非线性算子方程XAX=BX的可解性及其解集  

Solvability of the nonlinear operator equation XAX=BX and its solution set

作　　者：吴敬松 王华[1] WU Jing-song;WANG Hua(College of Sciences,Inner Mongolia University of Technology,Hohhot 010051,China)

机构地区：[1]内蒙古工业大学理学院数学系,内蒙古呼和浩特010051

出　　处：《高校应用数学学报（A辑）》2024年第4期464-472,共9页Applied Mathematics A Journal of Chinese Universities(Ser.A)

基　　金：国家自然科学基金(11961052,12261065);内蒙古自然科学基金(2022MS01005,2023LHMS01016);内蒙古自治区直属高校基本科研业务费(JY20220084,JY20220151)。

摘　　要：该文讨论Hilbert空间中算子方程XAX=BX解的存在性及其解集的形式.首先,在*-偏序B≤A条件下给出算子方程XAX=BX存在非平凡解的充分条件和必要条件,并得到此方程的非平凡解、正规解和自伴解的解集形式.其次,在*-偏序B≤A且B是正规算子的条件下,给出算子方程XAX=BX=XB存在非平凡解的充分条件和必要条件以及方程的解集形式.In this paper, the solvability of the operator equation XAX = BX and its solution set are discussed in Hilbert space. Firstly, sufficient conditions and necessary conditions are presented for the existence of nontrivial solutions to the equation XAX = BX under B≤A, and then solution set forms are obtained for nontrivial, normal and self-adjoint solutions,respectively. Secondly, sufficient conditions and necessary conditions are given for the existence of nontrivial solutions to the operator equation XAX = BX = XB when B≤A and B is normal. Moreover, the solution set of the equation is also given.

关 键 词：算子方程 *-偏序 不变子空间 

分 类 号：O151.1[理学—数学] O177.1[理学—基础数学]