无限维Hilbert空间上一类算子方程的解

The positive operator solutions to a class of operator equations in infinite dimension Hilbert space

在无限维Hilbert空间上研究非线性算子方程X-A^(*)X^(-t)A=Q的正算子解问题,寻求此类方程正算子解存在的必要条件和充分条件.利用算子谱理论、数值域特征以及构造有效的迭代序列,给出算子方程X-A^(*)X^(-t)A=Q有正算子解时方程中各算子之间的代数关系,以及有正算子解的一些必要条件和充分条件,特别给出了当A为正规算子且t=2m(其中m为正整数)时该方程有正解的条件.说明了当方程中给定的算子A,Q满足一定的条件时,算子方程X-A^(*)X^(-t)A=Q存在正算子解.

In this paper,the positive operator solution to nonlinear operator equation X-A^(*)X^(-t)A=Q was studied in infinite dimensional Hilbert space,and the necessary and sufficient conditions for the existence of positive operator solution to this kind of equation were obtained.By using the operator spectrum theory,the characteristics of numerical domain and the construction of effective iterative sequences,the algebraic relations between the operators in the equation X-A^(*)X^(-t)A=Q were given when the equation had positive operator solution.As well as some necessary and sufficient conditions for the solution to the positive operator were given especially when A was a normal operator and t=2m(where m was a positive integer).It was shown that,when the given operator A and Q satisfied certain conditions,there existed positive operator solutions to the operator equation X-A^(*)X^(-t)A=Q.