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

吴敬松 ¹王华¹

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作者信息

1. 内蒙古工业大学理学院数学系,内蒙古呼和浩特 010051
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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.

Abstract

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

Key words

operator equation/*-order/invariant subspace

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出版年

2024

高校应用数学学报

浙江大学中国工业与应用数学学会

高校应用数学学报

CSTPCD北大核心

影响因子：0.396

ISSN：1000-4424

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