, 2022, 42(6): 1640-1652.

Xiaolin Sun,Hua Wang. Solvability of Several Kinds of Ggeneralized Sylvester Operator Equations[J]. Acta mathematica scientia,Series A, 2022, 42(6): 1640-1652.

参考文献

1	Sylvester J . Sur l'équation en matrices px-xq. Comptes Rendus Mathematique, 1884, 99, 67- 71
2	Roth W E . The equation AX-YB=C and AX-XB=C in matrices. Proceedings of the American Mathematical Society, 1952, 3 (3): 392- 396
3	Wimmer H K . The matrix equation X - AXB = C and analogue of Roth's theorem. Linear Algebra and its Applications, 1988, 109, 145- 147
4	Lee S G , Vu Q P . Simultaneous solutions of matrix equations and simultaneous equivalence of matrices. Linear Algebra and its Applications, 2012, 437 (9): 2325- 2339
5	Dmytryshyn A , Ka?m B . Coupled Sylvester-type matrix equations and block diagonalization. SIAM Journal on Matrix Analysis and Applications, 2015, 36 (2): 580- 593
6	Eric C . The solution of the matrix equations AXB-CXD=E and (YA-DZ, YC-BZ)=(E, F). Linear Algebra and its Applications, 1987, 93 (87): 93- 105
7	Baksalary J K , Kala R . The matrix equation AXB+CYD=E. Linear Algebra and its Applications, 1980, 30 (1): 141- 147
8	Rehman A , Wang Q W , Ali I , et al. A constraint system of generalized Sylvester quaternion matrix equations. Advances in Applied Clifford Algebras, 2017, 27 (11): 3183- 3196
9	He Z H , Wang Q W , Zhang Y . A system of quaternary coupled Sylvester-type real quaternion matrix equations. Automatica, 2018, 87 (7): 25- 31
10	Liu X , Song G J , Zhang Y . Determinantal representations of the solutions to systems of generalized Sylvester equations. Advances in Applied Clifford Algebras, 2019, 30, Article number 12
11	Djordjevi B D . On a singular Sylvester equation with unbounded self-adjoint A and B. Complex Analysis and Operator Theory, 2020, 14, Article number 43
12	Chen H X , Wang L , Li T T . A note on the solvability for generalized Sylvester equations. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturaless. Serie A. Matemáticas, 2021, 115, Article number 64
13	Wang Q W , Sun J H , Li S Z . Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra. Linear Algebra and its Applications, 2002, 353 (1): 169- 182
14	Wang Q W , He Z H . Solvability conditions and general solution for mixed Sylvester equations. Automatica, 2013, 49 (9): 2713- 2719
15	Wang Q W , He Z H , Zhang Y . Constrained two-sided coupled Sylvester-type quaternion matrix equations. Automatica, 2019, 101, 207- 213
16	Wang Q W , Wang X , Zhang Y S . A constraint system of coupled two-sided Sylvester-like quaternion tensor equations. Computational and Applied Mathematics, 2020, 39, Article number 317
17	Wang Q W , Wang X . A system of coupled two-sided Sylvester-type tensor equations over the quaternion algebra. Taiwanese Journal of Mathematics, 2020, 24 (6): 1399- 1416
18	Rosenblum M . The operator equation BX-XA=Q with selfadjoint A and B. Proceeding of the American Mathematical Society, 1969, 20 (1): 115- 120
19	Schweinsberg A . The operator equation AX-XB=C with normal A and B. Pacific Journal of Mathematics, 1982, 102 (2): 447- 453
20	Mansour A . Solvability of AXB-CXD = E in the operators algebra B(H). Lobachevskii Journal of Mathematics, 2010, 31 (3): 257- 261
21	童裕孙. 关于算子方程AXB-X=C. 数学年刊, 1986, 7 (3): 325- 337
21	Tong Y S . On the operator equation AXB-X=C. Chinese Ann Math Ser A, 1986, 7 (3): 325- 337
22	严绍宗, 李绍宽. 关于Putnam-Fuglede定理. 中国科学(A辑), 1984, 9, 775- 783
22	Yan S Z , Li S K . On Putnam-Fuglede theorem. Science China(Series A), 1984, 9, 775- 783
23	Weiss G . The Fuglede commutativity theorem modulo the Hilbert-Schmidt class and generating functions for matrix operators Ⅱ. Journal of Operator Theory, 1981, 5 (1): 3- 16

[1]	刘月园,王凯,秦树娟,李姣芬. 列正交约束下广义Sylvester方程极小化问题的有效算法[J]. 数学物理学报, 2021, 41(2): 479-495.
[2]	王月虎, 张从军. 广义均衡问题的算法及强收敛性——基于Wiener-Hopf方程技巧[J]. 数学物理学报, 2015, 35(4): 695-709.
[3]	张云, 陈冬君, 叶永升. 具有公共的零点特征投影的有界线性算子的表示[J]. 数学物理学报, 2011, 31(3): 729-736.
[4]	杨凯凡, 杜鸿科. 关于算子方程X+A*X^-tA=Q的正算子解的研究[J]. 数学物理学报, 2009, 29(2): 359-364.
[5]	苏华; 刘立山; 吴从炘. 一类抽象算子方程组的迭代解及其应用[J]. 数学物理学报, 2007, 27(3): 449-455.
[6]	张晓燕, 孙经先. 一类非线性算子方程解的存在唯一性及其应用[J]. 数学物理学报, 2005, 25(6): 846-851.
[7]	崔明根, 李云晖. 二次非线性算子方程精确解的表示[J]. 数学物理学报, 2005, 25(3): 421-427.
[8]	杨宏奇, 侯宗义. 非线性不适定算子方程的双参数正则化方法[J]. 数学物理学报, 2004, 24(2): 129-134.
[9]	朱传喜. 一类随机算子方程随机解的存在性[J]. 数学物理学报, 2003, 23(3): 276-279.
[10]	唐建国. 基于正交多项式的解不适定算子方程的隐式迭代法[J]. 数学物理学报, 2003, 23(3): 265-275.
[11]	张晓燕, 刘立山. 非线性算子方程迭代解的存在性定理及其应用[J]. 数学物理学报, 2001, 21(3): 398-404.
[12]	杨宏奇侯宗义. 非线性第一类单调算子方程正则解的收敛率讨论[J]. 数学物理学报, 2000, 20(3): 360-363.
[13]	闫玉斌, 李春利. 第一类不适定算子方程的解的表示[J]. 数学物理学报, 1998, 18(3): 264-269.
[14]	张克梅. 非线性非单调算子方程的解[J]. 数学物理学报, 1998, 18(2): 164-168.
[15]	娄本东, 宋光兴. 一类非线性算子方程解的存在唯一性定理及其应用[J]. 数学物理学报, 1997, 17(3): 308-313.